UNIVERSITE NICE-SOPHIA ANTIPOLIS - UFR SCIENCESEcole Doctorale Sciences Fondamentales et Appliquees

THESEPour obtenir le titre de

Docteur en Sciencesde l’Universite Nice Sophia-Antipolis

Specialite : MATHEMATIQUES

Presente et soutenue publiquement par

CHADI TAHER

Titre de la these

CALCULATING THE PARABOLIC CHERN CHARACTER

OF A LOCALLY ABELIAN PARABOLIC BUNDLE -

THE CHERN INVARIANTS

FOR PARABOLIC BUNDLES AT MULTIPLE POINTS.

These dirigee par Professeur CARLOS SIMPSONSoutenue le 16 Mai 2011

a la Faculte des Sciences de l’Universite de Nice

Membre du jury:

Mr.Tony PANTEV Professeur, Universite de Pennsylvania USA RapporteurMrs.Jaya IYER Professeur, Universite de Hyderabad India RapporteurMr.Alexandru DIMCA Professeur, Universite Nice Sophia-Antipolis France ExaminateurMr.Sorin DUMITRESCU Professeur, Universite Nice Sophia-Antipolis France ExaminateurMr.Bertrand Toen Directeur de Recherche, Universite Montpellier 2 France ExaminateurMr.Carlos SIMPSON DR1 CNRS, Universite Nice Sophia-Antipolis France Directeur

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Acknowledgment

First of all I would like to express my gratitude and my deepest thanks to professor Car-los Simpson for his supervision, advice, and guidance from the very early stage of thisresearch as well as giving me extraordinary experiences through out the work. Duringthese years i have benefited from his experience and his vast knowledge mathematics.While working under his supervision, he always offered unlimited support to bring forththe best possible work. He has a special method to simplify every thing in a way thatmade me capable of achieving more and more. I am very grateful to the care and theattention that he gave during this work. I feel unable to find the appropriate words toexpress my gratitude to him properly. I am indebted to him more than he knows. Thankyou very much.

I am very grateful to Professor Alexandre Dimca, who introduced me to AlgebraicGeometry and Singularity Theory. I thank him for agreeing to be the rapporteur andto the jury. Also a lot of thanks go to the professors who accepted to judge this work.I thank Professor.Tony Pantev, Professor.Jaya Iyer, Professor.Sorin Dumitrescu, Pro-fessor.Bertrand Toen. I gratefully thank for the panel who gave me the opportunity torepresent this work and a big honor by their present.

Many thanks go in particular to Professor Nicole Simpson, for her valuable advicein science discussion.

Words fail me to express my appreciation to my wife Pharmacy Doctor.MarwaAwada Taher(My Love), whose dedication, love and persistent confidence in me, hastaken the load off my shoulder. I owe her for being unselfishly let her intelligence, pas-sions, and ambitions collide with mine. Therefore, I would also thank Made Awada’sfamily for letting me take her hand in marriage, and accepting me as a member ofthe family. I would like to thank My uncle Dr.Hassan Awada, Mrs.Joumana Baraket,Mrs.Souna Awada, Mrs.Kawthar Awada, Mr.Ali Awada, and finally Mr.Moussa Hassan.

Where would I be without my family? My parents deserve special mention for theirinseparable support and prayers. My Father Hassan Taher in the first place is the per-son who put the fundament my learning character, showing me the joy of intellectualpursuit ever since I was a child. My Mother Samia El-hajj, is the one who sincerelyraised me with her caring and gently love. A Lot of thanks to my sisters and brothers.First I gratefully thank my sister Professor.Fadia Taher for her support me and advise,guidance, from the first year at the university I feel unable to find the appropriate wordsto express my gratitude to her properly. Thank you very much. Dr.Nachaat Mansour,

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Dr.Ismat Taher, Dr.Hanadi Taher(cuty nana), Dr.Isam Taher, Dr.Tania Taher, Dr.AdelTaher, Mrs.Lama Akel, Dr.Fadi Taher and Mrs.Rola Kassem. Thanks for being support-ive and caring siblings.

My special thanks to Mr.Mostapha El-Solh Honorary Consul of Lebanon in Monacofor giving me the opportunity to work with him as assistant during the period of study. Ifeel unable to find the appropriate words to express my gratitude to him properly. Thankyou very much. I would also acknowledge Dr.Samih Beik El-Solh, and Mrs.SouadMikati El-Solh, Mr.Marek Sinno, Mrs.Maya El-Solh Sinno, Mr.Mohamad El-Solh, andMrs.Cecile EL-Solh.

It is a pleasure to express my gratitude wholeheartedly to Mikati’s family. Manythanks to Mr.Taha Mikati, Mr.Najib Mikati the Prime minister of lebanon, Mrs.NadaMiskawi Mikati, Mrs.May Mikati, Mr.Azmi Mikati, Mr.Maher Mikati, Mr.Fouad Mikati,Mr.Malek Mikati, Mr.Ali Bdeir, Mrs.Mira Azmi Mikati, Mrs.Mira Mikati Bdeir, andMrs.Dana Mikati.

I would like also to express my gratitude and my deepest thanks to Colonel.RamezKhamiss and Colonel. Pierre Neghawi.

I want to thank also My University Nice Sophia-Antipolis, from which I, as well asthousands of students, have graduated and to my teachers in Master who gave me thechance to get the proper and high level of education.

Collective and individual acknowledgments are also owed to my colleagues at theUniversity of Nice Sophia-Antipolis. Many thanks go in particular to Dr.MohamedSarrage, Dr.Osman Khodor, Dr.Mouhamad Hanzal, Dr.Hayssam, Dr.Samer Alouch,Dr.Hamad hazim, Dr.Brahim Benzeghli... Also i would like to thank Mr.Samir Chahine,Dr.Kifah Yehya, Mrs.Rouba Yehya, Mr.Mazen Yehya, Mr.Jad Abou Khater, Mr.MaherRaed, Dr.Ali Hamzi, Mr.Ali Mousawi, Mrs.Manar Akel, Mr.Youssef Hachouch, Dr.RolaAbou-Taam, Dr.Hassan Kalakech, Mrs.Nancy Kalakech, Mr.Samer Shahine, Dr.KamelKalakech, Mr.Abdalah Rammel, Mr.Ramzi Abi Haydar, Mr.Ziad Dagher, Mr.Dani Kan-daleft, Captain.Abdalah Charkawi and Mr.Nouhad Kechle(Abou Taha)

Finally, I would like to thank everybody who was important to the successful real-ization of thesis, as well as expressing my apology that I could not mention personallyone by one.

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Contents

1 Introduction 11.1 Algebraic geometry background . . . . . . . . . . . . . . . . . . . . . 6

1.1.1 Affine varieties . . . . . . . . . . . . . . . . . . . . . . . . . . 61.1.2 Projective varieties . . . . . . . . . . . . . . . . . . . . . . . . 71.1.3 Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.1.4 Divisors on curves . . . . . . . . . . . . . . . . . . . . . . . . 81.1.5 Sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.1.6 Vector bundle . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.2 Intersection theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.2.1 Chow group . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.2.2 Cartier divisor . . . . . . . . . . . . . . . . . . . . . . . . . . 141.2.3 Chern classes . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.2.4 Segre and Chern classes vector bundles . . . . . . . . . . . . . 181.2.5 Statement of the Hirzeburch-Rimann-Roch theorem . . . . . . 241.2.6 Parabolic bundles . . . . . . . . . . . . . . . . . . . . . . . . . 331.2.7 Sections of the line bundle Lpar . . . . . . . . . . . . . . . . . 36

1.3 Blowing up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411.3.1 Elementary transformation of algebraic bundles . . . . . . . . . 431.3.2 Generalization of elementary transformation . . . . . . . . . . 45

2 Calculating the parabolic Chern character of a locally abelian parabolicbundle 472.1 Quasi-parabolic structures . . . . . . . . . . . . . . . . . . . . . . . . 51

2.1.1 Index sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512.1.2 Two approaches . . . . . . . . . . . . . . . . . . . . . . . . . . 512.1.3 Locally abelian condition . . . . . . . . . . . . . . . . . . . . . 53

2.2 Weighted parabolic structures . . . . . . . . . . . . . . . . . . . . . . . 582.2.1 Riemann-Roch theorem . . . . . . . . . . . . . . . . . . . . . 65

2.3 Computation of parabolic Chern characters of a locally abelian parabolicbundle E in codimension one and two chPar1 (E), chPar2 (E) . . . . . . . 69

2.3.1 The characteristic numbers for parabolic bundles in codimen-sion 1 and 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

3 Parabolic Chern character of a locally abelian parabolic bundle E in codi-mension 3, chPar3 (E) 783.1 The characteristic number for a parabolic bundle in codimension 3 . . . 80

4 Chern invariants for parabolic bundles at multiple points 824.1 Calculating the invariant ∆ of a locally abelian parabolic bundle . . . . 834.2 Parabolic bundles with full flags . . . . . . . . . . . . . . . . . . . . . 864.3 Resolution of singular divisors . . . . . . . . . . . . . . . . . . . . . . 924.4 Local Bogomolov-Gieseker inequality . . . . . . . . . . . . . . . . . . 934.5 Modification of filtrations due to elementary transformations . . . . . . 984.6 The local parabolic invariant . . . . . . . . . . . . . . . . . . . . . . . 1004.7 Normalization via standard elementary transformations . . . . . . . . . 1044.8 The rank two case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

4.8.1 Panov differentiation . . . . . . . . . . . . . . . . . . . . . . . 1094.8.2 The Bogomolov-Gieseker inequality . . . . . . . . . . . . . . . 111

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Abstract

In this thesis we calculate the parabolic Chern character of a bundle with locally abelianparabolic structure on a smooth strict normal crossings divisor, using the definition interms of Deligne-Mumford stacks. We obtain explicit formulas for ch1, ch2 and ch3,and verify that these correspond to the formulas given by Borne for ch1 and Mochizukifor ch2.

The second part of the thesis we take D ⊂ X is a curve with multiple points ina surface, a parabolic bundle defined on (X,D) away from the singularities can beextended in several ways to a parabolic bundle on a resolution of singularities. Weinvestigate the possible parabolic Chern classes for these extensions.

Chapter 1

Introduction

The first part of this thesis we suppose X be a smooth projective variety with a strictnormal crossings divisor D = D1 + . . . + Dn ⊂ X . The aim of this paper is to give anexplicit formula for the parabolic Chern character of a locally abelian parabolic bundleon (X,D) in terms of:—the Chern character of the underlying usual vector bundle,—the divisor components Di in the rational Chow groups of X ,—the Chern characters of the associated-graded pieces of the parabolic filtration alongthe multiple intersections of the divisor components, and—the parabolic weights.

After giving a general formula, we compute explicitly the parabolic first, second,third parabolic Chern characters chPar1 (E), chPar2 (E) and chPar3 (E).

The basic idea is to use the formula given in [IS2]. However, their formula didnot make clear the contributions of the different elements listed above. In order toadequately treat this question, we start with a somewhat more general framework ofunweighted quasi-parabolic sheaves [Se]. These are like parabolic sheaves except thatthe real parabolic weights are not specified. Instead, we consider linearly ordered setsΣi indexing the parabolic filtrations over the components Di. Let Σ′i denote the linearlyordered set of links or adjacent pairs in Σi. We also call these “risers” as Σ can bethought of as a set of steps. The parabolic weights are then considered as functionsαi : Σ′i → (−1, 0] ⊆ R. This division allows us to consider separately some Chernclass calculations for the unweighted structures, and then the calculation of the parabolicChern character using the parabolic weights.

A further difficulty stems from the fact that there are classically two different waysto give a parabolic structure: either as a collection of sheaves included in one another; orby fixing a bundle E (typically the zero-weight sheaf) plus a collection of filtrations ofE|Di . The formula of [IS2] is expressed in terms of the collection of sheaves, whereaswe look for a formula involving the filtrations. Thus, our first task is to investigate therelationship between these two points of view.

An important axiom concerning the parabolic structures considered here, is thatthey should be locally abelian. This means that they should locally be direct sums ofparabolic line bundles. It is a condition on the simultaneous intersection of three or morefiltrations; up to points where only two divisor components intersect, the condition isautomatic. This condition has been considered by a number of authors (Borne [Bo1][Bo2], Mochizuki [Mo2], Iyer-Simpson [IS1], Steer-Wren [Sr-Wr] and others) and isnecessary for applying the formula of [IS2].

A quasi-parabolic sheaf consists then of a collection of sheaves Eσ1,...,σn with σi ∈Σi on X , whereas a quasi-parabolic structure given by filtrations consists of a bundle Eon X together with filtrations F i

σi⊂ E|Di of the restrictions to the divisor components.

In the locally abelian case, these may be related by a long exact sequence (2.1.5):

0 −→ Eσ1,...,σn → E →n⊕i=1

(ξi)?(Liσi

)→⊕i<j

(ξij)?(Lijσi,σj

)→ . . .→ Lσ1,...,σn −→ 0.

Where Li1,...,iqσi1 ,...,σiqdenote the quotient sheaves supported on intersections of the divisors

Di1 ∩ ... ∩Diq .Using this long exact sequence we get a formula (2.1.6) for the Chern characters

of Eσ1,...,σn in terms of the Chern character of sheaves supported on intersection of thedivisors Di1 ∩ ... ∩Diq of the form:

chV b(Eσ1,...σn) = chV b(E) +n∑q=1

(−1)q∑

i1<i2<...<iq

ch(ξI,?(L

Iσi1 ,...,σiq

)).

The notion of parabolic weight function is then introduced, and the main work ofthis paper begins: we obtain the Chern characters for the Eα1,...,αn for any αi ∈ (−1, 0];these are then put into the formula of [IS2], and the result is computed. This computationrequires some combinatorial manipulations with the linearly ordered sets Σi notably theassociated sets of risers Σ′i in the ordering. It yields the following formula (2.2.2) ofTheorem 2.2.4:

chPar(E) = chV b(E)eD+

eD.n∑q=1

(−1)q∑

i1<i2<...<iq

∑λij∈Σ′ij

ch(ξI,?(Gr

i1,...,iqλi1 ,...,λiq

)) q∏j=1

[eDij (1− e−(αij (λij )+1)Dij )

eDij − 1

].

In this formula, the associated-graded sheaves corresponding to the multiple filtrationson intersections of divisor components DI = Di1 ∩· · ·∩Diq are denoted by Gri1,...,iqλi1 ,...,λiq

.These are sheaves on DI but are then considered as sheaves on X by the inclusionξI,? : DI → X . The Chern character ch

(ξI,?(Gr

i1,...,iqλi1 ,...,λiq

))

is the Chern character ofthe coherent sheaf on X . This is not satisfactory, since we want a formula involving the

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Chern characters of theGri1,...,iqλi1 ,...,λiqonDI . Therefore in §2.2.1 we use the Grothendieck-

Riemann-Roch theorem to interchange ch and ξI,?, leading to the introduction of Toddclasses of the normal bundles of the DI . Another difficulty is the factor of eD mul-tiplying the term chV b(E); we would like to consider the parabolic Chern class as aperturbation of the Chern class of the usual vector bundle chV b(E). Using the sameformula for the case of trivial parabolic weights, which must give back chV b(E) as ananswer, allows us to rewrite the difference between chV b(E) and chV b(E)eD in a waycompatible with the rest of the formula. After these manipulations the formula becomes(2.2.5) of 2.2.14:If X be a smooth projective variety with a strict normal crossings divisor D = D1 +. . .+Dn ⊂ X . Then the explicit formula for the parabolic Chern character of a locallyabelian parabolic bundle on (X,D) in terms of:—the Chern character of the underlying usual vector bundle,—the divisor components Di in the rational Chow groups of X ,—the Chern characters of the associated-graded pieces of the parabolic filtration alongthe multiple intersections of the divisor components, and—the parabolic weights, is defined as follows:

chPar(E) = chV b(E) −

eD.n∑q=1

(−1)q∑

i1<i2<...<iq

∑λij∈Σ′ij

q∏j=1

(1− e−DijDij

).ξI,?

(ch(Gr

i1,...,iqλi1 ,...,λiq

))

+

eD.n∑q=1

(−1)q∑

i1<i2<...<iq

∑λij∈Σ′ij

q∏j=1

[(1− e−(αij (λij )+1)Dij

Dij

)].ξI,?

(ch(Gr

i1,...,iqλi1 ,...,λiq

)).

Finally, we would like to compute explicitly the terms chPar1 (E), chPar2 (E) andchPar3 (E). For these, we expand the different terms

eD,

q∏j=1

[(1− e−(αij (λij )+1)Dij

Dij

)],

q∏j=1

(1− e−DijDij

)

in low-degree monomials of Dij , and then expand the whole formula dividing the termsup according to codimension. Denoting by S := 1, . . . , n the set of indices for divisorcomponents, we get the following formulae:

chPar0 (E) := rank(E).[X]

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chPar1 (E) := chV b1 (E) −∑i1∈S

∑λi1∈

∑′i1

αi1(λi1).rank(Gri1λi1).[Di1 ]

chPar2 (E) := chV b2 (E) −∑i1∈S

∑λi1∈

∑′i1

αi1(λi1).(ξi1)?

(cDi11 (Gri1λi1

))

+1

2

∑i1∈S

∑λi1∈

∑′i1

α2i1

(λi1).rank(Gri1λi1).[Di1 ]2

+∑i1<i2

∑λi1∈Σ′

i1λi2∈Σ′

i2

∑p∈Irr(Di1∩Di2 )

αi1(λi1).αi2(λi2).rankp(Gri1,i2λi1 ,λi2

).[Dp].

For chPar3 (E), see Chapter 3.

The formula for chPar1 (E) is well-known (Seshadri et al) and, in terms of the definitionof Chern classes using Deligne-Mumford stacks, it was shown by Borne in [Bo1]. Theformula for chPar2 (E) was given by Mochizuki in [Mo2], and also stated as a definitionby Panov [Pa]. In both cases these coincide with our result (see the discussion on page76). As far as we know, no similar formula for chPar3 (E) has appeared in the literature.

There are some of the motivations for the present work. In heterotic string theory[OPP] physicists look for a vector bundle with specific Chern classes. The third Chernclass corresponds to numbers of families of quarks and leptons on the observable brane.In future works where these vector bundles might be replaced by parabolic or orbifoldbundles, it would be important to have the formula for chPar3 (E). The BogomolovGieseker inequality says that ch2(E) ≥ 0 where E is a stable bundle with ch1(E) = 0.Donaldson’s theorem says that in case of equality one gets a flat unitary connection.These facts have been extended to the parabolic case notably in work of Li, Panov andMochizuki ([Mo2], [Pa], [L]). Our calculations confirm their formulas for ch2(E)—getting the right formula is essential for applying the Bogomolov-Gieseker inequality.The formula for chPar2 (E) will be useful in the Donagi-Pantev approach to the geometricLanglands program [DP]. Iyer and Simpson have pointed out that Reznikov’s theoremof vanishing of certain regulations of flat bundles, extends to the parabolic case, and forapplications it would be important to know the explicit formulas.

Mochizuki defines the Chern classes using the curvature of an adapted metric andobtains his formula as a result of a difficult curvature calculation. It should be notedthat our formula concerns the classes defined via Deligne-Mumford stacks in the ra-tional Chow groups of X whereas Mochizuki’s definition involving curvature can only

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define a class in cohomology. The identity of the two formulas shows that the curvaturedefinition and the stack definition give the same result up to degree 2. Of course theymust give the same result in general: to prove this for the higher Chern classes this is aninteresting question for further study.

Another question for further study is to calculate the Chern character defined byToen [To] using the intertia stack of the corresponding Deligne-Mumford stack. Thisaspect is not treated here.

In the second part of this thesis we Suppose X is a smooth surface and D = D1 + . . .+Dk is a divisor with each Di smooth. Suppose E is a bundle provided with filtrationsF i· along the Di, and parabolic weights αi· . If D has normal crossings, this defines a

locally abelian parabolic bundle on (X, D) and the parabolic Chern classes have beencalculated in [Mo2], [Pa] and [Ta].

Suppose that the singularities of D contain some points of higher multiplicity. Forthe present work we assume that these are as easy as possible, namely several smoothbranches passing through a single point with distinct tangent directions. The first basiccase is a triple points.

Let ϕ : X → X denote this birational transformation, and let Di ⊂ X denote thestrict transforms of the Di. Assuming for simplicity that there is a single multiple point,denote by D0 the exceptional divisor. Now D = D0 + · · ·+Dk is a divisor with normalcrossings. Suppose E is a vector bundle on X with

E|X−D0 = ϕ∗(E)|X−D0 .

The filtrations F i· induce filtrations of ϕ∗(E)|Di and hence of E|Di−Di∩D0 , which then

extend uniquely to filtrationsF i· ofE|Di . Associate to these filtrations the same parabolic

weights as before.Up until now we have already made a choice of extension of the bundle E. Choose

furthermore a filtration F 0· of E|D0 and parabolic weights associated to D0. Having

made these choices we get a parabolic bundle on the normal crossings divisor (X,D),which determines parabolic Chern classes. We are particularly interested in the invariant∆ which combines c1 and c2 in such a way as to be invariant by tensoring with a linebundle.

The goal of this paper is to provide a convenient calculation of ∆ and then investigateits dependence on the choices which have been made above. In particular we would liketo show that ∆ achieves its minimum and calculate this minimum, which can be thoughtof as the Chern invariant associated to the original parabolic structure on the multiplepoint singularity (X, D).

The main difficulty is to understand the possible choices for E. For this we use thetechnical of Ballico-Gasparim [Ba] [BG1] [BG2].

We would soon make explicit calculations for the case of 3 and 4 lines that meet at

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a point. We have not yet had time to do the Calculations.

1.1 Algebraic geometry background

1.1.1 Affine varietiesDefinition 1.1.1. Let k be an algebraically closed field. We define the affine n-spaceover k, denoted Ank or simply An, to be the set of all n-tuples of elements of k:

An := (a1, ..., an); ai ∈ k for 1 ≤ i ≤ n.

The elements of the polynomial ring

k[x1, ..., xn] := polynomials in the varaibles x1, ..., xn over k

= ∑I

aIxI ; aI ∈ k

(with the sum taken over all multi-indices I = (i1, ..., in) with ij ≥ 0 for all 1 ≤ j ≤n) define functions on An in the obvious way. For a given set S ⊂ k[x1, ..., xn] ofpolynomials, we define the zero set of S to be the common zeros of all the elements ofS, namely

Z(S) := P ∈ An; f(P ) = 0 for all f ∈ S ⊂ An.

Definition 1.1.2. A subset Y of An is an algebraic set if there exists a subset S ⊆k[x1, ..., xn] such that Y = Z(S).

Proposition 1.1.3. The union of two algebraic sets is an algebraic set. The intersectionof any family of algebraic sets is an algebraic set. The empty set and the whole spaceare algebraic sets.

Definition 1.1.4. We define the Zariski topology on An by taking the open subsets tobe the complements of the algebraic sets. This is a topology, because according to theproposition, the intersection of two open sets is open, and the union of any family ofopen sets is open. Furthermore, the empty set and the whole space are both open.

Definition 1.1.5. (i) A nonempty subset Y of a topological space X is said to bereducible if it can be written as a union Y = Y1 ∪ Y2 of two proper subsets, whereY1 and Y2 are (non-empty) closed subsets of Y not equal to Y . It is called irreducibleotherwise. An irreducible algebraic subset inAn is called affine variety. An open subsetof an affine variety is a quasi-affine variety.

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(ii) A nonempty subset Y of a topological spaceX is called disconnected if it can bewritten as a disjoint union Y = Y1 ∪ Y2 of (non-empty) closed subset of Y . It is calledconnected otherwise.

1.1.2 Projective varietiesTo define projective varieties, we proceed in a manner analogous to the definition ofaffine varieties, except that we work in projective space.

let k be a fixed algebraically closed field. We defined projective n-space over k, de-noted Pnk , or simply Pn, to be the set of equivalence classes of (n+ 1)-tuples (a0, ..., an)of element k, not all zero, under the equivalence relation given by (a0, ..., an) ∼ (λa0, ..., λan)for all λ ∈ k, λ 6= 0. Another way of saying this is that Pn as a set the quotient of theset An+1 − (0, ..., 0) under the equivalence relation which identifies points lying onthe same line through the origin.

Definition 1.1.6. For any polynomial f ∈ k[x1, ..., xn] denoted by f (1) the linear partof f . For an ideal I ∈ k[x1, ..., xn] denote by I(1) = f (1); f ∈ I the vector spaceof all linear parts of the elements of I; this is by definition a vector subspace of then-dimensional space k[x1, ..., xn](1) of all linear forms

a1x1 + ...+ anxn; ai ∈ k.

The zero locus Z(I(1)) is then a linear subspace of An. It is canonically dual (as avector space) to k[x1, ..., xn](1)/I(1), since the pairing

k[x1, ..., xn](1)/I(1) × Z(I(1)) −→ k, (f, P ) 7−→ f(P )

is obviously non-degenerate.Now let X ⊂ An be a variety. By a linear change of coordinates, assume that

P = (0, ...0) ∈ X . Then the linear space Z(I(X)(1)) is called the tangent space to Xat P and denoted TX,P .

Definition 1.1.7. A variety X is called smooth at the point P ∈ X if the tangent spaceTX,P to X at P has dimension (at most) dimX . It is called singular at P otherwise. Wesay that X is smooth if it is smooth at all points P ∈ X; otherwise X is singular.

1.1.3 SchemesDefinition 1.1.8. Let R be a ring (commutative with identity, as always). We defineSpecR to be the set of all prime ideals of R. (As usual, R itself does not count as aprime ideal, but (0) does if R is a domain.) We call SpecR the spectrum of R, or theaffine scheme associated to R. For every P ∈ SpecR, i.e. P ⊂ R a prime ideal, letk(P) be the quotient field of the domain R/P .

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Definition 1.1.9. A locally ringed space is a ringed space (X,OX) such that at eachpoint P ∈ X the stalk OX,P is a local ring. The maximal ideal of OX,P will be denotedby mX,P , and the residue field OX,P/mX,P will be denoted k(P ).

A morphism of locally ringed spaces from (X,OX) to (Y,OY ) is given by the fol-lowing data:

• a continuous map f : X −→ Y ,• for every open subsetU ⊂ Y a ring homomorphism f ∗U : OY (U) −→ OX(f−1(U)),

such that f ∗V ρU,V = ρf−1(U),f−1(V ) f ∗U for all V ⊂ U ⊂ Y (i.e. the f ∗ are compatiblewith the restriction maps) and (f ∗P )−1(mX,P ) = mY,f(P ), where the f ∗P : OY,f(P ) −→OX,P , are the maps induced on the stalks, as explained above. We will often omit theindex of the various pull-back maps f ∗ if it is clear from the context on which spacesthey act.

A morphism of affine schemes is a morphism as locally ringed spaces.

Definition 1.1.10. A scheme is a locally ringed space (X,OX) that can be covered byopen subsets Ui ⊂ X such that (Ui, OX |Ui) is isomorphic to an affine scheme SpecRi

for all i. A morphism of schemes is a morphism as locally ringed spaces.

1.1.4 Divisors on curvesDefinition 1.1.11. Let C ⊂ Pn be a smooth irreducible projective curve. A divisor onC is a formal finite linear combination D = a1P1 + ... + amPm of points Pi ∈ C withinteger coefficients ai. Obviously, divisors can be added and subtracted. The group ofdivisors on C is denoted Div C.

Equivalently, Div C is the free abelian group generated by the points of C.The degree deg D of a divisor D = a1P1 + ... + amPm is defined to be the integer

a1 + ...+ am. Obviously, the degree function is a group homomorphism

deg : DivC −→ Z.

Definition 1.1.12. The Picard group (or divisor class group) PicC ofC is defined to bethe group DivC modulo the subgroup of all divisors of the form (ϕ) for ϕ ∈ K(C)\0.If f ∈ S(C)(d), we will usually still write (f) for the divisor class in Pic C associatedto f . Two divisors D1 and D2 are said to be linearly equivalent if D1 −D2 = 0 ∈ PicC, i.e. if they define the same divisor class.

1.1.5 SheavesDefinition 1.1.13. A presheaf F of rings on a topological space X consists of the data:

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(a) for every open subset U ⊂ X , a ring F(U) (think of this as the ring of functionson U ),

(b) for every inclusion U ⊂ V of open subsets in X , a ring homomorphismρV U : F(V ) → F(U), called the restriction map (think of this as the usual restric-

tion of functions to a subset),

such that

(0) F(∅) = 0, where ∅ is the empty set,(1) ρUU is the identity map F(U) −→ F(U), for all U ,(2) for any inclusionU ⊂ V ⊂ W are three open subsets inX , we have ρV UρWV =

ρWU .

The elements ofF(U) are usually called the sections ofF overU , and the restrictionmaps ρV U are written as f 7−→ f |U . The space of global sectionsF(X) is often denotedΓ(F).

A presheaf F of rings is called a sheaf of rings if it satisfies the following glueingproperty: if U ⊂ X is an open subset, Ui an open cover of U and fi ∈ F(Ui) sectionsfor all i such that fi|Ui∩Uj = fj|Ui∩Uj for all i, j, then there is a unique f ∈ F(U)such that f |Ui = fi for all i. In other words, sections of a sheaf can be patched fromcompatible local data.

The same definition applies equally to categories other than rings, e.g. we can definesheaves of Abelian groups, k-algebras, and so on. For a ringed space (X,OX), e.g. ascheme, we can also define sheaves of OX-modules in the obvious way: every F(U)is required to be an OX(U)-module, and these module structures have to be compatiblewith the restriction maps in the obvious sense.

Definition 1.1.14. Let X be a topological space. A morphism f : F1 −→ F2 ofpresheaves of abelian groups (or rings, sheaves of OX-modules etc.) on X is a collec-tion of group homomorphisms (resp. ring homomorphisms, OX(U)-module homomor-phisms etc.) fU : F1(U) −→ F2(U) for every open subset U ⊂ X that commute withthe restriction maps, i.e. the restriction maps, i.e. the diagram

F1(U)ρUV //

fU

F1(V )

fV

F2(U)ρUV // F2(V )

is required to be commutative.

Definition 1.1.15. If f : X −→ Y is a morphism of topological spaces and F is asheaf on X , then we define the push-forward f?F of F to be the sheaf on Y given byf?F(U) = F(f−1(U)) for all open subsets U ⊂ Y .

9

Definition 1.1.16. Let f : F1 −→ F2 be a morphism of sheaves of e.g. Abelian groupson a topological space X . We define the kernel kerf of f by setting

(kerf)(U) = ker(fU : F1(U) −→ F2(U)).

We claim that kerf is a sheaf on X . In fact, it is easy to see that kerf with the obviousrestriction maps is a presheaf. Now let Ui be an open subset U ⊂ X , and assumewe are given ϕi ∈ ker(F1(Ui) −→ F2(Ui)) that agree on the overlaps Ui ∩ Uj . Inparticular, the ϕi are then in F1(Ui), so we get a unique ϕ ∈ F1(U) with ϕ |Ui= ϕi asF1 is a sheaf. Moreover, f(ϕi) = 0, so (f(ϕ)) |Ui= 0 by definition 1.1.14. As F2 is asheaf, it follows that f(ϕ) = 0, so ϕ ∈ kerf .

Remark 1. Now consider the dual case to definition 1.1.16, namely cokernels. Againlet f : F1 −→ F2 be a morphism of sheaves of e.g. Abelian groups on a topologicalspace X . As above we define a preasheaf coker′ f by setting

(coker′f)(U) = coker(fU : F1(U) −→ F2(U)) = F2(U)/imfU .

Definition 1.1.17. Let X be a topological space, P ∈ X , and F a (pre)-sheaf on X .Consider pairs (U,ϕ) where U is an open neighborhood of P and ϕ ∈ F(U) a sectionof F over U . We call two such pairs (U,ϕ) and (U ′, ϕ′) equivalent if there is an openneighborhood V of P with V ⊂ U ∩ U ′ such that ϕ |V = ϕ′ |V . (this is in fact anequivalence relation). The set of all such pairs modulo this equivalence relation iscalled the stalk FP of F at P , its elements are called germs of F .

Definition 1.1.18. Let F ′ be a preasheaf on a topological space X . The sheafificationof F ′, or the sheaf associated to the presheaf F ′, is defined to be the sheaf F such that

F(U) := ϕ = (ϕP )P∈U with ϕP ∈ F ′P for all P ∈ U

such that for every P ∈ U there is a neighborhood V ∈ U

and a section ϕ′ ∈ F ′(V ) with ϕQ = ϕ′Q ∈ F ′Q for all Q ∈ V .

Lemma 1.1.19. Let F ′ be a presheaf an a topological space X , and let F be its sheafi-fication.

(i) The stalks FP and F ′P agree at every point P ∈ X .(ii) If F ′ is a sheaf, then F = F ′.

Definition 1.1.20. Let f : F1 −→ F2 be a morphism of sheaves of e.g. Abelian groupson a topological space X .

(i) The cokernel cokerf of f is defined to be the sheaf associated to the presheafcoker′f .

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(ii) The morphism f is called injective if kerf = 0. It is called surjective if cokerf= 0.

(iii) If the morphism f is injective, its cokernel is also denoted F2/F1 and called thequotient of F2 by F1.

(iv) As usual, a sequence of sheaves and morphisms

... −→ Fi−1 −→ Fi −→ Fi+1 −→ ...

is called exact if ker(Fi −→ Fi+1) = im(Fi−1 −→ Fi) for all i.

Definition 1.1.21. Let R be a ring, X = SpecR and let M be an R-module. We definea sheaf of OX-module M on X by setting

M(U) := ϕ = (ϕP)P∈U with ϕP ∈MP for all P ∈ U

such that ϕ is locally of the formm

rfor m ∈M, r ∈ R

= ϕ = (ϕP)P∈U with ϕP ∈MP for all P ∈ U

such that for every P ∈ U there is a neighborhood V ∈ U

and m ∈M, r ∈ R with r /∈ q and ϕq =m

r∈Mq for all q ∈ V .

Definition 1.1.22. Let X be a scheme, and let F be a sheaf of OX-module. We say thatF is quasi-coherent if for every affine open subset U = SpecR ⊂ X the restrictedsheaf F |U is of the form M for some R-module M .

Definition 1.1.23. Let X be a scheme. A sheaf of OX-modules F is called locally freeof rank r if there is an open cover Ui of X such that F|Ui ∼= O⊕rUi for all i. Obviously,every locally free sheaf is also quasi-coherent.

1.1.6 Vector bundleDefinition 1.1.24. A vector bundle of rank r on a schemeX over a field k is a k-schemeF and a k-morphism π : F −→ X , together with the additional data consisting of anopen covering Ui of X and isomorphisms ψi : π−1(Ui) −→ Ui × Ark over Ui, suchthat the automorphism Ψi Ψ−1

j of (Ui ∩ Uj) × Ar is linear in the coordinates of Arfor all i, j. In other words, the morphism π : F −→ X looks locally like the projectionmorphism U × Ark −→ U for sufficiently small open subsets U ⊂ X .

We claim that there is one-to-one correspondence

vector bundles π : F −→ X of rank r ←→ locally free sheaves F of rank r on X

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given by the following constructions:

(i) Let π : F −→ X be a vector bundle of rank r. Define a sheaf F on X by

F(U) = k −morphims s : U −→ F such that π s = idU

(This is called the sheaf of section of F ). Note that this has a natural structure of a sheafofOX-modules (over every point inX we can multiply a vector with a scalar–doing thison an open subset means that we can multiply a section in F(U) with a regular functionin OX(U)).

Locally, on an open subset U on which π is of the form U×Ark −→ U , we obviouslyhave

F(U) = k −morphisms s : U −→ Ark,

so sections are just given by r independent functions. In other words,F |U is isomorphicto O⊕rU . So F is locally free by definition.

(ii) Conversely, let F be a locally free sheaf. Take an open cover Ui of X suchthat there are isomorphisms Ψi : F |Ui−→ O⊕rUi . Now consider the schemes Ui×Ark andglue them together as follows: for all i, j we glue Ui ×Ark and Uj ×Ark on the commonopen subset (Ui ∩ Uj)× Ark along the isomorphism

(Ui ∩ Uj)× Ark −→ (Ui ∩ Uj)× Ark (P, s) 7−→ (P,Ψi Ψ−1j ).

Note that isomorphisms Ψi Ψ−1j is an isomorphism of sheaves of OX-modules and

therefore linear in the coordinates of Ark.It is obvious that this gives exactly the inverse construction to (i).

Remark 2. Let π : F −→ X be a vector bundle of rank r, and let P ∈ X be a point.We call π−1(P ) the fiber of F over P ; it is an r-dimensional vector space. If F is thecorresponding locally free sheaf, the fiber can be realized as i∗F where i : P −→ Xdenotes the inclusion morphism (note that i∗F is a sheaf on a one-point space, so itsdata consists only of one k-vector space (i∗F)(P ), which is precisely the fiber FP ).

Definition 1.1.25. Let X be a scheme, and let F be a (quasi-coherent) sheaf on X . Fixan affine open cover Uii∈I of X , and assume for simplicity that I is an ordered set.For all p ≥ 0 we define the Abelian group

Cp(F) =∏

i0<...<ip

F(Ui0 ∩ ... ∩ Uip).

In other words, an element α ∈ Cp(F) is a collection α = (αi0,...,ip) of section Fover all intersections of p + 1 sets taken from the cover. These sections can be totally

12

unrelated. For every p ≥ 0 we define a boundary operator dp : Cp(F) −→ Cp+1(F) by

(dpα)i0,...,ip+1 =

p+1∑k=0

(−1)kαi0,...,ik−1,ik+1,...,ip+1|Ui0∩...∩Uip+1.

Note that this makes sense as the αi0,...,ik−1,ik+1,ip+1 are sections ofF on Ui0∩...∩Uik−1∩

Uik+1∩ ... ∩ Uip+1 which contains Ui0 ∩ .. ∩ Uip+1 as an open subset.

By abuse of notation we will denote all these operators simply by d if it is clear fromthe context on which Cp(F) they act.

Lemma 1.1.26. LetF be a sheaf on a schemeX . Then dp+1dp : Cp(F) −→ Cp+2(F)is the zero map for all p ≥ 0.

Proof. This statement is essentially due to the sign in the definition of dα: for everyα ∈ Cp(F) we have

(dp+1dpα)i0,...,ip+2 =

p+2∑k=0

(−1)k(dα)i0,...,ik−1,ik+1,...,ip+2

=

p+2∑k=0

k−1∑m=0

(−1)k+mαi0,...,im−1,im+1,...,ik−1,ik+1,...,ip+2

+

p+2∑k=0

p+2∑m=k+1

(−1)k+m−1αi0,...,ik−1,ik+1,...,im−1,im+1,...,ip+2

= 0

We have thus defined a sequence of Abelian groups and homomorphisms

C0(F)d0

−→ C1(F)d1

−→ C2(F)d2

−→ ...

such that dp+1 dp = 0 at every step. Such a sequence is usually called a complex ofAbelian groups. The maps dp are then called the boundary operators.

Definition 1.1.27. Let F be a sheaf on a scheme X . Pick an affine open cover Uiof X and consider the associated groups Cp(F) and homomorphisms dp : Cp(F) −→Cp+1(F) for p ≥ 0. We define the p-th cohomology group of F to be

Hp(X,F) = kerdp/imdp−1

with the convention that Cp(F) and dp are zero for p < 0. Note that this is well-definedas im dp−1 ⊂ ker dp by lemma 1.1.26. If X is a scheme over a field k then cohomologygroups will be vector spaces over k. The dimension of the cohomology groupsH i(X,F)as a k-vector space is then denoted hi(X,F).

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1.2 Intersection theory

1.2.1 Chow groupDefinition 1.2.1. Let X be a variety, and let V ⊂ X be subvariety of codimension 1,and set R = OX,V . For every non-zero f ∈ R ⊂ K(X) we define the order of f at V tobe the integer ordV (f) := lR(R/(f)). If ϕ ∈ K(X) is a non-zero rational function we

write ϕ =f

gwith f, g ∈ R and define the order of ϕ at V to be

ordV (ϕ) := ordV (f)− ordV (g).

Definition 1.2.2. Let X be a scheme. For K ≥ 0 denote by Zk(X) the free Abeliangroup generated by the k-dimensional subvarieties of X . In other words, the elementsZk(X) are finite formal sums

∑i

ni[Vi], where ni ∈ Z and the Vi, are k-dimensional

(closed) subvarieties of X . The element of Zk(X) are called cycles of dimension k.

For any (k+ 1)-dimensional subvariety W of X and any non-zero rational functionϕ on W we define a cycle of dimension k on X by

div(ϕ) =∑V

ordV (ϕ)[V ] ∈ Zk(X),

called the divisor of ϕ, where the sum is taken over all codimension-1 subvarieties Vof W . Note that this sum is always finite: it suffices to check this on a finite affine opencover Ui of W and for ϕ ∈ OUi(Ui), where it is obvious as Z(ϕ) is closed and Ui isNoetherian.

Let Bk(X) ⊂ Zk(X) be the subgroup generated by all cycles of the form div(ϕ)for all W ⊂ X and ϕ ∈ K(W )∗ as above. We define the group of k-dimensionalcycle classes to be the quotient Ak(X) = Zk(X)/Bk(X). These groups are usuallycalled Chow groups of X . Two cycles in Zk(X) that determine the same element inAk(X) are said to be rationnally equivalent.

We set Z∗(X) =⊕k≥0

Zk(X) and A∗(X) =⊕k≥0

Ak(X).

1.2.2 Cartier divisorDefinition 1.2.3. Let X be a scheme.

(i) If X has pure dimension n a Weil divisor on X is an element Zn−1(X). Obvi-ously, the Weil divisors form an Abelien group. Two weil divisors are called linearly equivalentif they define the same class inAn−1(X). The quotient groupAn−1(X) is called the group

14

of Weil divisor classes.

(ii) Let KX be the sheaf of rational functions on X , and denoted by K∗X the sub-sheaf of invertible elements (i.e. of those functions that are not identically zero on anycomponent of X). Note that K∗X is a sheaf of Abelian groups, with the group structuregiven by multiplication of rational functions; Similarly, let O∗X be the sheaf of invertibleelements of OX (i.e. of the regular functions that are nowhere zero). Note that O∗X is asheaf of Abelian groups under multiplication as well. In fact, O∗X is a sheaf of K∗X .

A Cartier divisor on X is a global section of the sheaf K∗X/O∗X . Obviously, theCartier divisors form an Abelian group under multiplication, denoted DivX . In anal-ogy to Weil divisors the group structure on DivX is usually written additively however.A Cartier divisor is called linearly equivalent to zero if it is induced by a global sectionof K∗X . Two Cartier divisors are linearly equivalent if their difference (i.e. quotient, seeabove) is linearly equivalent to zero. The quotient group Pic := Γ(K∗X/O∗X)/Γ(K∗X) iscalled the group of Cartier divisor classes.

Definition 1.2.4. Let X be a scheme, let V ⊂ X be a k-dimensional subvariety withinclusion morphism i : V −→ X , and let D be a Cartier divisor on X . We define theintersection product D.V ∈ Ak−1(X) to be

D.V = i∗[i∗OX(D)],

where OX(D) is the line bundle on X associated to the Cartier divisor D, i∗ denotesthe pull-back of line bundles, [i∗OX(D)] is the Weil divisor class associated to the linebundle i∗OX(D) (note that V is irreducible), and i∗ denotes the proper push-forward.

Lemma 1.2.5. Let f : X −→ Y be a proper surjective morphism of schemes. Letα ∈ Ak(X) be a k-cycle on X , and let D ∈ PicY be a Cartier divisor(class) on Y .Then

f∗(f∗D.α) = D.f∗α ∈ Ak−1(Y ).

1.2.3 Chern classesFor any vector bundle π : F −→ X of rank r on a scheme X we define an associatedprojetive bundle p : P(F ) −→ X whose fibers p−1(P ) are just the projectivizationsof the affine fibers π−1(p). We construct natural line bundles OP(F )(d) on P(F ) for alld ∈ Z that correspond to the standard line bundle O(d) on projective spaces. As inthe case of vector bundles there are pull-back homomorphisms A∗(X) −→ A∗(P(F ))between the Chow groups.

For a bundle as above we define the i-th Segre class si(F ) : A∗(X) −→ A∗−i(X)by si(F ).α = p∗(D

r−1+iF .p∗α), where DF denotes the Cartier divisor associated to the

line bundle OP(F )(1). The Chern classes ci(F ) are defined to be the inverse of the Segre

15

classes. Segre and Chern classes are commutative; they satisfy the projection formulafor proper push-forwards and are compatible with pull-backs. They are multiplicativeon exact sequences. Moreover, ci(F ) = 0 for i > r. The stop Chern class cr(F ) hasthe additional geometric interpretation as the zero locus of a section of F . Using thetechnique of Chern roots one can compute the Chern classes of almost any bundle thatis constructed from known bundles in some way (e.g. by means of direct sums, tensorproducts, dualizing, exact sequences, symmetric and exterior product).

The Chern character ch(F ) and Todd class td(F ) are defined to be certain polyno-mial combinations of the Chern classes of F . The Hirzebruch-Riemann-Rock theoremstates that

∑i

hi(X,F ) = deg(ch(F ).td(TX)) for any vector bundle F on a smooth

projective scheme X .

Projective bundle. Recall that for any line bundle L on a variety X there is aCartier divisor onX correponding to L that in turn defines intersection homomorphismsAk(X) −→ Ak−1(X). These homomorphisms can be thought of as intersecting a k-cycle on X with the divisor of any rational section of L. We now want to generalizethis idea from line bundles to vector bundles. To do so, we need some preliminaries onprojective bundles first.

Roughly speaking, the projective bundle P(E) associated to a vector bundle E ofrank r on a scheme X is simply obtained by replacing the fibers (that are all isomorphicto Ar) by the corresponding projective spaces Pr−1 = (Ar\0)/k∗. Let us give theprecise definition.

Definition 1.2.6. Let π : F −→ X be a vector bundle of rank r on a scheme X , thereis an open covering Ui of X such that

(i) there are isomorphisms Ψi : π−1(Ui) −→ Ui × Ar, over Ui,(ii) on the overlaps Ui ∩ Uj the compositions

Ψi Ψ−1j : (Ui ∩ Uj)× Ar −→ (Ui ∩ Uj)× Ar

are linear in the coordinates of Ar, i.e. they are of the form

(P, x) 7−→ (P,Ψi,jx)

where P ∈ U, x = (x1, x2, ..., xr) ∈ Ar, and the Ψi,j are r × r matrices with entries inOX(Ui ∩ Uj).

Then the projective bundle P(F ) is defined by glueing the patches Ui × Pr−1 alongthe same transition functions , i.e. by gleuing Ui× Pr−1 to Uj × Pr−1 along the isomor-phisms

(Ui ∩ Uj)× Pr−1 −→ (Ui ∩ Uj)× Pr−1, (P, x) 7−→ (P,Ψi,jx)

16

for all i, j, where P ∈ Ui ∩ Uj and x = (x1 : ... : xr) ∈ Pr−1. We say that P(F ) is aprojective bundle of rank r − 1 on X .

Note that in the same way as for vector bundles there is a natural projective mor-phism p : P(F ) −→ X that sends a point (P, x) to P .

Construction 1.2.7. Let p : P(F ) −→ X be a projective bundle over a schemeX , givenby open cover Ui of X and transition matrices Ψi,j . Consider the vector bundle p∗Fon P(F ). It is given by glueing the patches Ui × Pr−1 × Ar along the isomorphisms

(Ui ∩ Uj)× Pr−1 × Ar −→ (Ui ∩ Uj)× Pr−1 × Ar, (P, x, y) 7−→ (P,Ψi,jx ,Ψi,jy),

where x = (x1 : ... : xr) are projective coordinates on Pr−1, and y = (y1, ..., yr) areaffine coordinates on Ar. Now consider the subbundle S of p∗F given locally by theequations xiyj = xjyi for all i, j = 1, ..., r, i.e. the subbundle of p∗F consisting ofthose (y1, ..., yr) that are scalar multiples of (x1 : ... : xr). Obviously, S is a line bundleon P(F ) contained in p∗F . Geometrically, the fiber of S over a point (P, x) ∈ P(F ) isprecisely the line in the fiber Fp whose projectivization is the point x. The line bundleS ⊂ p∗F is called the tautological subbundle on P(F ).

Lemma 1.2.8. Let F be a vector bundle on a scheme X of rank r + 1, and let p :P(F ) −→ X be the associated projective bundle of rank r. Then there are pull-backhomorphisms

p∗ : Ak(X) −→ Ak+r(P(F )), [V ] 7−→ [p−1(V )]

for all k, satisfying the following compatibilities with our earlier constructions:

(i) (Compatibility with proper push-forward) Let f : X −→ Y be a proper mor-phism, and let F be a vector bundle of rank r + 1 on Y . Form the fiber diagram

P(f ∗F )f ′ //

p′

P(F )

p

X

f // Y.

Then p∗f∗ = f ′∗p′∗ as homomorphisms Ak(X) −→ Ak+r(P(F )).

(ii) (Compatibility with intersection products) Let F be a vector bundle of rank r+1on X , and let D ∈ PicX be a Cartier divisor(class). Then

p∗(D.α) = (p∗D).(p∗α)

in Ak+r−1(P(F )) for every k-cycle α ∈ Ak(X).

17

1.2.4 Segre and Chern classes vector bundlesLetX be a scheme, and let F be a vector bundle of rank r onX . Let p : P(F ) −→ X bethe projection from corresponding projective bundle. Note that we have the followingconstructions associated to p:

(i) push-forward homomorphisms p∗ : Ak(P(F )) −→ Ak(X) since p is proper,(ii) pull-back homomorphisms p∗ : Ak(X) −→ Ak+r−1(P(F )) by lemma 1.2.8,(iii) a line bundle OP(F )(1) on P(F ) (the dual of the tautological subbundle).We can now combine these three operations to get homomorphisms of the Chow

groups of X that depend on the vector bundle F .

Definition 1.2.9. Let X be a scheme, and let F be a vector bundle of rank r on X . Letp : P(F ) −→ X be the projection map from the associated projective bundle. Assumefor simplicity thatX (and hence P(F )) is irreducible (see below), so that the line bundleOP(F )(1) correspond to a Cartier divisorDF on P(F ). Now for all i ≥ −r+1 we defineSegre class homomorphisms by the formula

si(F ) : Ak(X) −→ Ak−i(X), α 7−→ si(F ).α := p∗(Dr−1+iF .p∗α).

Proposition 1.2.10. Let X and Y be schemes.

(i) For any vector line bundle F on X we have•si(F ) = 0 for i < 0,•s0(F ) = id.

(ii) For any line bundle L on X we have si(L).α = (−1)iDi.α for i ≥ 0 and allα ∈ A∗(X), where D is the Cartier divisor class associated to the line bundle L.

(iii) (Commutativity) If F1 and F2 are vector bundles on X , then

si(F1).sj(F2) = sj(F2).si(F1)

as homomorphisms Ak(X) −→ Ak−i−j(X) for all i, j (where the dot denotes the com-position of the two homomorphisms).

(iv) (projection formula) If f : X −→ Y is proper, F is a vector bundle on Y , andα ∈ A∗(X), then

f∗(si(f∗F ).α) = si(F ).f∗α.

(V) (Compatibility with pull-back) If f : X −→ Y is a morphism for which a pull-back f ∗ : A∗(Y ) −→ A∗(X) exists, F is a vector bundle on Y , and α ∈ A∗(Y ) then

si(f∗F ).f ∗α = f ∗(si(F ).α).

18

Corollary 1.2.11. Let F be a vector bundle on a scheme X , and let p : P(F ) −→ Xbe the projection. Then p∗ : A∗(P(F )) −→ A∗(X) is surjective and p∗ : A∗(X) −→A∗(P(F )) is injective.

Definition 1.2.12. Let X be a scheme, and let F be a vector bundle of rank r on X . Thetotal Segre class of F is defined to be the formal sum

s(F ) =∑i≥0

si(F ) : A∗(X) −→ A∗(X).

Note that:

(i) All si(F ) can be recovered from the homomorphism s(F) by considering thegraded parts.

(ii) Although the sum over i in s(F ) is formally infinite, it has of course only finitelymany terms as Ak(X) is non-zero only for finitely many k.

(iii) The homomorphism s(F ) is in fact an isomorphism of vector spaces: by propo-sition 1.2.10(i) it is given by triangular matrix with ones on the diagonal (in the naturalgrading of A∗(X)).

By (iii) it makes sense to define the total Chern class of F

c(F ) =∑i≥0

ci(F )

to be the inverse homomorphism of s(F ). In order words, the Chern classes ci(F ) arethe unique homomorphisms ci(F ) : Ak(X) −→ Ak−i(X) such that

s(F ).c(F ) = (1 + s1(F ) + s2(F ) + ...).(c0(F ) + c1(F ) + c2(F ) + ...) = id.

Explicitly, the first few Chern classes are given by

c0(F ) = 1,

c1(F ) = −s1(F ),

c2(F ) = −s2(F ) + s1(F )2,

c3(F ) = −s3(F ) + 2s1(F )s2(F )− s1(F )3.

Proposition 1.2.13. Let X and Y be schemes.

(i) For any bundle L onX with associated Cartier divisor classD we have c(L).α =(1 + D).α. In other words, ci(L) = 0 for i > 1, and c1(L) is the homomorphism ofintersection with the Cartier divisor class associated to L. By obuse of notation, the

19

Cartier divisor class associated to L is often also denoted c1(L).

(ii) (Commutativity) If F1 and F2 are vector bundle on X , then

ci(F1).cj(F2) = cj(F2).ci(F1)

for all i, j.

(iii) (projection formula) If f : X −→ y is proper, F is a vector bundle on Y , andα ∈ A∗(X), then

f∗(ci(f∗F ).α) = ci(F ).f∗α.

(iv) (pull-back) If f : X −→ Y is a morphism for which a pull-back f ∗ : A∗(Y ) −→A∗(X) exists, F is a vector bundle on Y , and α ∈ A∗(Y ), then

ci(f∗F ).f ∗α = f ∗(ci(F ).α).

Proposition 1.2.14. Let 0 −→ F ′ −→ F −→ F ′′ −→ 0 be an exact sequence of vectorbundles an a scheme X . Then

c(F ) = c(F ′).c(F ′′).

Remark 3. The proposition 1.2.14 can be split up into graded parts to obtain the equa-tions

ck(F ) =∑i+j=k

ci(F′).cj(F

′′)

for all k ≥ 0 and any exact sequence 0 −→ F ′ −→ F −→ F ′′ −→ 0 of vector bundleon a scheme X .

Note moreover that by definition the same relation s(F ) = s(F ′).s(F ′′) then holdsfor the Segre classes.

Example 1.2.15. In this example we will compute the Chern classes of the tangentbundle TX of X = Pn. We have an exact sequence of vector bundles on X

0 −→ OX −→ OX(1)⊕(n+1) −→ TX −→ 0.

Moreover proposition 1.2.13(i) implies that c(OX) = 1 and c(OX(1)) = 1 + H , whereH is the divisor class of a hyperplane in X . So by proposition 1.2.14 it follows that

c(TX) = c(OX(1))n+1/c(OX) = (1 +H)n+1,

i.e. ck(TX) =

(n+ 1k

).Hk (where Hk is the class of a linear subspace of X of

comdimension k).

20

Remark 4. Note that proposition 1.2.14 allows us to compute the Chern classes of anybundle F of rank r on a scheme X that has a filtration

0 = F0 ⊂ F1 ⊂ ... ⊂ Fr−1 ⊂ Fr = F

by vector bundles such that the quotients Li := Fi/Fi−1 are all line bundles(i.e. Fi hasrank i for all i). In fact, in this case a recursive application of proposition 1.2.14 to theexact sequences

0 −→ Fi−1 −→ Fi −→ Li −→ 0

yields (toghether with proposition 1.2.13(i))

c(F ) =r∏i=1

(1 +Di)

where Di = c1(Li) is the divisor associated to the line bundle Li.Unfortunately, not every vector bundle admits such a filtration. Well will see now

however that for computations with Chern classes we can essentially pretend that sucha filtration always exists.

Lemma 1.2.16. (Splitting construction) Let F be a vector bundle of rank r on a schemeX . Then there is a scheme Y and a morphism f : Y −→ X such taht

(i) f admit push-forwards and pull-backs for Chow groups (in fact it will be aniterated projective bundle),

(ii) the push-forward f∗ is surjective,(iii) the pull-back f ∗ is injective,(iv) f ∗F has a filtration by a vector bundles

0 = F0 ⊂ F1 ⊂ ... ⊂ Fr−1 ⊂ Fr = f ∗F

such that the quotient Fi/Fi−1 are line bundle on Y .In other words, ”every vector bundle admits a filtration after pulling back to an

iterated projective bundle”.

Proof. We construct the morphism f by induction on rankF . There is nothing to doif rankF = 1. Otherwise set Y ′ = P(F∨) and let f ′ : Y ′ −→ X be the projection.Let L∨ ⊂ f ′∗F∨ be the tautological line bundle on Y ′. Then we have exact sequence ofvector bundles 0 −→ F −→ f ′∗F −→ L −→ 0 on Y ′, where rankF = rankF − 1.Hence by the inductive assumption there is a morphism f ′′ : Y −→ Y ′ such that f ′′∗Fhas a filtration (Fi) with line bundle quotients. If we set f = f ′ f ′′ it follows that wehave an induced filtration of f ∗F on Y

0 = F0 ⊂ F1 ⊂ ... ⊂ Fr−1 = f ′′∗F ⊂ f ∗F

21

with line bundle quotients. Moreover, f∗ is surjective and f ∗ is injective, as this is truefor f ′′ by the inductive assumption and for f ′ by corollary 1.2.18

Construction 1.2.17. (Splitting construction) Suppose one wants to prove a universalidentity among Chern classes of vector bundles on a scheme X , e.g. the statement thatci(F ) = 0 whenever i > rankF (see corollary 1.2.18 below). If the identity is invariantunder pull-backs (which it essentially always is because of proposition 1.2.13(iv)) thenone can assume that the vector bundles in question have filtrations with line bundlequotients. More precisely, pick a morphism f : Y −→ X as in lemma 1.2.16. We canthen show the identity for the pulled-back bundle f ∗F on Y , using the filtration. As thepull-back f ∗ is injective and commutes with the identity we want to show, the identitythen follows for F on X as well.

Corollary 1.2.18. Let F be a vector bundle of rank r on a scheme X . Then ci(F ) = 0for all i > r.

Proof. By the splitting construction 1.2.16 we can assume that F has a filtration with

line bundle quotient Li, i = 1, ..., r. But then c(F ) =r∏i=1

(1 + c1(Li)) by remark 4,

which obviously has no parts of degree bigger than r.

Remark 5. This vanishing of Chern classes beyond the rank of the bundle is a propertythat is not shared by the Segree classes (see e.g. Proposition 1.2.13(ii)). This is onereason why Chern classes are usually preferred over Segree classes in computations(although they carry the same information).

Remark 6. The splitting construction is usually formalized as follows. Let F be a vectorbundle of rank r on a scheme X . We write formally

c(F ) =r∏i=1

(1 + αi).

There are two ways to think of the α1, ..., αr:

• The αi are just formal ”variables” such that the k-th elementary symmetric poly-nomial in the αi is exactly ck(F ). So any symmetric polynomial in the αi is expressibleas a polynomial in the Chern classes of F in a unique way.

• After having applied the splitting construction, the vector bundle F has a filtrationwith line bundle quotient Li. Then we can set αi = c1(Li), and the decomposition

c(F ) =r∏i=1

(1 + αi) becomes an actual equation (and not just a formal one).

22

The αi are usually called the Chern roots of F . Using the splitting construction andChern roots, one can compute the Chern classes of almost any bundle that is constructedfrom other known bundles by standard operations:

Proposition 1.2.19. Let X be a scheme, and let F and F ′ be a vector bundles withChern roots (αi)i and (α′j)j , respectively. Then:

(i) F∨ has Chern roots (−αi)i.(ii) F ⊗ F ′ has Chern roots (αi + α′j)i,j .(iii) SkF has Chern roots (αi1 + ...+ αik)i1<...<ik .(iv) ∧kF has Chern roots (αi1 + ...+ αik)i1<...<ik .

Example 1.2.20. The results of proposition 1.2.19 can be restarted using Chern classesinstead of Chern roots. For example, (i) just that ci(F∨) = (−1)ici(F ). It is moredifficult to write down closed forms for the Chern classes in the cases (ii) to (iv). Forexample, if F ′ = L is a line bundle, then

c(F ⊗ L) =∏i

(1 + (αi + α′)) =∑i

(1 + ci(L))r−ici(F )

where r = rank F . So for 0 ≤ p ≤ r we have

cp(F ⊗ L) =

p∑i=0

(r − ip− i

)ci(F )c1(L)p−i.

Also, from part (iv) it follows immediately that c1(F ) = c1(∧rF ).

As a more complicated example, assume that F is a rank-2 bundle on a scheme Xand let us compute the Chern classes of S3F . Say F has Chern roots α1 and α2, so thatc1(F ) = α1 + α2 and c2(F ) = α1α2. Then by part (iii) a tedious but easy computationshows that

c(S3F ) = (1 + 3α1)(1 + 2α1 + α2)(1 + α1 + 2α2)(1 + 3α2)

= 1 + 6c1(F ) + 10c2(F ) + 11c1(F )2 + 30c1(F )c2(F )

+6c1(F )3 + 9c2(F )2 + 18c1(F )2c2(F ).

Splitting this up into graded pieces one obtains the individual Chern classes, e.g.

c4(S3F ) = 9c2(F )2 + 18c1(F )2c2(F ).

Proposition 1.2.21. Let F be a vector bundle of rank r on an n-dimensional scheme X .Let s ∈ Γ(F ) be a global section of F , and assume that its scheme-theoretic zero locusZ(s) has dimension n− r (as expected). Then [Z(s)] = cr(F ).[X] ∈ An−r(X).

23

1.2.5 Statement of the Hirzeburch-Rimann-Roch theoremDefinition 1.2.22. Let F be a coherent sheaf on a projective scheme X . Then thedimension hi(X,F) = dim(X,F) are all finite. We define the Euler characteristic ofF to be the integer

X (X,F) :=∑i≥0

(−1)ihi(X,F).

(Note that the sum is finite as hi(X,F) = 0 for i > dimX.)

The ”left hand side” of the Hirzebruch-Riemann-Rock theorem will just beX (X,F);this is the number that we want to compute. Recall that there were many ”vanishing theorems”, e.g. hi(X,F ⊗ OX(d)) = 0 for i > 0 and d 0. So in the cases whensuch vanishing theorems aplly the theorem will actually compute the desired numberh0(X,F).

The ”right hand side” of the Hirebruch-Riemann-Roch theorem is an intersection-theoretic expression that is usually easy to compute. It is a certain combination of theChern (resp. Segre) classes of the bundle F (corresponding to the locally free sheaf F)and the tangent bundle TX of X . These combinations will have rational coefficients, sowe have to tensor the Chow groups withQ (i.e. we consider formal linear combinationsof subvarieties with rational coefficients instead of integer ones).

Definition 1.2.23. Let F be a vector bundle of rank r with Chern roots α1, ..., αr on ascheme X . Then we define the Chern character ch(F ) : A∗(X) ⊗ Q −→ A∗(X) ⊗ Qto be

ch(F ) =r∑i=1

exp(αi)

and the Todd class td(F ) : A∗(X)⊗Q −→ A∗(X)⊗Q to be

td(F ) =r∏i=1

αi1− exp(−αi)

,

where the expressions in the αi are to be understood as formal power series, i.e.

exp(αi) = 1 + αi +1

2α2i +

1

6α3i + ...

andαi

1− exp(−αi)= 1 +

1

2αi +

1

12α2i + ....

24

Remark 7. As usual we can expand the definition of ch(F ) and td(F ) to get symmetricpolynomials in the Chern roots which can then be written as polynomials (with rationalcoefficients) in the Chern classes ci = ci(F ) of F . Explicitly,

ch(F ) = r + c1 +1

2(c2

1 − 2c2) +1

6(c3

1 − 3c1c2 + 3c3) + ...

andtd(F ) = 1 +

1

2c1 +

1

12(c2

1 + c2) +1

24c1c2 + ...

Remark 8. If 0 −→ F ′ −→ F −→ F ′′ −→ 0 is an exact sequence of vector bundleson X then the Chern roots of F are just the union of the Chern roots of F ′ and F ′′. Sowe see that

ch(F ) = ch(F ′) + ch(F ′′)

andtd(F ) = td(F ′).td(F ′′).

We can now state the Hirzebruch-Rieman-Roch theorem:

Theorem 1.2.24. (Hirzebruch-Riemann-Roch theorem) Let F be a vector bundle on asmooth projective variety X . Then

X (X,F) = deg(ch(F ).td(TX))

where deg(α) denotes the degree of the dimension-0 part of the (non-homogeneous)cycle α.

Example 1.2.25.

(i) Let F = L be a line on a smooth projective curveX of genus g. ThenX (X,L) =h0(X,L) − h1(X,L). On the right hand side, the dimension-0 part of ch(L).td(TX),i.e. its codimension-1 part, is equal to

deg(ch(L).td(TX)) = deg((1 + c1(L))(1 +1

2c1(TX))) by remark 7

= deg(c1(L)− 1

2(ΩX))

= degL− 1

2(2g − 2)

= degL+ 1− g,

(ii) If F is a vector bundle of rank r on a smooth projective curve X then we get inthe same way

25

h0(X,F )− h1(X,F ) = deg(ch(F ).td(TX))

= deg((r + c1(F ))(1 +1

2c1(TX)))

= deg c1(F ) + r(1− g).

(iii) Let L = OX(D) be a line bundle on a smooth projective surfaceX correspond-ing to a divisor D. Now the dimension-0 part of the right hand side has codimension 2,so the Hirzebruch-Riemann-Rock theorem states that

h0(X,L)− h1(X,L) + h2(X,L)

= deg(ch(F ).td(TX))

= deg

((1 + c1(L) +

1

2c1(L)2

)(1 +

1

2c1(TX) +

1

12(c1(TX)2 + c2(TX))

))

=1

2D.(D −KX) +

K2X + c2(TX)

12.

Note that:

• The numberX (X,OX) =K2X + c2(TX)

12is an invariant ofX that does not depend

on the line bundle. The Hirzebruch-Riemann-Rock theorem implies that it is always aninteger, i.e. that K2

X + c2(TX) is divisible by 12 (which is not at all obvious from thedefinitions).• If X has degree d and L = OX(n) for n 0 then h1(X,L) = h2(X,L) = 0.

Then we have D2 = dn2, so we get

h0(X,OX(n)) =d

2n2 +

1

2(H.KX).n+

K2X + c2(TX)

12,

where H denotes the class of a hyperplane (restricted to X).

Definition 1.2.26. LetX be a scheme. The Grothendieck group of vector bundlesK(X)

on X is defined to be the group of formal finite sums∑i

ai[Fi] where ai ∈ Z and the

Fi are vector bundles on X , modulo the relations [F ] = [F ′] + [F ′′] for every exact se-

quence 0 −→ F ′ −→ F −→ F ′′ −→ 0. (Of course we then also haver∑i=1

(−1)i[Fi] = 0

for every exact sequence

0 −→ F1 −→ F2 −→ ... −→ Fr −→ 0.)

26

Example 1.2.27. Definition 1.2.26 just says that every construction that is additive onexact sequences passes to the Grothendieck group. For example:

(i) If X is projective then the Euler characteristic of a vector bundle (see definition1.2.22) is additive on exact sequences by the long exact cohomology sequence of propo-sition. Hence the Euler characteristic can be thought of as a homomorphism of Abeliangroups

X : K(X) −→ Z, X ([F ]) = X (X,F ).

(ii) The Chern character of a vector bundle is additive on exact sequences remark 8. So we get a homomorphism

ch : K(X) −→ A∗(X)⊗Q, ch([F ]) = ch(F ).

(It can in fact be shown that this homomorphism gives rise to an isomorphism K(X)⊗Q −→ A∗(X)⊗Q if X is smooth.

(iii) Let X be a smooth projective variety. For the same reason as in (ii) the righthand side of the Hirzebruch-Riemann-Rock theorem gives rise to a homomorphism

τ : K(X) −→ A∗(X)⊗Q, τ(F ) = ch(F ).td(TX).

Lemma 1.2.28. Let X be a smooth projective scheme. Then for every coherent sheaf Fon X there is an exact sequence

0 −→ Fr −→ Fr−1 −→ ... −→ F0 −→ F −→ 0

where the Fi are vector bundles (i.e. locally free sheaves). We say that ”every coherentsheaf has a finite locally free resolution”. Morever, if X = Pn then the Fi can all bechosen to be direct sums of line bundles OX(d) for various d.

Proof. There is a (possibly infinite) exact sequence

... −→ Fr −→ ... −→ F1 −→ F0 −→ F −→ 0.

Now one can show that for an n-dimensional smooth scheme the kernel K of the mor-phism Fr−1 −→ Fr−2 is always a vector bundle. So we get a locally free resolution

0 −→ K −→ Fr−1 −→ Fr−2 −→ ... −→ F0 −→ F −→ 0

as required.

27

If X = Pn with homogeneous coordinate ring S = k[x0, ..., xn] then one can showthat a coherent sheaf F on X is nothing but a graded S-module M (in the same way thata coherent sheaf on an affine scheme Spec R is given by an R-module). By the famousHilbert syzygy theorem there is a free resolution of M

0 −→⊕i

Sn,i −→ ... −→⊕i

S1,i −→⊕i

S0,i −→M −→ 0

where each Sj,i is isomorphic to S, with the grading shifted by some constants aj,i. Thismeans exactly that we have a locally free resolution

0 −→⊕i

OX(an,i) −→ ... −→⊕i

OX(a1,i) −→⊕i

OX(a0,i) −→ F −→ 0

of F .

Corollary 1.2.29. The Hirzebruch-Riemann-Roch theorem 1.2.24 is true for any vectorbundle on Pn.

Remark 9. To study the Hirzebruch-Riemann-Roch theorem for general smooth pro-jective X let i : X −→ Pn be an embedding of X in projective space and consider thediagram see [G] Remark 10.5.5 page 205:

Let us first discuss the right square. The homomorphisms X and τ are explained inexample 1.2.27 , and deg denotes the degree of the dimension-0 part of a cycle class.The Hirzebruch-Riemann-Rock theorem for Pn of corollary 1.2.29 says precisely thatthis right square is commutative.

Now consider the left square. The homomorphism τ is as above, and the i∗ in thebottom row is the proper push-forward of cycles. We have to explain the push-forwardi∗ in the top row. Of course we would like to define to define i∗[F ] = [i∗F ] for anyvector bundle F on X , but we cannot do this directly as i∗F is not a vector bundle butonly a coherent sheaf on Pn. So instead we let

0 −→ Fr −→ Fr−1 −→ ... −→ F0 −→ i∗F −→ 0 (∗)

be a locally free resolution of the coherent sheaf i∗F on Pn and set

i∗ : K(X) −→ K(Pn), i∗([F ]) =r∑

k=0

(−1)k[Fk].

One can show that this is indeed a well-defined homomorphism of groups (i.e. that thisdefinition does not depend on the choice of locally free resolution). But in fact we do

28

not really need know this: we do know by the long exact cohomology sequence appliedto (∗) that

X (X,F ) =r∑

k=0

(−1)kX (Pn, Fk),

so is clear that at least the composition X i∗ does not depend on the choise of locallyfree resolution. The Hirzebruch-Riemann-Rock theorem on X is now precisely thestatement that the outer rectangle in the above diagram is commutative.

As we know already that the right square is commutative, it suffices therefore toshow that the left square is commutative as well (for any choice of locally free resolutionas above), i.e. that

r∑k=0

(−1)kch(Fk).td(TPr) = i∗(ch(F ).td(TX)).

As the Todd class is multiplicative on exact sequences by remark 8 we can rewrite thisusing the projection formula as:

r∑k=0

(−1)rch(Fk) = i∗.ch(F )

td(NX/Pn).

Summarizing our ideas we see that to prove the general Hirzebruch-Riemann-Rock the-orem it suffices to prove the following proposition (for Y = Pn):

Proposition 1.2.30. Let i : X −→ Y be a closed immersion of smooth projectiveschemes, and let F be a vector bundle on X . Then there is a locally free resolution

0 −→ Fr −→ Fr−1 −→ ... −→ F0 −→ i∗F −→ 0

of the coherent sheaf i∗F on Y such that

r∑k=0

(−1)rch(Fk) = i∗.ch(F )

td(NX/Y )

in A∗(Y )⊗Q.

Example 1.2.31. Before we give the general proof let us consider an example whereboth sides of the equation can be computed explicitly: let X be a smooth scheme, E avector bundle of rank r on X , and Y = P(E ⊕ OX). The embedding i : X −→ Y isgiven by X = P(0⊕OX) → P(E⊕OX). In order words, X is just ”the zero section ofa projective bundle”. The special features of this particular case that we will need are:

(i) There is a projection morphism p : Y −→ X such that p i = id.

29

(ii) X is the zero locus of a section of a vector bundle on Y : consider the exactsequence

0 −→ S −→ p∗(E ⊕OX) −→ Q −→ 0 ,

on Y , where S is the tautological subbundle of construction 1.2.7. The vector bundle Q(which has rank r) is usually called the universal quotient bundle. Note that we have aglobal section of p∗(E ⊕ OX) by taking the point (0, 1) in every fiber(i.e. 0 in the fiberof E and 1 in the fiber of OX). By definition of S the induced section s ∈ Γ(Q) vanishesprecisely on P(0⊕OX) = X .

(iii) Restricting (∗) to X (i.e. pulling the sequences back by i) we get the exactsequence

0 −→ i∗S −→ E ⊕OX −→ i∗Q −→ 0 (?)

on X . Note that the first morphism is given by λ 7−→ (0, λ) by construction, so weconclude that i∗Q = E.

(iv) As X is given in Y as the zero locus of a section of Q, the normal bundle of Xin Y is just NX/Y = i∗Q = E.

Let us now check proposition 1.2.30 in this case. Note that away from the zero locusof s there is an exact sequence

0 −→ OY.s−→ Q

∧s−→ ∧2Q∧s−→ ∧3Q −→ ... ∧r−1 Q

∧s−→ ∧rQ −→ 0

of vector bundles (which follows from the corresponding statement for vector spaces).Dualizing and tensoring this sequence this sequence with p∗F we get the exact se-

quence

0 −→ p∗F ⊗ ∧rQ∨ −→ p∗F ⊗ ∧r−1Q∨ −→ ... −→ p∗F ⊗Q∨ −→ p∗F −→ 0

again on Y \ Z(s) = Y \X . Let us try to extend this exact sequence to all of Y . Notethat the last morphism p∗F ⊗ Q∨ −→ p∗F is just induced by the evaluation morphisms : Q∨ −→ OY , so its cokernel is precisely the sheaf (p∗F ) |Z(s)= i∗F . One canshow that the order stages of the sequence remain indeed exact, so we get locally freeresolution

0 −→ p∗F⊗∧rQ∨ −→ p∗F⊗∧r−1Q∨ −→ ... −→ p∗F⊗Q∨ −→ p∗F −→ i∗F −→ 0

on Y . (This resolution is called the Koszul complex). So what we have to check is thatr∑

k=0

(−1)kch(p∗F ⊗ ∧kQ∨) = i∗ch(F )

td(i∗Q).

But note that

i∗ch(F )

td(i∗Q)=ch(p∗F )

td(Q).i∗[X] =

ch(p∗F )cr(Q)

td(Q)

30

by the projection formula and proposition 1.2.21. So by the additivity of Chern charac-ters it suffices to prove that

r∑k=0

(−1)kch(∧kQ∨) =cr(Q)

td(Q).

But this is easily done: if α1, ..., αr are the Chern roots of Q then the left hand side is

r∑k=0

(−1)k∑

i1<...<ik

exp(−αi1−...−αik) =r∏i=1

(1−exp(−αi)) = α1...αr.

r∏i=1

1− exp(−αi)αi

,

which equals the right hand side. It is in fact this formal identity that explains theappearance of Todd classes in the Hirzebruch-Riemann-Roch theorem.

Using this computation of this special example we can now give the general proofof the Hirzebruch-Riemann-Roch theorem.

Proof. (of proposition 1.2.30) We want to reduce the proof to the special case consid-ered in example 1.2.31 .

Let i : X −→ Y be any inclusion morphism of smooth projective varieties. Wedenote by M be the blow-up of Y × P1 in X × 0. The smooth projective schemeM comes together with a projection morphism q : M −→ P1. Its fibers q−1(P ) forP 6= 0 are all isomorphic to Y . The fiber q−1(0) however is reducible with two smoothcomponents: one of them (the exceptional hypersurface of the blow-up) is the projec-tivized normal bundle of X × 0 in Y × P1, and the other one is simply the blow-upY of Y in X . We are particularly interested in the first component. As the normal bun-dle of X × 0 in Y × P1 is NX/Y ⊗ OX this component is just the projective bundleP := P(NX/Y ⊕ OX) on X . Note that there is an inclusion of the space X × P1 in Mthat corresponds to the given inclusion X ⊂ Y in the fibers q−1(P ) for P 6= 0, and tothe ”zero section inclusion” X ⊂ P(NX/Y ⊕OX) = P as in example 1.2.31 in the fiberq−1(0).

The idea of the proof is now simply the following: we have to prove an equalityin the Chow groups, i.e. modulo rational equivalence. The fibers q−1(0) and q−1(∞)are rationally equivalent as they are the zero resp. pole locus of a rational function onthe base P1, so they are effectively ”the same” for intersection-theoretic purposes. Butexample 1.2.31 shows that the proposition is true in the fiber q−1(0), so it should be truein the fiber q−1(∞) as well.

To be more precise, let F be a sheaf on X as in the proposition. Denote by pX :X × P1 −→ X the projection, and by iX : X × P1 −→ M the inclusion discussed

31

above. Then iX∗p∗XF is a coherent sheaf on M that can be thought of as ”the sheaf F

on X in every fiber of q”. By lemma 1.2.28 we can choose a locally free resolution

0 −→ Fr −→ Fr−1 −→ ... −→ F0 −→ iX∗p∗XF −→ 0 (1)

on M .Note that the divisor [0]− [∞] on P1 is equivalent to zero. So it follows that

r∑k=0

(−1)kch(Fi).q∗([0]− [∞]) = 0

in A∗(M) ⊗ Q. Now by definition of the pull-back we have q∗[0] = [Y ] + [P ] andq∗[∞] = [Y ], so we get the equalityr∑

k=0

(−1)kch(Fi|Y ).[Y ]+r∑

k=0

(−1)kch(Fi|P ).[P ] =r∑

k=0

(−1)kch(Fi|Y ).[Y ] (2)

in A∗(M)⊗Q. But note that the restriction to Y of the sheaf iX∗p∗XF in (1) is the zero

sheaf as X × P1 ∩ Y = ∅ in M . So the sequence

0 −→ Fr|Y −→ ... −→ F1|Y −→ F0|Y −→ 0

is exact, which means that the first sum in (2) vanishes. The second sum in (2) is

preciselych(F )

td(NX/Y ).[X] by example 1.2.31. So we conclude that

r∑k=0

(−1)kch(Fi|Y ).[Y ] =ch(F )

td(NX/Y ).[X]

in A∗(M)⊗Q. Pushing this relation forward by the (proper) projection morphism fromM to Y then gives the desired equation.

This completes the proof of the Hirzebruch-Riemann-Roch theorem 1.2.24.

Remark 10. Combining proposition 1.2.30 with remark 9 we see that we have justproven the following statement: let f : X −→ Y be a closed immersion of smoothprojective schemes, and let F be a coherent sheaf on X . Then there is a locally freeresolution

0 −→ Fr −→ Fr−1 −→ ... −→ F0 −→ f∗F −→ 0

of the coherent sheaf f∗F on Y such thatr∑

k=0

(−1)kch(Fk).td(TY ) = f∗(ch(F ).td(TX)) ∈ A∗(Y )⊗Q.

32

This is often written as

ch(f∗F ).td(TY ) = f∗(ch(F ).td(TX))

In other words, ”the push-forward f∗ commutes with the operator τ of example 1.2.27(iii)”.

It is the statement of the Grothendieck-Riemann-Roch theorem that this relation isactually true for any proper morphism f of smooth projective schemes (and not just forclosed immersions).

1.2.6 Parabolic bundlesA quasi-parabolic bundle (E, (fp))p∈I) on X with quasi-parabolic structure at I is afiltration fp of locally free sheaves

E = E(p,1) ⊃ E(p,2) ⊃ ... ⊃ E(p,lp) ⊃ E(p,lp+1) = E(−p),

for all p ∈ I .The positive integers ni(p) = deg(E(p,i)/E(p,i+1)) are called multiplicities of (E, (Fp)p∈I)

at p and lp is the length of the filtration fp. Let ri(p) denotei∑

j=1

ni(p).

LetEq G be a quotient bundle of E. Then a quasi-parabolic structure onE induces

a quasi-parabolic structure onG: let hp,i be the injectionE(p,i) → E and denoteG(p,i) =Im(qhp,i). Then the quotient morphism induces a filtration at each parabolic point

G = G(p,1) ⊇ G(p,2) ⊇ ... ⊃ G(p,lp) ⊇ G(p,lp+1) = G(−p).

By considering the distinct locally free sheaves of each filtration, this defines a quasi-parabolic structure on G. It is induced by the one on E in the sense that the morphismq is naturally compatible with the filtrations. Dually, there is a natural induced quasi-

parabolic structure on a subbundle Hj→ E: if π = coker(j), then it is obtained by

letting H(p,i) = ker(πhp,i). In other words, H(p,i) = H ∩ E(p,i).

We denote by (V ′′, (fpV ′′)p∈I) (respectively, (V ′, (fV′

p )p∈I)) the quasi-parabolic struc-ture induced by (E, (fp)p∈I) on a quotient bundle V ′′ (respectively, a subbundle V ′).

A parabolic bundle E∗ on X is a quasi-parabolic bundle with, for all p ∈ I , asequence of real numbers

0 ≤ α1(p) ≤ α2(p) ≤ ... ≤ αlp(p) ≤ 1,

33

attached to the filtration at p. These numbers are called parabolic weight. It is con-venient to introduce Simpson definition [Si] of a parabolic bundle as a filtration as afiltered vector bundle. In the notations of [MY3], a parabolic bundle is

- for all α ∈ R, a locally free sheaf Eα on X and an isomorphism

jα : Eα(−∑p∈I

p)∼−→ Eα+1,

- for all α, β ∈ R such that α ≥ β, an injective morphism iα,βE∗ : Eα → Eβ , such that thediagram

Eα+1 i.α,α+1E∗ //

jα

Eα

id

Eα(−

∑p∈I p)

// Eα

commutes,

a sequence of real numbers 0 ≤ α1 < α2 < ... < αL < 1, such that i.α,αiE∗is an

isomorphism Eα ∼= Eαi for all α ∈]αi−1, αi].

As a convention , for a parabolic bundle E∗ and α ∈ R+, the sheaf E = E0 is calledthe underling vector bundle and the morphisms of the parabolic structure will be de-noted by iαE := iα,0E∗

and παE := coker(iαE).

Let E∗ and F∗ be two parabolic bundle on X , with parabolic structure at I . Amorphism ϕ : E −→ F is parabolic if, for all α ∈ R+, the composition παFϕi

αE is the

zero morphism. This produces a morphism

0 −→ EαiαE−→ E −→ E/Eα −→ 0

0 −→ Fα −→ FπαF−→ F/Fα −→ 0

that will be denoted ϕα : Eα → Fα. The notation ϕ∗ : E∗ → F∗, means that ϕ isparabolic.

Consider the sheaf defined by

Hom(E∗, F∗)(U) = ϕU∗ : E∗|U −→ F∗|U.

By definition of parabolic morphism, it is a subsheaf of Hom(E,F ) and for all opensubset U ⊂ X , such that I ∩U = ∅, it actually isHom(E∗, F∗)(U) = Hom(E,F )(U).

34

Thus the quotient sheaf is a torsion sheaf with support at I , that can be described, by[B-H], lemma 2.4, in terms of the parabolic structures of E∗ and F∗: suppose for sim-plicity that I = p and let (α1, ..., αl) be the weights, nE∗i = deg(Eαi/Eαi+1

) themultiplicities of E∗, (β1, ..., βh) the weights, nF∗j = deg(Fβj/Fβj+1

) the multiplicities ofF∗. Then there is a short exact sequence

0 −→ Hom(E∗, F∗) −→ Hom(E,F ) −→ τE∗,F∗ −→ 0, (1.2.1)

where τE∗,F∗ is a torsion sheaf supported at p of degree

h0(X, τE∗,F∗) =∑i,j

αi>βj

nE∗i nF∗j .

This is a consequence of the fact that a morphism ϕ : E −→ F is parabolic, if and onlyif the linear map over the parabolic point

ϕp = (ϕi,j) :⊕i

Eαi/Eαi+1−→

⊕j

Fβj/Fβj+1

is such that ϕi,j = 0 for all αi > βj . Hence the fiber of τE∗,F∗ at p is isomorphic to⊕αi>βj

(Eαi/Eαi+1)∨ ⊗ Fβi/Fβi+1

.

For a general parabolic subset I , the degree of the torsion sheaf τE∗,F∗ can be computedas

h0(X, τE∗,F∗) =∑p∈I

∑i,j

αi(p)>βj(p)

nE∗i (p)nF∗j (p).

Let χ(E∗, F∗) denote χ(Hom(E∗, F∗)). By Riemann-Roch formula and the exact se-quence (1.2.1), this Euler characteristic can be computed as

χ(E∗, F∗) = rk(E)deg(F )− rk(F )deg(E) + rk(E)rk(F )(1− g)− h0(τE∗,F∗).

The group of global sections H0(Hom(E∗, F∗)) is the group of parabolic morphisms,Hom(E∗, F∗). The first cohomology group H1(Hom(E∗, F∗)) is, by [Y] lemma 1.4,isomorphic to the group of isomorphim classes of parabolic extension of F∗ byE∗ and isdenoted by Ext1(E∗, F∗). By definition a parabolic extension is a short exact sequence

0 −→ E ′∗i∗−→ E∗

p∗−→ E ′′∗ −→ 0,

two parabolic extensions being isomorphic, if there is a parabolic isomorphism of ex-tensions.

35

Recall the definitions of the parabolic invariants of E∗. Let d = deg(E) and r =rk(E); the parabolic degree of E∗ is defined as the real number

deg(E∗) = deg(E) +∑p∈I

lp∑i=1

ni(p)αi(p)

that can also computed as the integral∫ 1

0

deg(Eα)dα + r|I| =∫ 0

−1

deg(Eα)dα.

The parabolic Hilbert polynomial is

P (E(m)∗) = deg(E(m)∗) + r(1− g) = deg(E∗) + r(m+ 1− g)

and the parabolic slope is defined as µ(E∗) =deg(E∗r

).

Let E∗ be a parabolic bundle and E G a quotient vector bundle. Consider theparabolic structure obtained by the induced quasi-parabolic structure, weighted as theone of E∗. This parabolic structure is said to be the one induced on G by E∗ and will bedenoted by G∗. Dually, all subbundle H → E has an induced parabolic structure, thatwill be denoted by H∗. Recall the notations of the induced quasi-parabolic structure.For all p ∈ I and i = 1, ..., lp, consider the integers n′′i (p) = deg(G(i)/G(i+1). Theyverify 0 ≤ n′′i (p) ≤ ni(p) and n′′1(p) + ...+ n′′lp(p) = rk(G). Then we can easily checkthe equality

deg(G∗) = deg(G) +∑p∈I

lp∑i=1

n′′i (p)αi(p).

Remark 11. For all parabolic structureG∗′ (respectively,H∗′), such thatE∗ G∗′ (re-spectively, H∗′ → E∗) is parabolic, it is deg(G∗) ≤ deg(G∗′) (respectively, deg(H∗) ≥deg(H∗′)).

Definition 1.2.32. A parabolic bundle E∗ is semistable if, for all quotient bundle E G it is µ(E∗) ≤ µ(G∗). A semistable bundle is stable if the inequality is strict, wheneverG is a nontrivial quotient of E.

1.2.7 Sections of the line bundle Lpar

LetMpar denote the moduli space of parabolic bundles onX of rank r, trinvial determi-nant and parabolic structure at the points I of multiplicities ((n1(p), ..., nlp(p))p∈I) andweights

0 ≤ α1(p) = 0 < α2(p) < α3(p) < ... < αlp(p) < 1.

36

Let ((d1(p), ..., dlp−1(p))p∈I , k) be strictly positive integers such that, for all j =

2, ..., lp the weights at p can be written as αj(p) =1

k

j−1∑h=1

dh(p).

A family E∗ of parabolic bundles at I , of rank r, trivial determinant, multiplicities((ni(p))P∈I) and weights ((di(p))P∈I , k) parametrized by a scheme S is a vector bundleE over X × S of rank r, such that det(E) = OX×S and, for all p ∈ I , quotient bundlesQi(p) of E|p×S , of rank ri(p) = n1(p) + ...+ ni(p), such that, by letting Ki(p) denotethe kernel

0 −→ Ki(p) = Ker(πi(p)) −→ E|p×Sπi(p)−→ Qi(p) −→ 0,

then Ki(p) ⊂ Ki−1(p) for all i = 1, ..., lp − 1. The family is parabolic in the sense that,for all s ∈ S the vector bundle Es, has the quasi-parabolic structure

Esp ⊃ K1(p)s ⊃ ... ⊃ Kl−1(p)s ⊃ 0

and weights ((di(p))P∈I , k). Actually the family E has a weighted filtration, induced bythe quotient Qi(p), that is

E = E0 ⊃ Eδ1(p) ⊃ ... ⊃ Eδ1(p)+...+δlp−1(p) ⊃ E1 = E ⊗ π∗XOX(−p),

where δj(p) =dj(p)

kandQj(p) ∼= E/Eδ1(p)+...+δj(p). Suppose that the family is semistable

and let φS : S −→Mpar the modular morphism. Suppose that1

r

∑p∈I

∑i>j

ni(p)dj(p) ∈ Z and let Lpar(E∗) be the line bundle on S defined as the

tensor product

Lpar(E∗) = (detRπSE)⊗k) ⊗⊗p∈I

⊗j

(detQj(p))⊗dj(p) ⊗ (detE|q×S)⊗e.

Here e is an integer depending on the parabolic structure defined by

e =1

r

∑p∈I

∑i>j

ni(p)dj(p) + k(1− g)−∑p∈I

∑j

dj(p).

Theorem 1.2.33. [N-Ra], theorem 1, for rank 2, [Pa1], theorem 3.3, for arbitrary rank)There exists a unique ample line bundle Lpar over Mpar such that, for all semistableparabolic family E∗ parametrized by S, it is φ∗SLpar = Lpar(E∗).

In the case of rank two parabolic bundles, Pauly gives in [Pa2] a method to producesections of Lpar of type theta. In what follows, we extend his method to the rank r case

37

and produce sections of Lpar⊗h, for all h ∈ N.

Let E∗,F∗ be families of parabolic bundles on I , parametrized by S, of quotientsand flags respectively

0 −→ Eαiα−→ E pα−→ Qα −→ 0 0 −→ Ki(p) −→ E|p×S −→ Qi(p) −→ 0,

0 −→ Fαi′α−→ E p′α−→ Q′α −→ 0 0 −→ K′i(p) −→ F|p×S −→ Q′i(p) −→ 0.

A morphism ϕ : E −→ F of vector bundles is parabolic if, for all α ∈ R+, the compo-sition p′αϕiα is the zero morphism. The sheaf of parabolic homomorphisms is a locallyfree subsheaf of Hom(E ,F), that will be denoted by Hom(E∗,F∗). The quotient sheafcoker (H(E∗,F∗) → Hom(E ,F)) is a family of torsion sheaves parametrized by Swhose support is contained in the parabolic subset I , that we denote by TE∗,F∗ .

The sheaf Hom(E∗,F∗), is the family on S parametrizing the sheaves of parabolicmorphisms between bundles of the families, that is for all s ∈ S there is natural isomor-phism

Hom(E∗s,F∗s) ∼= Hom(E∗,F∗)s.let F be a vector bundle on X of rank hk such that

deg(F ) =h

r

∑p∈I

∑i>j

ni(p)dj(p) + hk(g − 1)

and let F∗ be a parabolic structure at I of multiplicities

((hd1(p), ..., hdlp−1(p), hk − h∑j

dj(p))p∈I)

and weights ((dj(p))p∈I , k). Let π∗XF∗ be the constant family of parabolic bundles ofvalue F∗, parametrized by S. For s ∈ S, let E∗s denote the parabolic bundle of E∗ overs. Then it is

χ(E∗s, F∗) = rhk(µ(E∗s) + (g−1))−hkdeg(Es) + rhk(1− g)−∑p∈I

∑i>j

nE∗si (p)nF∗j (p)

= h∑p∈I

∑i>j

ni(p)dj(p)−∑p∈I

∑i>j

ni(p)hdj(p) = 0.

Fix a basis of Fp =⊕j

Fp,j/Fp,j+1 and let TE∗,F∗ denote the family of torsion sheaves

of the short exact sequence of parabolic morphisms

0 −→ Hom(E∗, π∗XF∗) −→ Hom(E , π∗XF )p−→ TE∗,F∗ −→ 0 (1.2.2)

38

Lemma 1.2.34. With the notations above, it is

detRπSHom(E∗, π∗XF∗) ∼= Lpar(E∗)⊗h.

Proof. by the short exact sequence (1.2.2) there is a natural isomorphism

detRπSHom(E∗, π∗XF∗) ∼= (detRπSHom(E , π∗XF ))⊗ (detRπSTE∗,F∗)∨.

By Serre duality theorem, [Pa2], lemma 3.4, there is an isomorphism

detRπSHom(E , π∗XF∗) = detRπSE∨ ⊗ π∗XF ∼= detRπSE ⊗ π∗X(F∨ ⊗KX).

The vector bundle detE|q×S is independent of q ∈ X by [Pa2], lemma 3.5, it followsthat

detRπSE ⊗ π∗X(F∨ ⊗KX) ∼= (detRπSE)⊗hk ⊗ (detE|q×S)⊗−deg(F∨⊗KX).

Now, since the degree of F∨ ⊗KX can be computed as

deg (F∨ ⊗KX) = hkdeg (KX)− deg(F ) = hk(g − 1)− h

r

∑p∈I

∑i>j

ni(p)dj(p),

the first determinant bundle is isomorphic to

detRπSHom(E , π∗XF ) ∼= (detRπSE)⊗hk ⊗ (detE|q×S)⊗hr

∑p∈I

∑i>j ni(p)dj(p)+hk(1−g).

The sheaf TE∗,F∗ is a family of skyscraper sheaves supported at I , hence the sheafR1πS∗TE∗F∗ is zero and there is an isomorphism

πS∗TE∗F∗ ∼=⊕p∈I

lp−1⊕j=1

Kj(p)∨ ⊗Ohdj(p)S .

Thus the second determinant can be computed as follows

(detRπS∗TE∗F∗)∨ ∼= det⊕p∈I

lp−1⊕j=1

Kj(p)∨ ⊗Ohdj(p)S

∼=⊗p∈I

lp−1⊗j=1

det(Kj(p)∨ ⊗Ohdj(p)S ) ∼=

⊗p∈I

lp−1⊗j=1

(detKj(p)∨)⊗hdj(p).

By definition, it is Ki(p) = Ker(πi(p)) and this yields the isomorphism

⊗p∈I

lp−1⊗j=1

(detKj(p)∨)⊗hdj(p) ∼=⊗p∈I

lp−1⊗j=1

(detQj(p))⊗hdj(p) ⊗ (detE∨|p×S)⊗hdj(p).

39

The lemma then follows from the fact that, for all p ∈ I there is a natural isomorphismdetE|q×S ∼= detE|p×S , and the equality

he =h

r

∑p∈I

∑i>j

ni(p)dj(p) + hk(1− g)− h∑p∈I

∑j

dj(p).

Let F be a quasi-coherent OX×S-module, flat over S. Recall how one obtains acomplex that is quasi-isomorphic toRπSF . By the relative version of Segree A theorem,is an integer m0 such that, if m ≥ m0, the natural evaluation morphism

K0 = π∗XOX(−m)⊗ π∗XπX∗F(m)q F

is surjective. Since deg (OX(−m)) < 0, it is πS∗K0 = 0 and if we denote by K1 =kerq it is πS∗K1 = 0 as well. Moreover the higher direct image sheaves L1 = R1πS∗K1,L0 = R1πS∗K0 are locally free. Hence the short exact sequence

0 −→ K1i−→ K0

q−→ F −→ 0

yields the long exact sequence in cohomology

0 −→ πS∗F −→ R1πS∗K1R1πS∗i−→ R1πS∗K0 −→ R1πS∗F −→ 0

and there is a natural isomorphism detRπSF ∼= detL0 ⊗ detL∨1 .

Let 0 −→ L v−→ L0 −→ 0 be a complex of locally free sheaves on X × S, quasi-isomorphic to RπSHom(E∗, π∗XF∗). The hypothesis on the Euler characteristicX (E∗s, F∗) = 0, for all s ∈ S, is equivalent to the assumption that the locally

free sheaves Li have same rank and the morphism of vector bundles detυ : detL1 −→detL0 defines a section of (detL1)∨⊗detL0 = detRπSHom(E∗, π∗XF∗) ∼= Lpar(E∗)⊗h,that we denote by θE∗F∗ . This section is zero at a point s ∈ S, if and only if

dim(Hom(E∗s, F∗)) = dim(Ext1(E∗s, F∗)) 6= 0.

To show that this produces a section of the line bundle on the moduli spaces Mpar,recall its construction (see [Pa1], theorem 2.3). Let Q be the scheme of quotients ofrank r and trivial determinant ofOP (n)

X (−n), where P is the Hilbert polynomial of suchquotients and n is an integer, n 0. Let Ω denote the open subset of Q of locally freequotients, F the universal family of quotients on X×Ω and Fp the flag varieties bundleof multiplicities (n1(p), ..., nlp(p))

Fp = F lag(n1(p),...,nlp (p))(F|p×Ω)π(p) Ω.

40

Let Qi(p) denote the universal quotients on Fp and let R be the fibred product of theF,ps, for p ∈ I , over Ω. We still denote by F∗ and Qi(p) the universal families ob-tained by pullback to R. The parabolic family F∗, with parabolic quotient (Qi(p)), islocally a universal family of parabolic bundles. LetRss be the open subscheme ofR ofsemistable parabolic bundles. The MPar is obtained as the good quotient ofRss for thenatural action of SL(P (n)).

Consider the line bundle LPar(F∗) on Rss. By [Pa1], theorem 3.3 it descends tothe moduli space MPar. The section θF∗F∗ is SL(P (n))-invariant, thus it descends to asection of Lpar⊗h, that will be called parabolic theta function (of order h) associatedwith the parabolic bundle F∗.

1.3 Blowing upLet us consider the projection map pa : Pn(K)\a −→ Pn−1(K). If n > 1 it isimpossible to extend it to the point a. However, we may try to find another projectiveset X which contains an open subset isomorphic to Pn(K)\a such that the map paextends to a regular map pa : X −→ Pn−1(K). The easiest way to do it is to con-sider the graph Γ ⊂ Pn(K)\a × Pn−1(K) of the map pa and take for X its closurein Pn(K) × Pn−1(K). The second projection map X −→ Pn−1(K). will solve ourproblem. It is easy to find the bi-homogeneous equations defining X . For simplic-ity we may assume that a = (1, 0, ..., 0) so that the map pa is given by the formula(x0, ..., xn) −→ (x1, ..., xn). Let Z0, ..., Zn be projective coordinates in Pn−1(K). Ob-viously, the graph Γ is contained in the closed set X defined by the equations

ZiTj − ZjTi = 0, i, j = 1, ..., n. (1.3.1)

The projection q : X −→ Pn−1(K) has the fibre over a point t = (t1, ..., tn) equal to thelinear subspace of Pn(K) defined by the equations

Zitj − Zjti = 0, i, j = 1, ..., n. (1.3.2)

Assume that ti = 1. Then the matrix of coefficients of the system of linear equations(1.3.1) contains n−1 unit columns so that its rank is equal to n−1. This shows that thefiber q−1(t) is isomorphic, under the first projection X −→ Pn(K), to the line spannedby the points (0, t1, ..., tn) and (1, 0, ..., 0). On the other hand the first projection is anisomorphism over Pn(K)\0. Since X is irreducible (all fibres of q are of the samedimension), we obtain that X is equal to the closure of Γ. By plugging z1 = ...zn inequations (1.3.1) we see that the fibre of p over the point a = (1, 0, ..., 0) is isomorphicto the projective Pn−1(K). Under the map q this fibre is mapped isomorphically toPn−1(K).

41

The pre-image of the subset Pn(K)\V (Z0) ∼= An(K) under the map p is isomorphicto the closed subvariety B of An(K)× Pn−1(K). The restriction of the map p to B is aregular map σ : B −→ An(K) satisfying the following properties

(i) σ|σ−1(An(K)\(0, ..., 0)) −→ An(K)\(0, ..., 0) is an isomorphism;(ii) σ−1(0, ..., 0) ∼= Pn−1(K).

We express this by saying that σ ”blows up” the origin. Of course if we take n = 1nothing happens. The algebraic set B is isomorphic to An(K). But if take n = 2, thenB is equal to the closed subset of A2(K)× P1(K) defined by the equation

Z2T0 − T1Z1 = 0.

It is equal to the union of two affine algebraic sets V0 and V1 defined by the conditionT0 6= 0 and T1 6= 0, respectively. We have

V0 = V (Z2 −XZ1) ⊂ A2(K)× P1(K)0, X = T1/T0,

V1 = V (Z2Y − Z1) ⊂ A2(K)× P1(K)1, Y = T0/T1,

If L : Z2 − tZ1 = 0 is the line in A2(K) through the origin ” with slope ” t, then thepre-image of this line under the projection σ : B −→ A2(K) consists of the union oftwo curves, the fibre E ∼= P1(K) over the origin, and the curve L isomorphic to L underσ. the curves L intersects E at the point ((0, 0), (1, t)) ∈ V0. The pre-image of each lineL with the equation tZ2 − Z1 consists of E and the curve intersecting E at the point((0, 0), (t, 1)) ∈ V1. Thus the points of E can be thought as the set of slopes of thelines through (0, 0). The ” infinite slope ” corresponding to the line Z1 = 0 is the point(0, 1) ∈ V1 ∩ E.

Let I be an ideal in a commutative ring A. Each power In of I is a A-module andInIr ⊂ In+r for every n, r ≥ 0. This shows that the multiplication maps In × In −→In+r define a ring structure on the direct sum of A-modules

A(I) =⊕n≥0

In.

Moreover, it makes this ring a graded algebra over A = A(I)0 = I0. Its homogeneouselements of degree n are elements of In.

Assume now that I is generated by a finite set f0, ..., fn of elements of A. Considerthe surjective homomorphism of graded A-algebras

φ : A[T0, ..., Tn] −→ A(I)

defined by sending Ti to fi. The kernel Ker(φ) is a homogeneous ideal in A[T0, ..., Tn].If we additionally assume that A is a finitely generated algebra over a field k, we can

42

interpret Ker(φ) as the ideal defining a closed subset in the product X × Pnk where Xia an affine algebraic variety with O(X) ∼= A. Let Y be the subvariety of X defined bythe ideal I .

Definition 1.3.1. The subvariety of X × Pnk defined by the ideal Ker(φ) is denoted byBY (X) and is called the blow-up of X along Y . The morphism σ : BY (X) −→ Xdefined by the projection X × Pnk −→ X is called the monoidal transformation or theσ-process or the blowing-up morphism along Y.

Let us fix an algebraically closed field K containing k and describe the algebraicset BY (X)(K) as a subset of X(K)× Pn(K). Let Ui = X × (Pn(K))i and BY (X)i =BY (X) ∩ Ui. This is an affine algebraic k-set with

O(BY (X)i) ∼= O(X)[T0/Ti, ..., Tn/Ti]/Ker(φ)i

where Ker(φ)i is obtained from the ideal Ker(φ) by dehomogenization with respect tothe variable Ti. The fact that the isomorphism classBY (X) is independent of the choiceof generators f0, ..., fn follows from following

Lemma 1.3.2. Let Y ⊂ X×Pnk(K) and Y ′ ⊂ X×Prk(K) be two closed subsets definedby homogeneous ideals I ⊂ O(X)[T0, ..., Tn] and J ⊂ O(X)[T ′0, ..., T

′r], respectively.

Let p : Y −→ X and p′ : Y ′ −→ X be the regular maps induced by the first projectionsX × Pnk(K) −→ X and X × Prk(K) −→ X . Assume that there is an isomorphism ofgraded O(X)-algebras ψ : O(X)[T ′0, ..., T

′r]/I

′ −→ O(X)[T0, ..., Tn]/I . Then thereexist an isomorphism f : Y −→ Y ′ such that p = p′ f .

Lemma 1.3.3. Let U = D(f) ⊂ X be a principal affine open subset of an affine set X ,then

BY ∩U ∼= σ−1(U).

Proposition 1.3.4. The blow-up σ : BY (X) −→ X induces an isomorphism

σ−1(X\Y ) ∼= X\Y.

1.3.1 Elementary transformation of algebraic bundlesIntroduction 1.3.5. Let E be a vector bundle on a locally noetherian scheme X and Fa vector bundle on an effective Cartier divisor Y ofX . If there is a surjective homomor-phism ψ : E −→ F of OX-modules, then E ′ = ker(ψ) is a vector bundle on X . Theprocedure to obtainE ′ fromE is called the (sheaf theoretic) elementary transformationof E along F and we denote it by E ′ = elmF (E). E ′ is said to be the elementary trans-form ofE along F . For the given ψ : E −→ F , we have the exact commutative diagramwhich is called the display of the elementary transformation. See [MY2], page 1. where

43

F ′ is the kernel of ψ|Y and E(−Y ) = E ⊗OX OX(−Y ). The leftmost vertical exactsequence gives us the inverse of the given transformation, that is, E(−Y ) = elmF ′(E

′).P(F ) and P(F ′) are closed subschemes of P(E) and P(E ′), respectively, because ψ

and ψ′ are surjective. We have a geometric interpretation of the above operation.

Theorem 1.3.6. Let f : W −→ P(E) (or, g : W ′ −→ P(E ′)) be the blowing-up alongP(F ) (or, P(F ′), resp.). Then we have an isomorphism ξ : W −→ W ′ of X-schemes.For the projection p : P(E) −→ X (or, p′ : P(E ′) −→ X), the proper transformf−1[p−1(Y )] (or, g−1[p′−1(Y )], rep.) is equal to the exceptional divisor of g (or, f ,resp.).

By using the above theorem we define a (geometric) elementary transformation as fol-lows.

Definition 1.3.7. The birational map gξ−1f−1 : P(E) −→ P(E ′) is called the ele-mentary transformation of P(E) along P(F ) and is denoted by elmP(F ). Obviouslyelm−1

P(F ) = elmP(F ′).

The elementary transformation suppplies us with a strong tool to construct manyvector bundles on quasi-projective varieties. Moreover, it is sometimes easy to study thestructure of the constructed bundle. For example, we have the following.

Theorem 1.3.8. Let (X,OX(1)) be a couple of a non-singular projective varietyX overan algebraically closed field and an ample line bundle OX(1) on X . Assume that dimX ≥ 2. LetD be a divisor onX and r an integer which is not smaller than dimX . Then,for every integer s, there is µ-stable vector bundle E such that r(E) = r, c(E) = D(rational equivalence) and d(c2(E),OX(1)) ≥ s, where d(∗,OX(1)) is the degree withrespect to OX(1).

In lower dimensional cases, all the vector bundles on a non-singular quasi-projectivevariety can be constructed by using elementary transformations.

Theorem 1.3.9. Let X be a non-singular quasi-projective variety over an algebraicallyclosed field. Assume that dimX ≤ 3. Then, for a given vector bundle E on X , there area line bundle L on X , a smooth effective Cartier divisor Y on X and a vector bundle Fon Y with r(F ) = r(E)− 1 such that E ⊗OX L ' elmF (O⊕r(E)

X ).

In other words, all the projective space bundles on X are obtained from the trivialbundle by elementary transformations.

44

1.3.2 Generalization of elementary transformationletE be a coherent sheaf on a locally noetherian schemeX and F a coherent sheaf on aneffective Cartier divisor Y Of X . Assume that there is a surjective OX-homomorphismψ : E −→ F . We shall denote E ′ = ker(ψ) by elmF (E) as in Introduction and wedefine F ′ and ψ′ as in the diagram (1.3.5). Let I be the defining ideal of Y . then, asOX-modules, I is isomorphic to OX(−Y ). For these data, we assume the following:

(∗) Supp(F ) = Y . For all positive integers m, the natural maps I ⊗OX Sm(E) −→ISm(E) and I ⊗OX Sm(E ′) −→ ISm(E ′) are injective.

The surjection ψ induces a graded surjective homomorphism S(ψ) of the symmetricalgebra S(E) to another symmetric algebra S(F ). The image K of E ′ ⊗OX S(E) toS(E) is contained in Ker(S(ψ)).

(∗∗) P(F )′ is the closed subscheme of P(E) = Proj(S(E)) defined by the ideal K.Similarly we define a closed subscheme P(F ′)′ of P(E ′) = Proj(S(E ′)) for the coupleE ′ and F ′

P(F ) itself is a subscheme of P(E). Note that P(F ) is a subscheme of P(F )′ and assubset of P(E),P(F ) coincides with P(F )′.

Theorem 1.3.10. Let ψ : E −→ F be the surjection in the above and let E ′, F ′, Y andI be as above. Assume that (∗) is satisfied. If f : W −→ P(E) (or, g : W ′ −→ P(E ′))is the blowing-up of P(E) (or, P(E ′), resp.) along P(F )′ (or, P(F ′)′, resp.), then thereis an isomorphism ξ : W −→ W ′ of X-schemes. For the projection p : P(E) −→ X(or, p′ : P(E ′) −→ X) the proper transform f−1[p−1(Y )] (or, g−1[p′−1(Y )], resp.) ofthe Cartier divisor p−1(Y ) (or, p′−1(Y ), resp.) is equal to the exceptional divisor of g(or, f, resp.).

Definition 1.3.11. Under the situation of theorem 1.3.10 the birational map gξ−1f−1 :P(E) −→ P(E ′) is called an elementary transformation along P(F )′, which is denotedby elmP(F )′ . P(E ′) is said to be the elementary transform of P(E) along P(F )′ and wedenoted P(E ′) = elmP(F )′P(E).

If bothE andE ′ are locally free and F is supported by an effective Cartier divisor Yin X , the condition (∗∗) is obviously satisfied. We have a slightly stronger result whenX is a regular scheme.

Proposition 1.3.12. Let X be a regular scheme and E a coherent torsion free sheafon X . Assume that E satisfies the Serre’s condition (Sn) and that the homologicaldimension of Ex is not greater than one at every closed point x of X . If r(E) + n ≥dimk(x)E(x) at every closed point x of X then Sm(E) is torsion free for every m > 0.

45

To define elementary transformation we used the closed subscheme P(F )′ of P(E).If E is a vector bundle on X and F is a vector bundle on the Cartier divisor Y , then itis obvious that P(F )′ coincides with P(F ) as subschemes of P(E). When is this true ingeneral or when is E ′ ⊗OX S(E) the kernel of the natural surjection of S(E) to S(F )

Proposition 1.3.13. Let X be a locally noetherian scheme, Y an effective Cartier divi-sor on X and F a coherent OY -module. Assume that Y has no embedded components,F is locally free on a dense open set of Y and Ass(F ) = Ass(OY ). If there is a sur-jection ψ of a locally free coherent OX-module E to F . Then the kernel of the naturalsurjection of S(E) to S(F ) coincides with E ′ ⊗OX S(E), where E ′ is the kernel of ψ.

46

Chapter 2

Calculating the parabolic Cherncharacter of a locally abelian parabolicbundle

In this paragraph, we calculate the parabolic Chern character of a bundle with locallyabelian parabolic structure on a smooth strict normal crossings divisor, using the def-inition in terms of Deligne-Mumford stacks. We obtain explicit formulas for ch1, ch2

and ch3, and verify that these correspond to the formulas given by Borne for ch1 andMochizuki for ch2.

Let X be a smooth projective variety with a strict normal crossings divisor D =D1 + . . . + Dn ⊂ X . The aim of this paper is to give an explicit formula for theparabolic Chern character of a locally abelian parabolic bundle on (X,D) in terms of:—the Chern character of the underlying usual vector bundle,—the divisor components Di in the rational Chow groups of X ,—the Chern characters of the associated-graded pieces of the parabolic filtration alongthe multiple intersections of the divisor components, and—the parabolic weights.

After giving a general formula, we compute explicitly the parabolic first, second,third parabolic Chern characters chPar1 (E), chPar2 (E) and chPar3 (E).

The basic idea is to use the formula given in [IS2]. However, their formula didnot make clear the contributions of the different elements listed above. In order toadequately treat this question, we start with a somewhat more general framework ofunweighted quasi-parabolic sheaves [Se]. These are like parabolic sheaves except thatthe real parabolic weights are not specified. Instead, we consider linearly ordered setsΣi indexing the parabolic filtrations over the components Di. Let Σ′i denote the linearlyordered set of links or adjacent pairs in Σi. We also call these “risers” as Σ can bethought of as a set of steps. The parabolic weights are then considered as functionsαi : Σ′i → (−1, 0] ⊆ R. This division allows us to consider separately some Chern

class calculations for the unweighted structures, and then the calculation of the parabolicChern character using the parabolic weights.

A further difficulty stems from the fact that there are classically two different waysto give a parabolic structure: either as a collection of sheaves included in one another; orby fixing a bundle E (typically the zero-weight sheaf) plus a collection of filtrations ofE|Di . The formula of [IS2] is expressed in terms of the collection of sheaves, whereaswe look for a formula involving the filtrations. Thus, our first task is to investigate therelationship between these two points of view.

An important axiom concerning the parabolic structures considered here, is thatthey should be locally abelian. This means that they should locally be direct sums ofparabolic line bundles. It is a condition on the simultaneous intersection of three or morefiltrations; up to points where only two divisor components intersect, the condition isautomatic. This condition has been considered by a number of authors (Borne [Bo1][Bo2], Mochizuki [Mo2], Iyer-Simpson [IS1], Steer-Wren [Sr-Wr] and others) and isnecessary for applying the formula of [IS2].

A quasi-parabolic sheaf consists then of a collection of sheaves Eσ1,...,σn with σi ∈Σi on X , whereas a quasi-parabolic structure given by filtrations consists of a bundle Eon X together with filtrations F i

σi⊂ E|Di of the restrictions to the divisor components.

In the locally abelian case, these may be related by a long exact sequence (2.1.5):

0 −→ Eσ1,...,σn → E →n⊕i=1

(ξi)?(Liσi

)→⊕i<j

(ξij)?(Lijσi,σj

)→ . . .→ Lσ1,...,σn −→ 0.

Where Li1,...,iqσi1 ,...,σiqdenote the quotient sheaves supported on intersections of the divisors

Di1 ∩ ... ∩Diq .Using this long exact sequence we get a formula (2.1.6) for the Chern characters

of Eσ1,...,σn in terms of the Chern character of sheaves supported on intersection of thedivisors Di1 ∩ ... ∩Diq of the form:

chV b(Eσ1,...σn) = chV b(E) +n∑q=1

(−1)q∑

i1<i2<...<iq

ch(ξI,?(L

Iσi1 ,...,σiq

)).

The notion of parabolic weight function is then introduced, and the main work ofthis paper begins: we obtain the Chern characters for the Eα1,...,αn for any αi ∈ (−1, 0];these are then put into the formula of [IS2], and the result is computed. This computationrequires some combinatorial manipulations with the linearly ordered sets Σi notably theassociated sets of risers Σ′i in the ordering. It yields the following formula (2.2.2) ofTheorem 2.2.4:

chPar(E) = chV b(E)eD+

eD.n∑q=1

(−1)q∑

i1<i2<...<iq

∑λij∈Σ′ij

ch(ξI,?(Gr

i1,...,iqλi1 ,...,λiq

)) q∏j=1

[eDij (1− e−(αij (λij )+1)Dij )

eDij − 1

].

48

In this formula, the associated-graded sheaves corresponding to the multiple filtrationson intersections of divisor components DI = Di1 ∩· · ·∩Diq are denoted by Gri1,...,iqλi1 ,...,λiq

.These are sheaves on DI but are then considered as sheaves on X by the inclusionξI,? : DI → X . The Chern character ch

(ξI,?(Gr

i1,...,iqλi1 ,...,λiq

))

is the Chern character ofthe coherent sheaf on X . This is not satisfactory, since we want a formula involving theChern characters of theGri1,...,iqλi1 ,...,λiq

onDI . Therefore in §2.2.1 we use the Grothendieck-Riemann-Roch theorem to interchange ch and ξI,?, leading to the introduction of Toddclasses of the normal bundles of the DI . Another difficulty is the factor of eD mul-tiplying the term chV b(E); we would like to consider the parabolic Chern class as aperturbation of the Chern class of the usual vector bundle chV b(E). Using the sameformula for the case of trivial parabolic weights, which must give back chV b(E) as ananswer, allows us to rewrite the difference between chV b(E) and chV b(E)eD in a waycompatible with the rest of the formula. After these manipulations the formula becomes(2.2.5) of Theorem 2.2.14:

If X be a smooth projective variety with a strict normal crossings divisor D =D1 + . . . + Dn ⊂ X . Then the explicit formula for the parabolic Chern character of alocally abelian parabolic bundle on (X,D) in terms of:—the Chern character of the underlying usual vector bundle,—the divisor components Di in the rational Chow groups of X ,—the Chern characters of the associated-graded pieces of the parabolic filtration alongthe multiple intersections of the divisor components, and—the parabolic weights, is defined as follows:

chPar(E) = chV b(E) −

eD.n∑q=1

(−1)q∑

i1<i2<...<iq

∑λij∈Σ′ij

q∏j=1

(1− e−DijDij

).ξI,?

(ch(Gr

i1,...,iqλi1 ,...,λiq

))

+

eD.n∑q=1

(−1)q∑

i1<i2<...<iq

∑λij∈Σ′ij

q∏j=1

[(1− e−(αij (λij )+1)Dij

Dij

)].ξI,?

(ch(Gr

i1,...,iqλi1 ,...,λiq

)).

Finally, we would like to compute explicitly the terms chPar1 (E), chPar2 (E) andchPar3 (E). For these, we expand the different terms

eD,

q∏j=1

[(1− e−(αij (λij )+1)Dij

Dij

)],

q∏j=1

(1− e−DijDij

)in low-degree monomials of Dij , and then expand the whole formula dividing the termsup according to codimension. Denoting by S := 1, . . . , n the set of indices for divisorcomponents, we get the following formulae:

49

chPar0 (E) := rank(E).[X]

chPar1 (E) := chV b1 (E) −∑i1∈S

∑λi1∈

∑′i1

αi1(λi1).rank(Gri1λi1).[Di1 ]

chPar2 (E) := chV b2 (E) −∑i1∈S

∑λi1∈

∑′i1

αi1(λi1).(ξi1)?

(cDi11 (Gri1λi1

))

+1

2

∑i1∈S

∑λi1∈

∑′i1

α2i1

(λi1).rank(Gri1λi1).[Di1 ]2

+∑i1<i2

∑λi1∈Σ′

i1λi2∈Σ′

i2

∑p∈Irr(Di1∩Di2 )

αi1(λi1).αi2(λi2).rankp(Gri1,i2λi1 ,λi2

).[Dp].

For chPar3 (E), see Chapter 3.

The formula for chPar1 (E) is well-known (Seshadri et al) and, in terms of the definitionof Chern classes using Deligne-Mumford stacks, it was shown by Borne in [Bo1]. Theformula for chPar2 (E) was given by Mochizuki in [Mo2], and also stated as a definitionby Panov [Pa]. In both cases these coincide with our result (see the discussion on page76). As far as we know, no similar formula for chPar3 (E) has appeared in the literature.

There are some of the motivations for the present work. In heterotic string theory[OPP] physicists look for a vector bundle with specific Chern classes. The third Chernclass corresponds to numbers of families of quarks and leptons on the observable brane.In future works where these vector bundles might be replaced by parabolic or orbifoldbundles, it would be important to have the formula for chPar3 (E). The BogomolovGieseker inequality says that ch2(E) ≥ 0 where E is a stable bundle with ch1(E) = 0.Donaldson’s theorem says that in case of equality one gets a flat unitary connection.These facts have been extended to the parabolic case notably in work of Li, Panov andMochizuki ([Mo2], [Pa], [L]). Our calculations confirm their formulas for ch2(E)—getting the right formula is essential for applying the Bogomolov-Gieseker inequality.The formula for chPar2 (E) will be useful in the Donagi-Pantev approach to the geometricLanglands program [DP]. Iyer and Simpson have pointed out that Reznikov’s theoremof vanishing of certain regulations of flat bundles, extends to the parabolic case, and forapplications it would be important to know the explicit formulas.

Mochizuki defines the Chern classes using the curvature of an adapted metric andobtains his formula as a result of a difficult curvature calculation. It should be noted

50

that our formula concerns the classes defined via Deligne-Mumford stacks in the ra-tional Chow groups of X whereas Mochizuki’s definition involving curvature can onlydefine a class in cohomology. The identity of the two formulas shows that the curvaturedefinition and the stack definition give the same result up to degree 2. Of course theymust give the same result in general: to prove this for the higher Chern classes this is aninteresting question for further study.

2.1 Quasi-parabolic structures

2.1.1 Index setsLet X be a smooth projective variety over an algebraically closed field of characteristiczero and let D be a strict normal crossings divisor on X . Write D = D1 + ... + Dn

where Di are the irreducible smooth components, meeting transversally. We sometimesdenote by S := 1, . . . , n the set of indices for components of the divisor D.

Definition 2.1.1. For i = 1, ..., n, let Σi be finite linearly ordered sets with notationsηi ≤ ... ≤ σ ≤ σ′ ≤ σ′′ ≤ ... ≤ τi where ηi is the smallest element of Σi and τi thegreatest element of Σi.

Let Σ′i be the set of connections between the σ’s i.e

Σ′i = (σ, σ′), s.t σ < σ′ and there exist no σ′′ with σ < σ′′ < σ′.

Consider the tread functions m+ : Σ′i → Σi and m− : Σ′i → Σi if λ = (σ, σ′) ∈ Σ′ithen σ = m−(λ), σ′ = m+(λ). In the other direction, consider the riser functionsC+ : Σi − τi → Σ′i and C− : Σi − ηi → Σ′i such that C+(σ) = (σ, σ′) whereσ′ > σ the next element and C−(σ) = (σ′′, σ) where σ′′ < σ the next smaller element.

One can think of the elements of Σi as the steps or “treads” of a staircase, with ηiand τi the lower and upper landings; then Σ′i is the set of risers between stairs. Thetread function sends a riser to the upper and lower treads, while the riser functions senda tread to the upper and lower risers. The upper riser of τi and the lower riser of ηi areundefined.

2.1.2 Two approachesDefinition 2.1.2. A quasi-parabolic sheaf E· on (X,D) is a collection of sheaves Eσindexed by multi-indices σ = (σ1, ..., σn) with σi ∈ Σi, together with inclusions ofsheaves

Eσ → Eσ′

51

whenever σi ≤ σ′i (a condition which we write as σ ≤ σ′ in what follows), subject to thefollowing hypothesis:

Eσ1,..., σi−1, ηi, σi+1,...,σn = Eσ1,..., σi−1, τi, σi+1,..., σn(−Di). (2.1.1)

Construction 2.1.3.

We have inclusions of sheaves

Eσ1,...,σn → Eτ1,...,τi−1,σi,τi+1,...,τn → Eτ1,...,τn .

Consider the exact sequence

0 −→ Eτ1,...,τi−1,σi,τi+1,...,τn −→ Eτ1,...,τn −→ Eτ1,...,τn/Eτ1,...,σi,...,τn −→ 0,

and put F iσi

=Eτ1,...,σi,...,τnEτ1,...,ηi,...,τn

⊂ Eτ1,...,τnEτ1,...,ηi,...,τn

= E |Di then we get the exact sequence

0 −→ Eσ1,...,σn −→ Eτ1,...,τn −→⊕i

Eτ1,...,τn/Eτ1,...,σi,...,τn =⊕i

Eτ1,...,τn/Eτ1,...,ηi,...,τnEτ1,...,σi,...,τn/Eτ1,...,ηi,...,τn

(2.1.2)which can be written as

0 −→ Eσ1,...,σn −→ Eτ1,...,τn −→⊕i

E |Di /F iσi.

Here, to E· we associate the usual vector bundle E := Eτ1,...,τn .

Definition 2.1.4. LetE be a locally free sheaf onX suppose we have a filtration denotedby F i = F i

σ ⊆ E |Di , σ ∈ Σi of E |Di where F iηi

= 0 and F iτi

= E |Di with theremaining terms being saturated subsheaves

0 = F iηi⊆ F i

σ ⊆ ... ⊆ F iτi

= E |Di

for each i = 1, ..., n. We call this a quasi-parabolic structure given by filtrations.

The construction 2.1.3 allows us to pass from a quasi-parabolic sheaf, to a quasi-parabolic structure given by filtrations. Suppose we are given a quasi-parabolic sheafEσ when all the component sheaves Eσ1,...,σn are vector bundles. Set E = Eτ1,...,τn and

E |Di= Eτ1,...,τn/Eτ1,...,τi−1,ηi,...,τn .

The image of Eτ1,...,τi−1,σi,...,τn in E |Di is a subsheaf F iσi

, and we assume that it is asaturated subbundle. This gives a quasi-parabolic structure given by filtrations.

We can also go in the opposite direction.

52

Construction 2.1.5.

Suppose (E, F iσ) is a quasi-parabolic structure given by filtrations. Consider the

kernel sheaves

0 −→ Eτ1,...,σi,...,τn −→ Eτ1,...,τn −→ E |Di /F iσi,

define a collection of sheaves

Eσ1,...,σn =⋂i

(Eτ1,...,σi,...,τn) ⊂ Eτ1,...,τn

with has the property that

Eσ1,...,σi−1,ηi,σi+1,...,σn = Eσ1,...,σi−1,τi,σi+1,...,σn(−Di).

Thus we get a quasi-parabolic sheaf.

2.1.3 Locally abelian conditionA quasi-parabolic line bundle is a quasi-parabolic sheaf F such that all the Fσ are linebundles. An important class of examples is obtained as follows: if σ′ is a multiindexconsisting of σ′i ∈ Σ′i then we can define a quasi-parabolic line bundle denoted

F := OX(σ′) (2.1.3)

by setting

Fσ1,...,σn := OX(n∑i=1

γiDi)

where each γi is equal to−1 or 0; with γi = −1 when σi < σ′i and γi = 0 when σi > σ′i.Note here that the relations <,> are defined between treads σi and risers σ′i.

On the other hand, if E is a locally -free sheaf on X then it may be consideredas a quasi-parabolic sheaf (we say with trivial parabolic structure) by setting Eσ to be

E(n∑i=1

γiDi) for γi = 0 if σi = τi and γi = −1 otherwise.

Definition 2.1.6. Suppose E is a vector bundle on X and σ′ is a multiindex consistingof σ′i ∈ Σ′i, we can define the quasi-parabolic bundle as follows:

E(σ′) := E ⊗OX(σ′).

Lemma 2.1.7. Any quasi-parabolic line bundle has the form L(σ) for some σ′ and L aline bundle on X .

53

Definition 2.1.8. A quasi-parabolic sheaf E·, or quasi-parabolic structure given byfiltrations (E,F ·· ), is called a locally abelian quasi-parabolic bundle if in a Zariskineighbourhood of any point x ∈ X there is an isomorphism between F and a directsum of unweighted quasi-parabolic line bundles.

Lemma 2.1.9. Suppose Eσ1,...,σn define a locally abelian quasi-parabolic bundle on Xwith respect to (D1, ..., Dn). Let E = Eτ1,...,τn , which is a sheaf on X. Then Eσ comesfrom the construction 2.1.5 as above using unique filtrations F i

σiof E |Di and we have

the follwing properties:1) the Eσ1,...,σn are locally free;2) for each q and collection of indices (i1, ..., iq) at each point in the q-fold intersec-

tion p ∈ Di1 ∩ ...∩Diq the filtrations F i1 , ..., F iq of Ep admit a common splitting, hencethe associated-graded

GrFi1

j1...GrF

iq

jq (Ep)

is independent of the order in which it is taken;3) the functions

P 7→ rk GrFi1

j1...GrF

iq

jq (Ep)

are locally constant functions of P on the multiple intersections Di1 ∩ ... ∩Diq .

Borne [Bo1] shows:

Theorem 2.1.10. Suppose given a quasi-parabolic structure which is a collection ofsheaves Eσ1,...,σn obtained from filtrations on a bundle E as above. If the sheaves satisfycondition 1), or if the filtrations satisfy 2) and 3) of the previous lemma, then the quasi-parabolic structure is a locally abelian quasi-parabolic bundle on (X, D).

Now we have two directions:

1- If we have a subsheaf structure

E(−D) = Eη1,...,ηn ⊆ Eσ1,...,σn ⊆ Eτ1,...,τn = E

we can define the filtration structures

F iσi⊆ E |Di

as in Construction 2.1.3 by using the exact sequence (2.1.2). Then, to calculate theChern character of Eσ1,...,σn in terms of filtration structure, we must be find an extensionfor the left exact sequence (2.1.2) to a long exact sequence.

2- Vice versa.

54

Suppose we have a locally abelian quasi-parabolic structure F i given by filtra-tions, on a vector bundle E with filtrations

0 = F iηi⊆ F i

σ ⊆ ... ⊆ F iτi

= E |Di .

Then for ηi ≤ σi ≤ τi define the quotient sheaves supported on Di

Liσi :=E |DiF iσi

and the parabolic structure Eσ is given by

Eσ1,...,σn = Ker(E −→n⊕i=1

Liσi). (2.1.4)

More generally define a family of multi-index quotient sheaves by

Liσi :=E |DiF iσi

on Di

Liσi,σj :=E |Di∩Dj

F iσi|Dij + F j

σj |Dijon Dij = Di ∩Dj

.

.

Lσ1,...,σn :=E |D1∩...∩Dn

F 1σ1

+ ....+ F nσn

on D1 ∩D2 ∩ ... ∩Dn.

In these notations we have ηi ≤ σi ≤ τi, for i = 1, ..., n.

If we consider quotient sheaves as corresponding to linear subspaces of the dualprojective bundle associated to E, then the multiple quotients above are multiple in-tersections of the Liσi . The formula (2.1.4) extends to a Koszul-style resolution of thecomponent sheaves of the parabolic structure.

Lemma 2.1.11. Suppose that the filtrations give a locally abelian quasi-parabolic struc-ture, in particular they satisfy the conditions of Lemma 2.1.9. Then for any ηi ≤ σi ≤ τithe following sequence is well defined and exact over X:

0 −→ Eσ1,...,σn → E →n⊕i=1

(ξi)?(Liσi

)→⊕i<j

(ξi,j)?(Li,jσi,σj

)→ . . .→ Lσ1→...,σn −→ 0.

(2.1.5)

55

Where E is a sheaf over X ,⊕i

(Liσi) are a sheaves over Di,⊕i<j

(Li,jσi,σj) are sheaves

over Di,j = Di ∩ Dj , etc.; ξi denotes the closed immersion Di → X , and(ξi)? : coh(Di) −→ coh(X) denotes the associated Gysin map. The general term isa sum over I = (i1, ..., iq), where LI are sheaves over DI = Di1 ∩ ... ∩ Diq pushedforward by the Gysin map (ξI)? : coh(DI) −→ coh(X).

Proof. The proof in [IS2] is modified to cover the unweighted quasi-parabolic case. Themaps in the exact sequence are obtained from the quotient structures of the terms withalternating signs like in the Cech complex. We just have to prove exactness. This a localquestion. By the locally abelian condition, we may assume that E with its filtrations isa direct sum of rank one pieces. The formation of the sequence, and its exactness, arecompatible with direct sums. Therefore we may assume that E has rank one, and in factE ∼= OX .

In the case OX(σ′) as in (2.1.3), the vector bundle E is the trivial bundle and thefiltration steps are either 0 or all of ODi . In particular, there is ηi ≤ σ′i ≤ τi such thatF ij = ODi for j ≥ σ′i and F i

j = 0 for j < σ′i. Then

Li1,...,iqσi1 ,...,σiq= ODi1 ,...,Diq

if σij < σ′ij for all j = 1, ..., q and the quotient is zero otherwise. The sequence isdefined for each multiindex σ1, ..., σn. Up to reordering the coordinates which doesn’taffect the proof, we may assume that there is p ∈ [0, n] such that σi < σ′i for i ≤ pbut σi > σ′i for i > p. In this case, the quotient is nonzero only when i1, ..., iq 6 p.Furthermore,

Eσ1,...,σn = O(−D1 − ...−Dp).

In local coordinates, the divisors D1, ..., Dp are coordinate divisors. Everything is con-stant in the other coordinate directions which we may ignore. The complex in questionbecomes

O(−D1 − ...−DP )→ O → ⊕1≤i≤pODi → ⊕1≤i≤j≤pODi∩Dj → ...→ OD1∩..∩Dp .

Etale locally, this is exactly the same as the exterior tensor product of p copies of theresolution ofOA1(−D) on the affine line A1 with divisor D corresponding to the origin,

OA1(−D)→ OA1 → OD → 0.

In particular, the exterior tensor product complex is exact except at the beginning whereit resolves O(−D1 − ...−Dp) as required.

Using the resolution of the above lemma we can compute the Chern character ofEσ1,...σn in terms of the Chern character of sheaves supported on intersection of the

56

divisors Di1∩, ..., Diq . This gives us

chV b(Eσ1,...,σn) = chV b(E) +n∑q=1

(−1)q∑

i1<i2<...<iq

ch(ξI,?(L

Iσi1 ,...,σiq

)). (2.1.6)

Definition 2.1.12. Let I = (i1, ..., iq) for 1 < i1 < ... < iq ≤ n and analyse thequotient Li1,...,iqσi1 ,...,σiq

along the multiple intersection Di1,...,iq . There, the sheaf E|Di1,...,iqhas q filtrations F ij

σij|Di1,...,Diq indexed by σij ∈ Σij leading to a multiple associated-

graded defined as follows. put

F i1,...,iqσi1 ,...,σiq

:= F i1σi1∩ ... ∩ F iq

σiq⊂ E|Di1,...,iq .

Where σij ∈ Σij , Fi1,...,iqηi1 ,...,ηiq

= 0 and F i1,...,iqτi1 ,...,τiq

= E|Di1,...,iq , we have F i1,...,iqσi1 ,...,σiq

⊂Fi1,...,iqσ′i1

,...,σ′iqif σi1 6 σ′i1 and σiq 6 σ′iq . Then for a multiindex of risers λij ∈ Σ′ij , define

Gri1,...,iqλi1 ,...,λiq

:=Fi1,...,iqm+(λi1 ),...,m+(λiq )∑q

j=1 Fi1,...,iqm+(λi1 ),...,m−(λik ),...,m+(λiq )

where the indices in the denominator are almost all m+(λij) but one m−(λik). If thequasi-parabolic structure is locally abelian then the filtrations admit a common splitingand we have

Gri1,...,iqλi1 ,...,λiq

= GrFi1

λi1GrF

i2

λi2...GrF

iq

λiq(E|Di1,...,iq ).

Lemma 2.1.13. Let U be a bundle over Y, and F p1 , F

p2 , ..., F

pq are the filtrations such

that ∃ common local bases. Then in group of Grothendieck we have

GrFiGrFj commute, and U ≈ GrF1GrF2 ...GrFq(U).

Proof. This may be proven by an inductive argument.

Theorem 2.1.14. Suppose given a locally abelian quasi-parabolic bundle. Locally overDI in the Zariski topology ∃ a finite set β(λi1 , ..., λiq) such that we have a base overE|Di of the form

eλi1 ,...,λiq ; b λij∈Σ′

ijb∈β(λi1

,...,λiq )

and for the filtrations F i1,...,iqσi1 ,...,σiq

admit a base of the form

eλi1 ,...,λiq ; bλij<σij iff m+(λij )≤σij

andeλi1 ,...,λiq ; bλij<Σ′ij

form a base of Gri1,...,iqλi1 ,...,λiq.

57

Proof. By the locally abelian condition, locally we may assume that the quasi-parabolicbundle is a direct sum of quasi-parabolic line bundles. For these, the bases have eitherzero or one elements and we can verify which are nonempty in terms of conditions onthe σi.

Corollary 2.1.15. In the Grothendieck group of sheaves on Di1 ∩ ... ∩Diq , we have anequivalence

F i1,...,iqσi1 ,...,σiq

≈∑

λi1<σi1···

λiq<σiq

Gri1,...,iqλi1 ,...,λiq

(a)

and

ξI,?

(Li1,...,iqσi1 ,...,σiq

)≈ ξI,?

∑σi1

<λi1···σiq<λiq

Gri1,...,iqλi1 ,...,λiq

(b)

Now apply the part (b) of the above corollary in equation (2.1.6) we get a formulafor the Chern character of Eσ1,...,σn in terms of the associated graded as follows :

chV b(Eσ1,...,σn) = chV b(E) +n∑q=1

(−1)q∑

i1<i2<...<iq

∑σi1

<λi1···σiq<λiq

ch(ξI,?(Gr

i1,...,iqλi1 ,...,λiq

)).

(2.1.7)

2.2 Weighted parabolic structuresThe next step is to introduce the notion of weight function, providing a real numberα(i)(λi) for each λi ∈ Σ′i. The weights naturally go with the “risers” of the linearlyordered sets, which is why we introduced the sets Σ′i above.

We prolong Σi by adding its Z-translates. Define

Φi = Z.Σi := Z× Σi ∼

s.t (k, τi) ∼ (k + 1, ηi), andΦ′i := Z× Σ′i.

Prolong the functionsC−, C+ : Z.Σi −→ Z× Σ′i

by setting C+(k, τ) = (k + 1, C+(η)), and C−(k, η) = (k − 1, C−(τ)).

58

For any quasi-parabolic sheaf we prolong the notation ofEσ1,...,σn to sheavesEϕ1,...,ϕn

defined for all ϕi = (ki, σi) ∈ Z.Σi as follows: define

Eϕ1,...,ϕn := Eσ1,...,σn(ΣkiDi).

This is well defined modulo the equivalence relation defining Φi, because of the condi-tion (2.1.1). This gives the property

Eϕ1+l1,...,ϕn+ln = Eϕ1,...,ϕn(l1D1 + ...+ lnDn)

where li ∈ Z.

Definition 2.2.1. A weight function is a collection of functions

α(i) : Σ′i → (−1, 0] ⊆ R

which are increasing, i.e. α(i)(λ′) ≤ α(i)(λ) when λ′ ≤ λ.

To transform from unweighted quasi-parabolic structure −→ weighted parabolicstructure we must extend the function α(i) to all of Z.Σ′i by :

α(i) : Σ′i −→ (−1, 0] −→ α(i) : Φ′i −→ R

s.t α(i)(k, σ) = k + α(i)(σ).

Now define intervals by :

Int (α(i), σi) = (α(i)C−(σi), α(i)C+(σi)] ,

andInt (α(i), ϕi) = (α(i)C−(ϕi), α(i)C+(ϕi)] .

We can now define the weighted or usual parabolic sheaf, associated to a quasi-parabolicsheaf and a weight function. Consider the sheaves Eβ1,...,βn for every βi ∈ Rn, givenβ1, ..., βn ∀ i ∃! ϕi ∈ Φ′i such that βi ∈ Int(α(i), ϕi), then we define

Eβ1,...,βn := Eϕ1,...,ϕn

This defines a parabolic sheaf in the usual sense [Ma-Yo] [Mo2] [Bo2] [IS2].

59

Theorem 2.2.2. SupposeE is a weighted parabolic bundle onX with respect toD1, ..., Dn.Then we have the following formula for the Chern character of E :

chPar(E) =

∫ 1

β1=0...∫ 1

βn=0e−ΣβiDichV b(Eβ1,...,βn)∫ 1

β1=0...∫ 1

βn=0e−ΣβiDi

. (2.2.1)

Proof. See [IS2] (15), p. 35.

In this formula note the exponentials of real combinations of divisors are interpretedas formal polynomials. The power series for the exponential terminates because theproduct structure of CH>0(X) is nilpotent.

If the weights are real, then we need the integrals as in the formula, and the resultis in CH(X) ⊗Z R. If the weights are rational, then the integrals may be replaced bysums as in [IS2, Theorem 5.8]. In this case the answer lies in CH(X) ⊗Z Q. In whatfollows, if we were to replace the integrals by corresponding sums the answer wouldcome out the same (a factor in the numerator depending on the denominator of therational weights which are used, will cancel out with the same factor in the numerator).In order to simplify notation we keep to the integral formula.

Let ϕi = (σi + 1) then

chPar(E) =∑

ϕ1...ϕn∈Φ1×...×Φn

∫β1∈Int(α(1),ϕ1)∩(0,1]

...∫βn∈Int(α(n),ϕn)∩(0,1]

e−ΣβiDichV b(Eϕ1,...,ϕn)∫ 1

0...∫ 1

0e−ΣβiDi

Remark 12. Int(α(i), ϕi)∩(0, 1] = ∅ if ϕi * im (1 × Σi)→ Φi i.e just if ϕi = σi+1for σi ∈ Σi.

Definition 2.2.3. Let ϕi = σi + 1, for σi ∈ Σi. Define domains by

Dom(α(i), σi) := Int(α(i), σi + 1) ∩ (0, 1],

then

chPar(E) =∑

σ1...σn∈Σ1×...×Σn

∫Dom(α(1),σ1)

...∫Dom(α(n),σn)

e−ΣβiDichV b(Eϕ1,...,ϕn)∫ 1

0...∫ 1

0e−ΣβiDi

.

We haveEϕ1,..,ϕn = Eσ1,...,σn(D1 + ...+Dn)

thenchV b(Eϕ1,..,ϕn) = chV b(Eσ1,...,σn)eD1+...+Dn

therefore

chPar(E) =

(1∫ 1

0...∫ 1

0e−ΣβiDi

)×

60

∑σ1...σn∈Σ1×...×Σn

chV b(Eσ1,...,σn).

(∫ α+(σ1)+1

β1=α−(σ1)+1

.

∫ α+(σ2)+1

β2=α−(σ2)+1

...

∫ α+(σn)+1

βn=α−(σn)+1

e−(ΣβiDi)+ΣDidβ

)

for i = 1, ..., n, where

α+(σi) =

α(i)(C+(σi)) or,0 if σi = τi

and

α−(σi) =

α(i)(C−(σi)) or,−1 if σi = ηi

so Dom(α(i), σi) = (α−(σi) + 1, α+(σi) + 1].

Then

chPar(E) =

(1∫ 1

0...∫ 1

0e−ΣβiDi

)×

∑σ1...σn∈Σ1×...×Σn

chV b(Eσ1,...,σn).

(∫ α+(σ1)+1

β1=α−(σ1)+1

.

∫ α+(σ2)+1

β2=α−(σ2)+1

...

∫ α+(σn)+1

βn=α−(σn)+1

e−Σ(βi−1)Didβ

).

Take γi = βi − 1 for i = 1, ..., n, within the integrals in the numerator. Leave βi asvariable of integration in the denominator of the constant term for the moment. Then

chPar(E) =

(1∫ 1

0...∫ 1

0e−ΣβiDi

)×

∑σ1...σn∈Σ1×..×Σn

chV b(Eσ1,...,σn)

(∫ α+(σ1)

γ1=α−(σ1)

e−γ1D1dγ1.

∫ α+(σ2)

γ2=α−(σ2)

e−γ2D2dγ2 ..

∫ α+(σn)

γn=α−(σn)

e−γnDndγn

)

61

and we have

chV b(Eσ1,...,σn) = chV b(E) +n∑q=1

(−1)q∑

i1<i2<...<iq

∑σi1

<λi1σiq<λiq

ch(ξI,?(Gr

i1,...,iqλi1 ,...,λiq

)).

Then

chPar(E) =

(1∫ 1

0...∫ 1

0e−ΣβiDi

)×

∑σ1...σn∈Σ1×...×Σn

chV b(E) +n∑q=1

(−1)q∑

i1<i2<...<iq

∑σi1

<λi1σiq<λiq

ch(ξI,?(Gr

i1,...,iqλi1 ,...,λiq

))×

(∫ α+(σ1)

γ1=α−(σ1)

e−γ1D1dγ1

∫ α+(σ2)

γ2=α−(σ2)

e−γ2D2dγ2 ...

∫ α+(σn)

γn=α−(σn)

e−γnDndγn

)

=

(chV b(E)∫ 1

0...∫ 1

0e−ΣβiDi

)×

(∑σ1∈Σ1

∫ α+(σ1)

γ1=α−(σ1)

e−γ1D1dγ1

)·

(∑σ2∈Σ2

∫ α+(σ2)

γ2=α−(σ2)

e−γ2D2dγ2

)· · ·

( ∑σn∈Σn

∫ α+(σn)

γn=α−(σn)

e−γnDndγn

)

+

(1∫ 1

0...∫ 1

0e−ΣβiDi

).

n∑q=1

(−1)q∑

i1<i2<...<iq

∑λi1 ...λiq

ch(ξI,?(Gr

i1,...,iqλi1 ,...,λiq

))×

∑σ1∈Σ1

σ1<λ1 if 1∈I

∫ α+(σ1)

γ1=α−(σ1)

e−γ1D1dγ1

· ∑

σ2∈Σ2σ2<λ2 if 2∈I

∫ α+(σ2)

γ2=α−(σ2)

e−γ2D2dγ2

· · ·

·

∑σn∈Σn

σn<λn if n∈I

∫ α+(σn)

γn=α−(σn)

e−γnDndγn

62

which can be written aschPar(E) = A+B

where

A =

(chV b(E)∫ 1

0...∫ 1

0e−ΣβiDi

)×

(∑σ1∈Σ1

∫ α+(σ1)

γ1=α−(σ1)

e−γ1D1dγ1

)·

(∑σ2∈Σ2

∫ α+(σ2)

γ2=α−(σ2)

e−γ2D2dγ2

)...

( ∑σn∈Σn

∫ α+(σn)

γn=α−(σn)

e−γnDndγn

)

One can note that∫ b

a

e−ρDdρ =e−aD − e−bD

D= e−aD

(1− e(a−b)D)

D= e−aD

(a− b)td((a− b)D)

where td is the Todd class.

We have∑σi∈Σi

∫ α+(σi)

γi=α−(σi)

e−γiDidγi =

∫ 0

−1

e−γiDidγi for i = 1, 2, ..., n and put β = γ as

integration variable in the denominator of the constant term,

then

A =

(chV b(E)∫ 1

0...∫ 1

0e−ΣγiDi

).

(∫ 0

−1

e−γ1D1dγ1

).

(∫ 0

−1

e−γ2D2dγ2

)...

(∫ 0

−1

e−γnDndγn

)

=

[(∫ 0

−1e−γ1D1dγ1∫ 1

0e−γ1D1dγ1

).

(∫ 0

−1e−γ2D2dγ2∫ 1

0e−γ2D2dγ2

)...

(∫ 0

−1e−γnDndγn∫ 1

0e−γnDndγn

)].chV b(E)

where

Ai =

∫ 0

−1e−γiDidγi∫ 1

0e−γiDidγi

=e−Di − 1

1− e−Di=eDi(1− e−Di)

1− e−Di= eDi

for i = 1, 2, ..., n

thereforeA = chV b(E).eD1 .eD2 ...eDn = chV b(E)eD

63

where D = D1 +D2 + ...+Dn

and (again replacing β by γ as name of the integration variable in the denominator ofthe constant term)

B =

(1∫ 1

0...∫ 1

0e−ΣγiDi

).

n∑q=1

(−1)q∑

i1<i2<...<iq

∑λi1λiq

ch(ξI,?(Gr

i1,...,iqλi1 ,...,λiq

))×

∑σ1∈Σ1

σ1<λ1 if 1∈I

∫ α+(σ1)

γ1=α−(σ1)

e−γ1D1dγ1

· ∑

σ2∈Σ2σ2<λ2 if 2∈I

∫ α+(σ2)

γ2=α−(σ2)

e−γ2D2dγ2

...

∑σn∈Σn

σn<λn if n∈I

∫ α+(σn)

γn=α−(σn)

e−γnDndγn

.

The sums of integrals can be expressed as single integrals; if i /∈ I , set formallyαi(λi) := 0 in the following expression:

B =

[(∫ 0

−1e−γ1D1dγ1)(

∫ 0

−1e−γ2D2dγ2)...(

∫ 0

−1e−γnDndγn)

(∫ 1

0e−γ1D1dγ1)(

∫ 1

0e−γ2D2dγ2)...(

∫ 1

0e−γnDndγn)

]×

n∑q=1

(−1)q∑

i1<i2<...<iq

∑λij∈Σ′ij

ch(ξI,?(Gr

i1,...,iqλi1 ,...,λiq

))×

[(∫ 0

−1e−γ1D1dγ1)(

∫ 0

−1e−γ2D2dγ2)...(

∫ 0

−1e−γnDndγn)

(∫ 0

−1e−γ1D1dγ1)(

∫ 0

−1e−γ2D2dγ2)...(

∫ 0

−1e−γnDndγn)

]×

[(∫ α1(λ1)

−1e−γ1D1dγ1)(

∫ α2(λ2)

−1e−γ2D2dγ2)...(

∫ αn(λn)

−1e−γnDndγn)

(∫ 0

−1e−γ1D1dγ1)(

∫ 0

−1e−γ2D2dγ2)...(

∫ 0

−1e−γnDndγn)

])

64

= eD.

n∑q=1

(−1)q∑

i1<i2<...<iq

∑λij∈Σ′ij

ch(ξI,?(Gr

i1,...,iqλi1 ,...,λiq

)) q∏j=1

[∫ αij(λij)

−1 e−ρDij dρ∫ 0

−1e−ρDij dρ

]

= eD.

n∑q=1

(−1)q∑

i1<i2<...<iq

∑λij∈Σ′ij

ch(ξI,?(Gr

i1,...,iqλi1 ,...,λiq

)) q∏j=1

[eDij − e−(αij (λij ))Dij

eDij − 1

]

= eD.n∑q=1

(−1)q∑

i1<i2<...<iq

∑λij∈Σ′ij

ch(ξI,?(Gr

i1,...,iqλi1 ,...,λiq

)) q∏j=1

[eDij (1− e−(αij (λij )+1)Dij )

eDij − 1

].

Therefore we have proven the following

Theorem 2.2.4.chPar(E) = chV b(E)eD+

eD.n∑q=1

(−1)q∑

i1<i2<...<iq

∑λij∈Σ′ij

ch(ξI,?(Gr

i1,...,iqλi1 ,...,λiq

)) q∏j=1

[eDij (1− e−(αij (λij )+1)Dij )

eDij − 1

].

(2.2.2)

2.2.1 Riemann-Roch theoremThe next step is to use Riemann-Roch theory to interchange ch and ξI,?. Let KDI

denote the Grothendieck group of vector bundles(locally free sheaves) on DI . Eachvector bundleE determines an element, denoted [E], inKDI . KDI is the free abeliangroup on the set of isomorphism classes of vector bundles, modulo the relations

[E] = [E ′] + [E ′′]

whenever E ′ is a subbundle of a vector bundle E with quotient bundle E ′′ = E/E ′. Thetensor product makes KDI a ring: [E].[F ] = [E ⊗ F ].

Definition 2.2.5. For any morphism ξI : X → DI there is an induced homomorphism

ξ?I : KDI → KX,

taking [E] to [ξ?IE], where ξ?IE is the pull-back bundle.

65

The Grothendieck group of coherent sheaves on DI , denoted by KDI , is definedto be the free abelian group on the isomorphism class [F ] of coherent sheaves on DI ,modulo the relaions

[F ] = [F ′] + [F ′′]

for each exact sequence0→ F ′ → F → F ′′ → 0

of coherent sheaves on DI . Tensor product makes KDI a KDI-module:

KDI ⊗KDI → KDI ,

[E].[F ] = [E ⊗ODI F ].

Definition 2.2.6. For any proper morphism ξI : DI → X , there is a homomorphism

ξI,? : KDI → KX

which takes [F ] to∑i>0

(−1)i[RiξI,?F ].where RiξI,?F is Grothendieck higher direct im-

age sheaf, the sheaf associated to the presheaf

U → H i(ξ−1I (U),F)

on X .

It is a basic fact the RiξI,?F are coherent when F is coherent and ξI is proper. Thefact that this push-forward ξI,? is well-defined on KDI results from the long excatcohomology sequence for the RiξI,?.

Proposition 2.2.7. The push-forward and pull-back are related by the usual projectionformula:

ξI,?(ξ?Ia.b) = a.(ξI,?b)

for ξI : DI → X proper, a ∈ KX , b ∈ KDI .

Theorem 2.2.8. On any DI there is a canonical duality homomorphism

KDI → KDI

which takes a vector bundle to its sheaf of sections. When DI is non-singular, thisduality map is an isomorphism.

66

Definition 2.2.9. Consider DI which are smooth over a given ground field C. For suchDI we identify KDI and KDI , and write simply K(DI). There is a homomorphism,called the Chern character

ch : K(DI)→ A(DI)Q

determined by the following properties:i) ch is a homomorphism of rings;ii) if ξI : X → DI , ch ξ?I = ξ?I ch;iii) if l is a line bundle on DI , ch[DI ] = exp(c1(l)) =

∑i>0

(1/i!)c1(l)i

Theorem 2.2.10. Let ξI : DI → X be a smooth projective morphism of non-singularquasi-projective varieties. Then for any GrIλ ∈ K(DI) we have

ξI,?

(ch(Gr

i1,...,iqλi1 ,...,λiq

).td(TDI ))

= td(TX).ch(ξI,?(Gr

i1,...,iqλi1 ,...,λiq

))

in A(X)⊗Q. where Td(X) = td(TX) ∈ A(X)Q is the relative tangent sheaf of ξI .

See [Fu, pp 286-287].

Theorem 2.2.11. If DI is a non-singular variety set

Td(DI) = td(TDI ) ∈ A(DI)Q

then

if ξI : DI → X , is a closed imbedding of codimension q, and DI is the intersection ofq Cartier divisors Di1 , ..., Diq on X , then

Td(DI) = ξ?I

[Td(X).

q∏j=1

(1− e−DijDij

)].

See [Fu, p. 293].

So

ξI,?

(ch(Gr

i1,...,iqλi1 ,...,λiq

).td(TDI ))

= ξI,?

[ch(Gr

i1,...,iqλi1 ,...,λiq

).ξ?I

(Td(X).

q∏j=1

(1− e−DijDij

))]

= Td(X).

q∏j=1

[1− e−DijDij

].ξI,?

(ch(Gr

i1,...,iqλi1 ,...,λiq

)).

67

Corollary 2.2.12.

ch(ξI,?(Gr

i1,...,iqλi1 ,...,λiq

))

=

q∏j=1

[1− e−DijDij

].ξI,?

(ch(Gr

i1,...,iqλi1 ,...,λiq

))

=

q∏j=1

[(e−Dij (eDij − 1)

Dij

)].ξI,?

(ch(Gr

i1,...,iqλi1 ,...,λiq

)).

Theorem 2.2.13. Apply the above corollary in the equation (2.2.2) of Theorem 2.2.4 weget

chPar(E) = chV b(E)eD+

eD.n∑q=1

(−1)q∑

i1<i2<...<iq

∑λij∈Σ′ij

q∏j=1

[(1− e−(αij (λij )+1)Dij

Dij

)].ξI,?

(ch(Gr

i1,...,iqλi1 ,...,λiq

)).

(2.2.3)

Now return to the equation (2.1.7) but apply it to the bottom value of σi = ηi. Recallthat Eη1,...,ηn = E(−D); but on the other hand ηij < λij for any λij ∈ Σ′ij . Thus we get

chV b(E(−D)) = chV b(E) +n∑q=1

(−1)q∑

i1<i2<...<iq

∑λij∈Σ′ij

ch(ξI,?(Gr

i1,...,iqλi1 ,...,λiq

)).

=⇒

chV b(E) = chV b (E(−D))−n∑q=1

(−1)q∑

i1<i2<...<iq

∑λij∈Σ′ij

ch(ξI,?(Gr

i1,...,iqλi1 ,...,λiq

)).

(2.2.4)

Put equation (2.2.4) in (2.2.3) and use Corollary 2.2.12, to get

chPar(E) = chV b (E(−D)) eD−

eD.

n∑q=1

(−1)q∑

i1<i2<...<iq

∑λij∈Σ′ij

q∏j=1

(1− e−DijDij

).ξI,?

(ch(Gr

i1,...,iqλi1 ,...,λiq

))

+

68

eD.

n∑q=1

(−1)q∑

i1<i2<...<iq

∑λij∈Σ′ij

q∏j=1

[(1− e−(αij (λij )+1)Dij

Dij

)].ξI,?

(ch(Gr

i1,...,iqλi1 ,...,λiq

))

We have chV b (E(−D)) eD = chV b(E), therefore:

Theorem 2.2.14. If X be a smooth projective variety with a strict normal crossingsdivisor D = D1 + . . . + Dn ⊂ X . Then the explicit formula for the parabolic Cherncharacter of a locally abelian parabolic bundle on (X,D) in terms of:—the Chern character of the underlying usual vector bundle,—the divisor components Di in the rational Chow groups of X ,—the Chern characters of the associated-graded pieces of the parabolic filtration alongthe multiple intersections of the divisor components, and—the parabolic weights, is defined as follows:

chPar(E) = chV b(E) −

eD.n∑q=1

(−1)q∑

i1<i2<...<iq

∑λij∈Σ′ij

q∏j=1

(1− e−DijDij

).ξI,?

(ch(Gr

i1,...,iqλi1 ,...,λiq

))

+

eD.n∑q=1

(−1)q∑

i1<i2<...<iq

∑λij∈Σ′ij

q∏j=1

[(1− e−(αij (λij )+1)Dij

Dij

)].ξI,?

(ch(Gr

i1,...,iqλi1 ,...,λiq

)).

(2.2.5)

2.3 Computation of parabolic Chern characters of a lo-cally abelian parabolic bundleE in codimension oneand two chPar1 (E), chPar2 (E)

Proposition 2.3.1. For an n-dimensional, non singular variety Y , set

ApY = An−pY,

where p denotes the codimension, and n − p the dimension. With this indexing bycodimension, the product x⊗ y → x.y, reads

ApY ⊗ AqY → Ap+qY,

i.e, the degrees add. Let 1 ∈ A0Y denote the class corresponding to [Y ] in AnY , andset A∗Y =

⊕ApY .

69

Return to the equation (2.2.5)

chPar(E) = chV b(E) −

eD.

n∑q=1

(−1)q∑

i1<i2<...<iq

∑λij∈Σ′ij

q∏j=1

(1− e−DijDij

).ξI,?

(ch(Gr

i1,...,iqλi1 ,...,λiq

))

+

eD.

n∑q=1

(−1)q∑

i1<i2<...<iq

∑λij∈Σ′ij

q∏j=1

[(1− e−(αij (λij )+1)Dij

Dij

)].ξI,?

(ch(Gr

i1,...,iqλi1 ,...,λiq

))

take

1− e−(αij (λij )+1)Dij

Dij

=1−

(1− (αij(λij) + 1)Dij +

(αij (λij )+1)2

2D2ij− (αij (λij )+1)3

6D3ij

+ ...)

Dij

=(αij(λij) + 1

)−(αij(λij) + 1

)2

2Dij +

(αij(λij) + 1

)3

6D2ij− ...

Let GrIλ be a vector bundles over DI for I = (i1, ..., iq) and λ = (λi1 , ..., λiq) with rankr, we have

ξI,?(ch(GrIλ)

)= ξI,?

(chDI0 (GrIλ) + chDI1 (GrIλ)

+chDI2 (GrIλ) + chDI3 (GrIλ) + ...) /X

= ξI,?(chDI0 (GrIλ)

)/X + ξI,?

(chDI1 (GrIλ)

)/X

+ξI,?(chDI2 (GrIλ)

)/X + ξI,?

(chDI3 (GrIλ)

)/X + ....

sochDI0 (GrIλ) = (rank(GrIλ))/DI =

∑p∈Irr(DI)

rankp(GrIλ).[Dp]/DI

∈ A0(DI) = AdimDI (DI) =⊕

A0(Dp) =⊕

Q.[Dp]

where DI =⋃

p∈Irr(DI)

Dp and Irr(DI) denotes the set of irreducible components of

Di1 ∩Di2 ∩ ... ∩Diq then

70

ξI,?

∑p∈Irr(DI)

rankp(GrIλ).[Dp]/DI

=

∑p∈Irr(DI)

rankp(GrIλ).[Dp]/X

∈ Aq(X)

which is of codimension q,

chDI1 (GrIλ) = cDI1 (GrIλ) ∈ A1(DI) = AdimDI−1(DI) then

ξI,?(cDI1 (GrIλ)

)/X ∈ A1+q(X) which is of codimension q + 1,

chDI2 (GrIλ) =1

2

[(cDI1 )2(GrIλ)− 2cDI2 (GrIλ)

]∈ A2(DI) = AdimDI−2(DI) then

ξI,?

(1

2[(cDI1 )2(GrIλ)− 2cDI2 (GrIλ)]

)/X ∈ A2+q which is of codimension q + 2,

chDI3 (GrIλ) =1

6

[(cDI1 )3(GrIλ)− 3cDI1 (GrIλ)c

DI2 (GrIλ)] + 3cDI3 (GrIλ)

]∈ A3(DI) then

ξI,?

(1

6

[(cDI1 )3(GrIλ)− 3(cDI1 (GrIλ)c

DI2 (GrIλ) + 3cDI3 (GrIλ)

])/X ∈ A3+q

which is of codimension q + 3. Therefore

chPar(E) = chV b(E) + eD.n∑q=1

(−1)q∑

i1<i2<...<iq

∑λij∈Σ′ij

q∏j=1

[(αij(λij) + 1

)−(αij(λij) + 1

)2

2Dij +

(αij(λij) + 1

)3

6D2ij− ...

].

∑p∈Irr(DI)

rankp(GrIλ).[Dp]/X + ξI,?(c

DI1 (GrIλ))/X + ξI,?

(1

2[(cDI1 )2(GrIλ)− 2cDI2 (GrIλ)]

)/X+

71

ξI,?

(1

6

[(cDI1 )3(GrIλ)− 3cDI1 (GrIλ)c

DI2 (GrIλ) + 3cDI3 (GrIλ)

])/X

]

− eD.n∑q=1

(−1)q∑

i1<i2<...<iq

∑λij∈Σ′ij

q∏j=1

(1−

Dij

2+D2ij

6− ...

)·

∑p∈Irr(DI)

rp(GrIλ).[Dp]/X+

ξI,?(cDI1 (GrIλ)

))/X + ξI,?

(1

2[(cDI1 )2(GrIλ)− 2cDI2 (GrIλ)]

)/X+

ξI,?

(1

6[(cDI1 )3(GrIλ)− 3cDI1 (GrIλ)c

DI2 (GrIλ) + 3cDI3 (GrIλ)]

)/X

](2.3.1)

we have

chV b(E) = rank(E).[X] + chV b1 (E) + chV b2 (E) + chV b3 (E) + ...

where rank(E).[X] ∈ A0, chV b1 (E) ∈ A1, chV b2 (E) ∈ A2, chV b3 (E) ∈ A3,

and eD = 1 +D +D2

2+D3

6+ ... where 1 ∈ A0, D ∈ A1,

D2

2∈ A2,

D3

6∈ A3

for k = 0, 1, ..., n, chPark (E) = chPar0 (E) + chPar1 (E) + chPar2 (E) + chPar3 (E) + ...

then

chPar0 (E) + chPar1 (E) + chPar2 (E) = rank(E).[X]/A0 + chV b1 (E)/A1

+ chV b2 (E)/A2 −∑i1∈S

∑λi1∈

∑′i1

(αi1(λi1) + 1).rank(Gri1λi1).[Di1 ]/A1

−∑i1∈S

∑λi1∈

∑′i1

(αi1(λi1) + 1).(ξi1)?

(cDi11 (Gri1λi1

))/A2

72

−∑i1∈S

∑λi1∈

∑′i1

(αi1(λi1) + 1).rank(Gri1λi1).[Di1 ].[D]/A2

+∑i1<i2

∑λi1λi2

∑p∈Irr(Di1∩Di2 )

(αi1(λi1) + 1)(αi2(λi2) + 1).rankp(Gri1,i2λi1 ,λi2

).[Dp]/A2

+1

2

∑i1∈S

∑λi1∈

∑′i1

(αi1(λi1) + 1)2.rank(Gri1λi1).[Di1 ]2/A2

+∑i1∈S

∑λi1∈

∑′i1

rank(Gri1λi1).[Di1 ]/A1

+∑i1∈S

∑λi1∈

∑′i1

(ξi1)?

(cDi11 (Gri1λi1

))/A2

+∑i1∈S

∑λi1∈

∑′i1

rank(Gri1λi1).[Di1 ].[D]/A2

−∑i1<i2

∑λi1λi2

∑p∈Irr(Di1∩Di2 )

rankp(Gri1,i2λi1 ,λi2

).[Dp]/A2

− 1

2

∑i1∈S

∑λi1∈

∑′i1

rank(Gri1λi1).[Di1 ]2/A2

Lemma 2.3.2. In the Grothendieck group, for all i1, i2 ∈ S, for all λi1 ∈ Σ′i1 , for allp ∈ Irr(Di1 ∩Di2), we have:

rank(Gri1λi1) =

∑λi2∈Σ′i2

rankp(Gri1,i2λi1 ,λi2

).

73

Now we have D =∑i

Di and [Di1

].[Di2 ] =∑

p∈Irr(Di1∩Di2 )

[Dp] then

[D].[Di1 ] =∑i1∈S

[Di1 ]2 +∑i1 6=i2

[Di1 ].[Di2 ] =∑i1∈S

[Di1 ]2 +∑i1 6=i2

∑p∈Irr(Di1∩Di2 )

[Dp].

Then

chPar0 (E) + chPar1 (E) + chPar2 (E) = rank(E).[X]/A0 + chV b1 (E)/A1

+ chV b2 (E)/A2 −∑i1∈S

∑λi1∈

∑′i1

αi1(λi1).rank(Gri1λi1).[Di1 ]/A1

−∑i1∈S

∑λi1∈

∑′i1

αi1(λi1).(ξi1)?

(cDi11 (Gri1λi1

))/A2

−∑i1∈S

∑λi1∈

∑′i1

αi1(λi1).rank(Gri1λi1).[Di1 ]2/A2

−∑i1∈S

∑i1 6=i2

∑λi1∈

∑′i1

∑λi2∈Σ′i2

∑p∈Irr(Di1∩Di2 )

αi1(λi1).rankp(Gri1,i2λi1 ,λi2

).[Dp]/A2

+∑i1<i2

∑λi1λi2

∑p∈Irr(Di1∩Di2 )

αi1(λi1).rankp(Gri1,i2λi1 ,λi2

).[Dp]/A2

+∑i1<i2

∑λi1λi2

∑p∈Irr(Di1∩Di2 )

αi2(λi2).rankp(Gri1,i2λi1 ,λi2

).[Dp]/A2

74

+∑i1<i2

∑λi1λi2

∑p∈Irr(Di1∩Di2 )

αi1(λi1).αi2(λi2).rankp(Gri1i2λi1 ,λi2

).[Dp]/A2

+1

2

∑i1∈S

∑λi1∈

∑′i1

α2i1

(λi1).rank(Gri1λi1).[Di1 ]2/A2

+∑i1∈S

∑λi1∈

∑′i1

αi1(λi1).rank(Gri1λi1).[Di1 ]2/A2

2.3.1 The characteristic numbers for parabolic bundles in codimen-sion 1 and 2

-For any parabolic bundle E in codimension one, and two, the parabolic first, secondChern characters chPar1 (E), and chPar2 (E), are obtained as follows:

chPar0 (E) := rank(E).[X]

chPar1 (E) := chV b1 (E) −∑i1∈S

∑λi1∈Σ′i1

αi1(λi1).rank(Gri1λi1).[Di1 ]

chPar2 (E) := chV b2 (E) −∑i1∈S

∑λi1∈

∑′i1

αi1(λi1).(ξi1)?

(cDi11 (Gri1λi1

))

+1

2

∑i1∈S

∑λi1∈

∑′i1

α2i1

(λi1).rank(Gri1λi1).[Di1 ]2

+∑i1<i2

∑λi1λi2

∑p∈Irr(Di1∩Di2 )

αi1(λi1).αi2(λi2).rankp(Gri1,i2λi1 ,λi2

).[Dp].

75

In order to compare with Mochizuki’s formula, note that

∑i1<i2

∑λi1λi2

∑p∈Irr(Di1∩Di2 )

αi1(λi1).αi2(λi2).rankp(Gri1,i2λi1 ,λi2

).[Dp] =

1

2

∑i1 6=i2

∑λi1λi2

∑p∈Irr(Di1∩Di2 )

αi1(λi1).αi2(λi2).rankp(Gri1,i2λi1 ,λi2

).[Dp]

therefore our formula may be written:

chPar2 (E) := chV b2 (E) −∑i1∈S

∑λi1∈

∑′i1

αi1(λi1).(ξi1)?

(cDi11 (Gri1λi1

))

+1

2

∑i1∈S

∑λi1∈

∑′i1

α2i1

(λi1).rank(Gri1λi1).[Di1 ]2

+1

2

∑i1 6=i2

∑λi1λi2

∑p∈Irr(Di1∩Di2 )

αi1(λi1).αi2(λi2).rankp(Gri1,i2λi1 ,λi2

).[Dp].

This coincides exactly with the formula given by Mochizuki in [Mo2, §3.1.5, p. 30].It is also the same as the definition given by Panov [Pa]. Note that in Panov’s generaldefinition the sum for the last term is written as ∑i,j without the factor of 1/2 but later heuses it as a sum over i < j, so our formula and Mochizuki’s also coincide with Panov’sformula in the way he uses it.

Mochizuki’s formula was for the Chern character in cohomology, which he definedas the integral of the Chern form of the curvature of an adapted metric. Our calculationverifies that this gives the same answer as the method using Deligne-Mumford stacksof [Bi] [Bo1] [IS1] for rational weights. Our formula is valid for the Chern character inthe rational or real Chow ring.

Here we explain some of the notation: chV b1 (E), chV b2 (E) denotes the first, second, Chern character of vector bundles E. Irr(DI) denotes the set of the irreducible components ofDI := Di1∩Di2∩...∩Diq .

76

ξI denotes the closed immersion DI −→ X , and ξI,? : Ak(DI) −→ Ak+q(X)denotes the associated Gysin map. Let p be an element of Irr(Di ∩Dj). Then rankp(GrIλ) denotes the rank of GrIλ

as an Op-module. [Dij ] ∈ A1(X)⊗Q, and [Dp] ∈ A2(X)⊗Q denote the cycle classes given by Dij

and Dp respectively.

77

Chapter 3

Parabolic Chern character of a locallyabelian parabolic bundle E incodimension 3, chPar3 (E)

By the same method of computation as above, we get the following formula, which hasnot been considered elsewhere in the literature.

chPar3 (E) = chV b3 (E)− 1

2

∑i1∈S

∑λi1∈

∑′i1

(αi1(λi1) + 1).rank(Gri1λi1).[Di1 ].[D]2

+1

2

∑i1∈S

∑λi1∈

∑′i1

(αi1(λi1) + 1)2.(ξi1)?

(cDi11 (Gri1λi1

)).[Di1 ]

−1

6

∑i1∈S

∑λi1∈

∑′i1

(αi1(λi1) + 1)3.rank(Gri1λi1).[Di1 ]3

−∑i1∈S

∑λi1∈

∑′i1

(αi1(λi1) + 1).(ξi1)?

(cDi11 (Gri1λi1

)).[D]

+1

2

∑i1∈S

∑λi1∈

∑′i1

(αi1(λi1) + 1)2.rank(Gri1λi1).[Di1 ]2.[D]

−1

2

∑i1∈S

∑λi1∈

∑′i1

(αi1(λi1) + 1).(ξi1)?

((cDi11 )2(Gri1λi1

)− 2cDi12 (Gri1λi1

))

−1

2

∑i1<i2

∑λi1λi2

∑p∈Irr(Di1∩Di2 )

(αi1(λi1) + 1).(αi2(λi2) + 1)2.rankp(Gri1,i2λi1 ,λi2

).[Di2 ].[Dp]

−1

2

∑i1<i2

∑λi1λi2

∑p∈Irr(Di1∩Di2 )

(αi1(λi1) + 1)2.(αi2(λi2) + 1).rankp(Gri1,i2λi1 ,λi2

).[Di1 ].[Dp]

+∑i1<i2

∑λi1λi2

∑p∈Irr(Di1∩Di2 )

(αi1(λi1) + 1).(αi2(λi2) + 1).rankp(Gri1,i2λi1 ,λi2

).[D].[Dp]

+∑i1<i2

∑λi1λi2

(αi1(λi1) + 1).(αi2(λi2) + 1).(ξi1,i2)?

(cDi1∩Di21 (Gri1,i2λi1 ,λi2

))

−∑

i1<i2<i3

∑λi1λi3

∑p∈Irr(Di1∩Di2∩Di3 )

(αi1(λi1)+1).(αi2(λi2)+1).(αi3(λi3)+1).rankp(Gri1,i2,i3λi1 ,λi2 ,λi3

).[Dp]

+1

2

∑i1∈S

∑λi1∈

∑′i1

rank(Gri1λi1).[Di1 ].[D]2 − 1

2

∑i1∈S

∑λi1∈

∑′i1

(ξi1)?

(cDi11 (Gri1λi1

)).[Di1 ]

+1

6

∑i1∈S

∑λi1∈

∑′i1

rank(Gri1λi1).[Di1 ]3 +

∑i1∈S

∑λi1∈

∑′i1

(ξi1)?

(cDi11 (Gri1λi1

)).[D]

−1

2

∑i1∈S

∑λi1∈

∑′i1

rank(Gri1λi1).[Di1 ]2.[D] +

1

2

∑i1∈S

∑λi1∈

∑′i1

(ξi1)?

((cDi11 )2(Gri1λi1

)− 2cDi12 (Gri1λi1

))

79

−∑i1<i2

∑λi1λi2

∑p∈Irr(Di1∩Di2 )

rankp(Gri1,i2λi1 ,λi2

).[D].[Dp]−∑i1<i2

∑λi1λi2

(ξi1,i2)?

(cDi1∩Di21 (Gri1,i2λi1 ,λi2

))

+∑

i1<i2<i3

∑λi1λi3

∑p∈Irr(Di1∩Di2∩Di3 )

rankp(Gri1,i2,i3λi1 ,λi2 ,λi3

).[Dp]

+1

2

∑i1<i2

∑λi1λi2

∑p∈Irr(Di1∩Di2 )

rankp(Gri1,i2λi1 ,λi2

).[Di2 ].[Dp]

+1

2

∑i1<i2

∑λi1λi2

∑p∈Irr(Di1∩Di2 )

rankp(Gri1,i2λi1 ,λi1

).[Di1 ].[Dp]

3.1 The characteristic number for a parabolic bundle incodimension 3

-For any parabolic bundle E in codimension 3, the parabolic third Chern characterchPar3 (E), is obtained as follows:

chPar3 (E) = chV b3 (E)− 1

2

∑i1∈S

∑λi1∈

∑′i1

αi1(λi1).rank(Gri1λi1).[Di1 ].[D]2

+1

2

∑i1∈S

∑λi1∈

∑′i1

[α2i1

(λi1) + 2αi1(λi1)].(ξi1)?

(cDi11 (Gri1λi1

)).[Di1 ]

−1

6

∑i1∈S

∑λi1∈

∑′i1

[α3i1

(λi1) + 3α2i1

(λi1) + 3αi1(λi1)].rank(Gri1λi1).[Di1 ]3

−∑i1∈S

∑λi1∈

∑′i1

αi1(λi1).(ξi1)?

(cDi11 (Gri1λi1

)).[D]

80

+1

2

∑i1∈S

∑λi1∈

∑′i1

[α2i1

(λi1) + 2αi1(λi1)].rank(Gri1λi1).[Di1 ]2.[D]

−1

2

∑i1∈S

∑λi1∈

∑′i1

αi1(λi1).(ξi1)?

((cDi11 )2(Gri1λi1

))

+∑i1∈S

∑λi1∈

∑′i1

αi1(λi1).(ξi1)?

(cDi12 (Gri1λi1

))

−1

2

∑i1<i2

∑λi1λi2

∑p∈Irr(Di1∩Di2 )

[α2i2

(λi2).αi1(λi1)+2αi1(λi1).αi2(λi2)+αi1(λi1)+α2i2

(λi2)+

2αi2(λi2)].rankp(Gri1,i2λi1 ,λi2

).[Di2 ].[Dp]

−1

2

∑i1<i2

∑λi1λi2

∑p∈Irr(Di1∩Di2 )

[α2i1

(λi1).αi2(λi2)+2αi1(λi1).αi2(λi2)+αi2(λi2)+α2i1

(λi1)+

2αi1(λi1)].rankp(Gri1,i2λi1 ,λi2

).[Di1 ].[Dp]

+∑i1<i2

∑λi1λi2

∑p∈Irr(Di1∩Di2 )

[αi1(λi1).αi2(λi2)+αi1(λi1)+αi2(λi2)].rankp(Gri1,i2λi1 ,λi2

).[D].[Dp]

+∑i1<i2

∑λi1λi2

[αi1(λi1).αi2(λi2) + αi1(λi1) + αi2(λi2)].(ξi1,i2)?

(cDi1∩Di21 (Gri1,i2λi1 ,λi2

))

−∑

i1<i2<i3

∑λi1λi3

∑p∈Irr(Di1∩Di2∩Di3 )

[αi1(λi1).αi2(λi2).αi3(λi3) + αi1(λi1).αi2(λi2)+

αi2(λi2).αi3(λi3)+αi1(λi1).αi3(λi3)+αi1(λi1)+αi2(λi2)+αi3(λi3)].rankp(Gri1,i2,i3λi1 ,λi2 ,λi3

).[Dp].

81

Chapter 4

Chern invariants for parabolic bundlesat multiple points

If D ⊂ X is a curve with multiple points in a surface, a parabolic bundle definedon (X,D) away from the singularities can be extended in several ways to a parabolicbundle on a resolution of singularities. We investigate the possible parabolic Chernclasses for these extensions.

Suppose X is a smooth surface and D = D1 + . . . + Dk is a divisor with each Di

smooth. Suppose E is a bundle provided with filtrations F i· along the Di, and parabolic

weights αi· . If D has normal crossings, this defines a locally abelian parabolic bundleon (X, D) and the parabolic Chern classes have been calculated as explained in theprevious part.

Suppose that the singularities of D contain some points of higher multiplicity. Forthe present work we assume that these are as easy as possible, namely several smoothbranches passing through a single point with distinct tangent directions. The first basiccase is a triple point.

Let ϕ : X → X denote this birational transformation, and let Di ⊂ X denote thestrict transforms of the Di. Assuming for simplicity that there is a single multiple point,denote by D0 the exceptional divisor. Now D = D0 + · · ·+Dk is a divisor with normalcrossings. Suppose E is a vector bundle on X with

E|X−D0 = ϕ∗(E)|X−D0 .

The filtrations F i· induce filtrations of ϕ∗(E)|Di and hence of E|Di−Di∩D0 , which then

extend uniquely to filtrationsF i· ofE|Di . Associate to these filtrations the same parabolic

weights as before.Up until now we have already made a choice of extension of the bundle E. Choose

furthermore a filtration F 0· of E|D0 and parabolic weights associated to D0. Having

made these choices we get a parabolic bundle on the normal crossings divisor (X,D),which determines parabolic Chern classes. We are particularly interested in the invariant∆ which combines c1 and c2 in such a way as to be invariant by tensoring with a linebundle.

The goal of this paper is to provide a convenient calculation of ∆ and then investigateits dependence on the choices which have been made above. In particular we would liketo show that ∆ achieves its minimum and calculate this minimum, which can be thoughtof as the Chern invariant associated to the original parabolic structure on the multiplepoint singularity (X, D).

The main difficulty is to understand the possible choices for E. For this we use thetechnical of Ballico-Gasparim [Ba] [BG1] [BG2].

4.1 Calculating the invariant ∆ of a locally abelian parabolicbundle

Recall from the previous chapters the formulas for the parabolic first, second Cherncharacters of a locally abelian parabolic bundleE in codimension one and two, chPar1 (E),and chPar2 (E).

Let X be a smooth projective variety over an algebraically closed field of character-istic zero and let D be a strict normal crossings divisor on X . Write D = D1 + ...+Dn

where Di are the irreducible smooth components, meeting transversally. We sometimesdenote by S := 1, . . . , n the set of indices for components of the divisor D.

For i = 1, ..., n, let Σi be finite linearly ordered sets with notations ηi ≤ ... ≤ σ ≤σ′ ≤ σ′′ ≤ ... ≤ τi where ηi is the smallest element of Σi and τi the greatest element ofΣi.

Let Σ′i be the set of connections between the σ’s i.e

Σ′i = (σ, σ′), s.t σ < σ′ and there exist no σ′′ with σ < σ′′ < σ′.

Consider the tread functions m+ : Σ′i → Σi and m− : Σ′i → Σi if λ = (σ, σ′) ∈ Σ′ithen σ = m−(λ), σ′ = m+(λ). In the other direction, consider the riser functionsC+ : Σi−τi → Σ′i and C− : Σi−ηi → Σ′i such that C+(σ) = (σ, σ′) where σ′ > σthe next element and C−(σ) = (σ′′, σ) where σ′′ < σ the next smaller element.

For any parabolic bundle E in codimension one, and two, the parabolic first, secondChern characters chPar1 (E), and chPar2 (E), are obtained as follows:

83

chPar1 (E) := chV b1 (E) −∑i1∈S

∑λi1∈Σ′i1

αi1(λi1).rank(Gri1λi1).[Di1 ]

chPar2 (E) := chV b2 (E) −∑i1∈S

∑λi1∈

∑′i1

αi1(λi1).(ξi1)?

(cDi11 (Gri1λi1

))

+1

2

∑i1∈S

∑λi1∈

∑′i1

α2i1

(λi1).rank(Gri1λi1).[Di1 ]2

+1

2

∑i1 6=i2

∑λi1λi2

∑p∈Irr(Di1∩Di2 )

αi1(λi1).αi2(λi2).rankp(Gri1,i2λi1 ,λi2

).[Dp].

Where:

chV b1 (E), chV b2 (E) denotes the first, second, Chern character of vector bundles E. Irr(DI) denotes the set of the irreducible components ofDI := Di1∩Di2∩...∩Diq . ξI denotes the closed immersion DI −→ X , and ξI,? : Ak(DI) −→ Ak+q(X)

denotes the associated Gysin map. Let p be an element of Irr(Di ∩Dj). Then rankp(GrIλ) denotes the rank of GrIλ

as an Op-module. [Dij ] ∈ A1(X)⊗Q, and [Dp] ∈ A2(X)⊗Q denote the cycle classes given by Dij

and Dp respectively.

Definition 4.1.1. Let Gri1λi1 be a bundle over Di1 . Define the degree of Gri1λi1 to be

deg(Gri1λi1) := (ξi1)?

(cDi11 (Gri1λi1

)).

Definition 4.1.2. The invariant ∆, which is a normalized version of c2 designed to beindependent of tensorization by line bundles. It is defined by

∆ = c2 −r − 1

2rc2

1.

Recall that: ch2 =1

2c2

1 − c2 =⇒ c2 =1

2c2

1 − ch2. Therefore

∆ =1

2c2

1 − ch2 −1

2c2

1 +1

2c2

1 +1

2rc2

1 =1

2rch2

1 − ch2.

84

Then ∆Par(E) =1

2rchPar1 (E)2 − chPar2 (E)

=1

2r

chV b1 (E) −∑i1∈S

∑λi1∈Σ′i1

αi1(λi1).rank(Gri1λi1).[Di1 ]

2

− chV b2 (E) +∑i1∈S

∑λi1∈

∑′i1

αi1(λi1).deg(Gri1λi1)

− 1

2

∑i1∈S

∑λi1∈

∑′i1

α2i1

(λi1).rank(Gri1λi1).[Di1 ]2

− 1

2

∑i1 6=i2

∑λi1λi2

∑p∈Irr(Di1∩Di2 )

αi1(λi1).αi2(λi2).rankp(Gri1,i2λi1 ,λi2

).[Dp].

=1

2rchV b1 (E)2

− 1

r

[chV b1 (E)

]·∑i1∈S

∑λi1∈Σ′i1

αi1(λi1).rank(Gri1λi1).[Di1 ]

+1

2r

∑i1 6=i2

∑λi1λi2

∑p∈Irr(Di1∩Di2 )

αi1(λi1).αi2(λi2).rank(Gri1λi1)rank(Gri2λi2

).[Dp].

+1

2r

∑i1∈S

∑λi1∈Σ′i1

∑λ′i1∈Σ′i1

αi1(λi1).αi1(λ′i1).rank(Gri1λi1).rank(Gri1λ′i1

).[Di1 ]2

− chV b2 (E) +∑i1∈S

∑λi1∈

∑′i1

αi1(λi1).deg(Gri1λi1)

85

− 1

2

∑i1∈S

∑λi1∈

∑′i1

α2i1

(λi1).rank(Gri1λi1).[Di1 ]2

− 1

2

∑i1 6=i2

∑λi1λi2

∑p∈Irr(Di1∩Di2 )

αi1(λi1).αi2(λi2).rankp(Gri1,i2λi1 ,λi2

).[Dp].

Proposition 4.1.3. ∆Par(E) = ∆V b(E)

− 1

rchV b1 (E) ·

∑i1∈S

∑λi1∈Σ′i1

αi1(λi1).rank(Gri1λi1).[Di1 ]

+1

2r

∑i1 6=i2

∑λi1λi2

∑p∈Irr(Di1∩Di2 )

αi1(λi1).αi2(λi2).rank(Gri1λi1)rank(Gri2λi2

)[Dp].

+1

2r

∑i1∈S

∑λi1∈Σ′i1

∑λ′i1∈Σ′i1

αi1(λi1).αi1(λ′i1).rank(Gri1λi1).rank(Gri1λ′i1

).[Di1 ]2

+∑i1∈S

∑λi1∈

∑′i1

αi1(λi1).deg(Gri1λi1)

− 1

2

∑i1∈S

∑λi1∈

∑′i1

α2i1

(λi1).rank(Gri1λi1).[Di1 ]2

− 1

2

∑i1 6=i2

∑λi1λi2

∑y∈Irr(Di1∩Di2 )

αi1(λi1).αi2(λi2).ranky(Gri1,i2λi1 ,λi2

).[Dy].

4.2 Parabolic bundles with full flagsWe use the fact that X is a surface to simplify the above expressions, by assuming thatthe parabolic filtrations are full flags.

86

Proposition 4.2.1. If E ′ is a locally free sheaf over X − P, then ∃! extension to alocally free sheaf E over X s.t E|X−P = E ′.

Proposition 4.2.2. If we have a strict sub-bundle of E|Di − P then ∃! extension to astrict sub-bundle of E|Di.

Remark 13. It follows from these propositions that if (E ′, F ′iαi) is a parabolic structureover (X−P, D−P), then we obtain a bundle E over X with the filtrations F ′iαiof E|Di by a strict sub-bundles.

Definition 4.2.3. Hom(OP1(−mi),O⊕2P1 ) = H0(P1,O(−mi)⊗O)⊕2 = H0(P1,O(mi))

⊕2.

For example, subbundles of a rank two trivial bundle may be expressed very explic-itly.

Proposition 4.2.4. Consider the two polynomials (Ai, Bi) ∈ H0(P1,O(mi))⊕2, the

sub-sheaves are saturated iff mi = ((max(deg(Ai), deg(Bi)) and (Ai, Bi) = 1. Thenthere is an isomorphism

(Ai, Bi) : OP1(−mi) −→ O⊕2P1

Lemma 4.2.5. ∀ 0 ⊆ F iσi⊆ F i

σ′i⊆ .... ⊆ F i

τi⊆ E|Di , ∃ complet flags 0 ⊆ F1 ⊆

... ⊆ Fr = E|Di s.t F iσi

= Fk(σi). where ∀ σi ∈ Σi we have k(i) ∈ 0, 1, ..., r then ∃k : Σi −→ 0, 1, ..., r s.t k = rank(F i

σi).

In view of this lemma, we will now suppose that all the filtrations are completeflags. The weights should then form an increasing sequence but not necessarily strictlyincreasing.

In particular we will change notation and denote the filtration of E|Di by

0 = F i0 ⊆ F i

1 ⊆ .... ⊆ F ir = E|Di .

In this case Σi = σ0i , . . . , σ

ri and Σ′i = λ1

i , . . . , λri. These sets now have the same

number of elements for each i so we can return to a numerical indexation. We denote

Grik(E|Di) := Griλki(E|Di) = F i

k/Fik−1.

Proposition 4.2.6. rank(Griλi) = 1 ⇐⇒ the filtrations F iσ1i≤ F i

σ2i≤ ... ≤ F i

σriare

complet flags for i = 1, 2, ..., n.

Since we are on a surface, Di ∩ Dj is a finite collection of points. At each pointP ∈ Di ∩Dj we have two filtrations of EP coming from the parabolic filtrations alongDi and Dj . We are now assuming that they are both complete flags. The incidencerelationship between these filtrations is therefore encoded by a permutation.

87

Lemma 4.2.7. ∀ k ∃! k′ ∈ 1, ..., r s.t rank(GrikGr

ik′(EP )

)=

F ik ∩ F

jk′

F jk−1 ∩ F

jk′ + F i

k ∩ F ik′−1

=

1.

Definition 4.2.8. ∀ P ∈ Di∩Dj define the permutation σ(P, i, j) : 1, ..., r = Σ′i −→1, ..., r which sends k ∈ 1, ..., r to σ(P, i, j)(k) = k′ where k′ is the unique indexgiven in the previous lemma.

Lemma 4.2.9. ∀ k if k′′ 6= σ(P, i, j)(k) then rank(GrikGr

ik′′(EP )

)= 0.

Since the filtrations are full flags, there are r different indices λ1i , . . . , λ

ri for each

divisor Di. We introduce the notation α(Di, k) := αi(λki ).

With this notation we obtain the following expression for the term involvingGri1,i2λi1 ,λi2:

− 1

2

∑i1 6=i2

∑λi1λi2

∑p∈Irr(Di1∩Di2 )

αi1(λi1).αi2(λi2).ranky(Gri1,i2λi1 ,λi2

).[y].

= − 1

2

∑i 6=j

r∑k=1

∑y∈Irr(Di∩Dj)

α(Di, k).α(Dj, σ(y, i, j)(k)).[y].

On the other hand, all ranks of the graded pieces Gr(Di, k) := Griλkiare equal to 1.

They are line bundles on Di.

Definition 4.2.10. Suppose i 6= 0. Let Gr(Di, k) are line bundles over Di. Then wedefine the degGr(Di, k) to be:

deg (Gr(Di, k)) = (ξi)?(cDi1 (Gr(Di, k))

).

We can now rewrite the statement of Proposition.

Proposition 4.2.11. ∆Par(E) = ∆V b(E)

− 1

rchV b1 (E) ·

∑i∈S

r∑k=1

α(Di, k).[Di]

88

+1

2r

∑i 6=j

∑k,l∈[1,r]

∑y∈Irr(Di∩Dj)

α(Di, k).α(Dj, l)[y].

+1

2r

∑i∈S

∑k,l∈[1,r]

α(Di, k).α(Di, l).[Di]2

+∑i∈S

r∑k=1

α(Di, k).deg (Gr(Di, k))

− 1

2

∑i∈S

r∑k=1

α(Di, k)2.[Di]2

− 1

2

∑i 6=j

r∑k=1

∑y∈Irr(Di∩Dj)

α(Di, k).α(Dj, σ(y, i, j)(k)).[y].

We have α(Di, k) ∈ [−1, 0], define αtot(Di) :=r∑

k=1

α(Di, k). With this notation,

∆Par(E) = ∆V b(E)

− 1

rchV b1 (E) ·

∑i∈S

αtot(Di)[Di]

+1

2r

∑i 6=j

αtot(Di)αtot(Dj)[Di ∩Dj]

+1

2r

∑i∈S

αtot(Di)2.[Di]

2

+∑i∈S

r∑k=1

α(Di, k).

− 1

2

∑i∈S

r∑k=1

α(Di, k)2.[Di]2

89

− 1

2

∑i 6=j

r∑k=1

∑y∈Irr(Di∩Dj)

α(Di, k).α(Dj, σ(y, i, j)(k)).[y].

For more simplification of ∆Par(E), define β such that

β(Di, k) := α(Di, k)− αtot(Di)

r=⇒ α(Di, k) = β(Di, k) +

αtot(Di)

r.

Remark 14. We remark thatr∑

k=1

β(Di, k) = 0. Hence

r∑k=1

α(Di, k)2 =r∑

k=1

β(Di, k)2 +αtot(Di)

2

r,

and for i 6= j and y ∈ Irr(Di ∩Dj),

r∑k=1

α(Di, k).α(Dj, σ(y, i, j)(k)) =r∑

k=1

α(Di, k).α(Dj, σ(y, i, j)(k))+αtot(Di).α

tot(Dj)

r.

Furthermore note thatr∑

k=1

(ξi)?(cDi1 (Gr(Di, k))

)= (ξi)?

(cDi1 (E|Di)

)= cV b1 (E).[Di]

sor∑

k=1

α(Di, k).deg (Gr(Di, k)) =r∑

k=1

β(Di, k).deg (Gr(Di, k)) +αtot(Di)c

V b1 (E).[Di]

r.

Using these remarks and the previous formula we get

∆Par(E) = ∆V b(E)

− 1

rchV b1 (E) ·

∑i∈S

αtot(Di)[Di]

90

+1

2r

∑i 6=j

αtot(Di)αtot(Dj)[Di ∩Dj]

+1

2r

∑i∈S

αtot(Di)2.[Di]

2

+∑i∈S

r∑k=1

β(Di, k).deg (Gr(Di, k))

+1

r

∑i∈S

αtot(Di).cV b1 (E).[Di]

− 1

2

∑i∈S

r∑k=1

β(Di, k)2.[Di]2

− 1

2r

∑i∈S

αtot(Di)2.[Di]

2

− 1

2

∑i 6=j

r∑k=1

∑y∈Irr(Di∩Dj)

β(Di, k).β(Dj, σ(y, i, j)(k)).[y].

− 1

2r

∑i 6=j

r∑k=1

∑y∈Irr(Di∩Dj)

αtot(Di).αtot(Dj).[y].

The terms containing αtot(Di) all cancel out, giving the following formula.

Proposition 4.2.12. ∆Par(E) = ∆V b(E)

+∑i∈S

r∑k=1

β(Di, k).deg (Gr(Di, k))

− 1

2

∑i∈S

r∑k=1

β(Di, k)2.[Di]2

− 1

2

∑i 6=j

r∑k=1

∑y∈Irr(Di∩Dj)

β(Di, k).β(Dj, σ(y, i, j)(k)).[y].

91

The fact that ∆Par(E) is independent of αtot is the parabolic version of the in-variance of ∆ under tensoring with line bundles. Even though this is the theoreticalexplanation, for the proof it was more convenient to calculate explicitly the formula andnotice that the terms containing αtot cancel out, than to try to compute the tensor productwith a parabolic line bundle.

4.3 Resolution of singular divisorsNow we can consider a more general situation, where X is a smooth projective surface

but D =n⋃i=1

Di is a divisor which may have singularities worse than normal crossings.

Let P = P1, ..., Pr be a set of points. Assume that the points Pj are crossing pointsof Di, and that they are general multiple points, that is through a crossing point Pj wehave divisors Di1 , ..., Dim which are pairwise transverse. Assume that D has normalcrossings outside of the set of points P . We choose an embedded resolution given bya sequence of blowing-ups ϕ : X → X in r points P1, ..., Pr and P be the exceptionaldivisor on X , note that P is a sum of disjoint exceptional components Pi = ϕ−1(Pi)over the points Pi respectively. The pullback divisor may be written as D = D1 + · · ·+Da + P1 + · · · + Pb where Di is the strict transform of a component Di of the originaldivisor, and Pj are the exceptional divisors.

Definition 4.3.1. Let E be a bundle over X , and consider the inclusion i : U → X

where U = X −k⋃i=1

Pi be a smooth connected quasi-projective surface. Hence Pi = P1

and let the blowing-up ϕ : X −→ X . Define E as a unique bundle over X such that

E|U ∼= E|U ,

E is locally free.

This construction allows us to localize the contributions of the Chern classes of Ealong the exceptional divisors, by comparison with ϕ∗(E).

Definition 4.3.2. Let E be a bundle over X . Consider the inclusions ϕ?E → i?(E|U),where i?(E|U) is a quasi-coherent sheaves over X , and E → i?(E|U), where i : U →X . Define E ′′ to be the intersection of subsheaves ϕ?E and E of i?(E|U).

Lemma 4.3.3. E ′′ is a free locally coherent sheaf.

Definition 4.3.4. Consider the two exact sequences

0 −→ E ′′ −→ ϕ?E −→ Q′ −→ 0

92

0 −→ E ′′ −→ E −→ Q −→ 0

Let E/E ′′ = Q =k⊕i=1

Qi and ϕ?E/E ′′ = Q′ =k⊕i=1

Q′i. Define the local contribution

to be,chV b(E,P )loc := chV b(Q)− chV b(Q′)

Proposition 4.3.5. If ϕ : X −→ X . Let Pi the blowing-up of Pi where Pi is theexceptional divisor for i = 1, 2, ...k. Then

chV b(E) = chV b(ϕ?(E)) +k∑i=1

chV b(E,Pi)loc

We have chV b1 (E) ∈ A1(X). Let ϕ? : A1(X) −→ A1(X); where

A1(X) = A1(X)⊕k⊕i=1

.Z.[Pi]. We have

Pi.ϕ?(D) = 0 if D ∈ A1(X) and Pi.Pj = 0 if i 6= j. Then

chV b1 (E) = ϕ?chV b1 (E) +k∑i=1

ai[Pi] = ϕ?chV b1 (E) +k∑i=1

chV b1 (E,Pi)loc.

When we take the square, the cross-terms are zero, indeed chV b1 (E,Pi)loc is a mul-tiple of the divisor class [Pi] but [Pi].[Pj] = 0 for i 6= j, and [Pi].ϕ

∗[C] = 0 for anydivisor C on X . Therefore,

chV b1 (E)2 = ϕ?chV b1 (E)2 +k∑i=1

a2i [Pi]

2 = ϕ?chV b1 (E)2 +k∑i=1

chV b1 (E,Pi)2loc.

Lemma 4.3.6. If L is a line bundle over X , then

∆(E ⊗ L) = ∆(E).

4.4 Local Bogomolov-Gieseker inequalityThe classical Bogomolov-Gieseker inequality states that if X is projective and E is asemistable vector bundle then ∆(E) ≥ 0. We will see that a local version holds; the firstobservation is that the invariant ∆ can be localized, even though it involves a quadraticterm in ch1.

Definition 4.4.1.

∆V b(E,Pi)loc :=1

2rchV b1 (E,Pi)

2loc − chV b2 (E,Pi)loc

93

Lemma 4.4.2. If L = ϕ?L(∑

biPj) is a line bundle over X . Then

∆V b(E ⊗ L;Pi)loc = ∆V b(E,Pi)loc

Proposition 4.4.3.

∆V b(E) = ϕ?∆V b(E) +k∑i=1

∆V b(E,Pi)loc

In order to get a bound, the technique is to apply the Grothendieck decompositionto analyse more closely the structure of E near the exceptional divisors Pi, followingBallico [Ba] and Ballico-Gasparim [BG1] [BG2] and others.

Theorem 4.4.4. Every vector bundleE on P1 is of the formO(m1)r1⊕· · ·⊕O(ma)ra =

a⊕j=1

O(mj)rj , m1 < . . . < mr where mj ∈ Z, and the rj are positive integers with

r1 + . . .+ ra = r. This called the Grothendick decomposition and it is unique.

Apply this decomposition to the restriction of the bundle E to each exceptionaldivisor Pi ∼= P1. Thus

E|Pi = O(mi,1)ri,1 ⊕ ...⊕O(mi,ai)ri,ai =

ai⊕j=1

O(mi,j)ri,j

with mi,1 < . . . < mi,ai .

Proposition 4.4.5. Let E be a bundle over X , we have,

mi,j = 0⇐⇒ E ∼= ϕ?E ,

if E ′ = E(∑i

ki.Pi) then m′i,j = mi,j − ki, therefore

mi,j = ki ⇐⇒ E ∼= (ϕ?E)(−∑i

ki.Pi).

In this case we say that E is pure, it is equivalent to saying that ai = 1.

See Ballico-Gasparim [BG1].

Definition 4.4.6. Let E be a non trivial bundle, and E|P = O(m1)r1 ⊕ ... ⊕ O(ma)ra

be the restriction of the bundle E, for m1 < m2 < ... < ma. We define

min(E|P ) := m1 ,max(E|P ) := ma, and ϕ(E) = max(E|P )−min(E|P ).

94

Remark 15. If µ(E) = ma −m1 = 0. Then E|P = OP1(m1)r ; E = E∨(−m.P )

Lemma 4.4.7. If we have an exact sequence of bundles over P1

0 −→ U −→ V −→ W −→ 0

thenmin(V ) ≥ min(min(U),min(W )),

max(V ) ≤ max(max(U),max(W )).

Proof. Define

max(U) = maxn; s.t ∃ OP1(n)→ U nontrivial = maxn; s.t H0(U(−n)) 6= 0

min(U) = minn; s.t ∃ U → OPn(n)nontrivial = maxn; s.t H0(U∗(n)) 6= 0then

max(U) ≤ max(max(U),max(W ))

min(V ) ≥ min(min(U),min(W ))

Now we concentrate on one of the exceptional divisors Pi and supress the index ifrom the notation.

Now for 1 ≤ t ≤ r, suppose that E|P is not pure, and consider the exact sequence

0↑Q := O(m1)r1

↑E|Pi := O(m1)r1 ⊕O(m2)r2 ⊕ · · · ⊕ O(ma)

ra

↑K := O(m2)r2 ⊕ · · · ⊕ O(ma)

ra

↑0

Definition 4.4.8. Suppose X and D are smooth with Di∗→ X . Let E be a free locally

bundle over X . Suppose we have an exact sequence

0 −→ K −→ E|D −→ Q −→ 0

where Q = O(m1)r1 is called constant stabilizer. Define E ′ to be the elementary trans-formation of E by

E ′ := Ker(E → i∗Q).

Then the sequence0 −→ E ′ −→ E −→ i∗Q −→ 0.

is exact.

95

Lemma 4.4.9. We have an exact sequence

0 −→ Q(−D) −→ E ′ |P−→ K −→ 0.

ThenE ′(U) = S ∈ E(U)s.tS|(D∩U) ∈ K(D ∩ U).

Lemma 4.4.10. µ(E ′) ≤ µ(E)− 1 (if µ(E) ≥ 1).

Proof. We haveOP (−P ) = OP (i)

apply the Lemma 4.4.9 we get

0 −→ O(m1 + 1)r1 −→ E ′|P −→a⊕i=2

O(mi)ri −→ 0

apply Lemma 4.4.7, take min = max = m1 + 1 and min = m2 then max ≥ m1 + 1=⇒min(E ′ |P ) ≤ ma. Therefore

µ(E ′) ≤ µ(E)− 1.

Lemma 4.4.11.ch (OP (m1)) = (P + (m1 +

1

2)).

Proof. We haveO(m1) = O(−m1P ),

consider the exact sequence

0 −→ OX(−(m1 + 1)P ) −→ OX(−m1P ) −→ OP (−m1P ) −→ 0

then

ch(OP (−m1P )) = e−m1P −e(−m1+1)P = (1−m1P +m2

1

2P 2)− (P −m1P

2− P2

2)

= (P − (m1 +1

2)P 2), but P 2 = −1, then ch(OP (−m1)) = (P + (m1 +

1

2)).

Proposition 4.4.12. We have

chV b1 (E ′) = chV b1 (E)− r1P and chV b2 (E ′) = chV b2 (E)− (m1 +1

2)r1,

96

then∆V b(E ′, P ) =

1

2r(chV b1 (E)− r1P )2 − chV b2 (E) + (m1 +

1

2)r1,

therefore

∆V b(E ′, P ) = ∆V b(E,P )− r1

rchV b1 (E).P − r2

1

2r+ (m1 +

1

2)r1.

We can now calculate using the previous lemma.

E|P = O(m1)r1 ⊕O(m2)r2 ⊕ ...⊕O(mk)rk , where

k∑i=1

ri = r, we have

ch1(E).P = ξP,? (ch1(E |P )) = ch1

(k⊕i=1

O(mi)

)=

k∑i=1

miri, then

∆V b(E ′, P ) = ∆V b(E,P )− r1

r

k∑i=1

miri −r2

1

2r+m1r1 +

r1

2= ∆(E)− 1

rA, where

A =k∑i=2

mirir1 +m1r21 +

1

2r2

1 − (m1 +1

2)r1r for r = r1 + ....+ rk

=k∑i=2

mirir1 +m1r21 +

1

2r2

1 − (m1 +1

2)r2

1 −k∑i=2

(m1 +1

2)r1ri, then

A =k∑i=2

(mi −m1 −1

2)r1ri where mi > m1 + 1

Note that, with our hypothesis that E |P is not pure, we have mi ≥ m1 + 1 so A > 0.

Proposition 4.4.13. If E|P is not pure, then let E ′ be the elementary transformationconsidered above. The local invariant satisfies

∆V bloc(E

′, P ) = ∆V bloc(E,P )− 1

r

k∑i=2

(mi −m1 −1

2)r1ri where mi > m1 + 1.

In particular, ∆V bloc(E

′, P ) < ∆V bloc(E,P ).

IfE ′ is pure then ∆V bloc(E

′, P ) = 0, if not we can continue by applying the elementarytransformation process to E ′ and so on, until the result is pure. The resulting theoremcan be viewed as a local analogue of the Bogomolov-Gieseker inequality.

97

Theorem 4.4.14. If E is a vector bundle on X and P ∼= P1 ⊂ X is the exceptionaldivisor of blowing up a smooth point P ∈ X , then ∆V b

loc(E,P ) ≥ 0, and ∆V bloc(E,P ) = 0

if and only if E ∼= µ∗(E) is the pullback of a bundle from X .

The invariant ∆V bloc(X,P ) also provides a bound for mi −m1.

Corollary 4.4.15. If E |P=k⊕i=1

O(mi)ri , where m1 < m2 < ... < mk. Then

∆V bloc(E

′, P ) ≥ 0; ∆V bloc(E,P ) ≥ 1

r

k∑i=2

(mi −m1 −1

2)r1ri.

4.5 Modification of filtrations due to elementary trans-formations

Given two bundles E and F such that E|U ∼= F |U , then F may be obtained from E bya sequence of elementary transformations. We therefore analyse what happens to thefiltrations along the divisor components Di different from exceptional divisors Pu, inthe case of an elementary transformation.

Suppose E ′ is obtained from E by an elementary transformation.We have bundles Gr(Di, k;E) and Gr(Di, k;E ′) over Di. In order to follow the

modification of the formula for ∆ we need to consider this change.For the bundle E we have a filtration by full flags F i

k ⊂ E|Di . Suppose E|D0 → Qis a quotient (locally free on D0) and let E ′ be the elementary transformation fitting intothe exact sequence

0→ E ′ → E → Q→ 0.

Lemma 4.5.1. Suppose i 6= 0 so Di ∩D0 is transverse. Tensoring this exact sequencewith ODi yields an exact sequence

0→ E ′|Di → E|Di → Q|Di → 0.

Proof. In fact we get a long exact sequence

Tor1OX (ODi , Q)→ E ′ ⊗ODi → E ⊗ODi → Q⊗ODi → 0,

but the facts thatQ is locally free onD0 andDi is transverse toD0 imply that Tor1OX (ODi , Q) =

0.

This lemma says that E ′|Di is an elementary transformation of E|Di . Notice thatsince Di is a curve, Q|Di is a skyscraper sheaf.

98

Define F ′ik := F ik ∩ (E ′|Di). It is a subsheaf of (E ′|Di). Furthermore it is saturated,

that is to say the quotient is torsion-free. To show this, suppose s is a section of (E ′|Di)which is contained in F ′ik over an open set. Then it may be seen as a section of E|Diwhich is contained in F i

k over an open set, but F ik is saturated so the section is contained

in F ik. Hence by definition the section is contained in F ′ik .Thus, we have defined a filtration F ′ik by sub-vector bundles. The same argument

says that F ′ik is saturated in F ′ik+1, so the quotients Gr(Di, k;E ′) = F ′ik+1/F′ik are locally

free; since they are line bundles over the open set, they are line bundles on Di.Consider the morphism, induced by (E ′|Di)→ (E|Di):

F ′ik+1/F′ik → F i

k+1/Fik.

By the definition of F ′ik it is seen that this morphism is an injection of sheaves. Considerthe cokernel. If s is a section of F i

k+1/Fik and if z is a local coordinate on Di such that

z = 0 defines the intersection point D0∩Di, then we claim that zs must be in the imageof F ′ik+1/F

′ik . Lift s to a section also denoted s of F i

k+1. Then, thought of as a sectionof E|Di , notice that zs projects to 0 in Q|Di . This may be seen by further extending toa section of E and extending z to a coordinate function defining D0; noting that Q issupported scheme-theoretically on D0 so zs projects to 0 in Q.

From the exact sequence, we conclude that zs is in the image of F ′ik+1/F′ik . Hence,

there are two cases:(1) the map F ′ik+1/F

′ik → F i

k+1/Fik is an isomorphism; or

(2) we have F ′ik+1/F′ik = F i

k+1/Fik ⊗ODi ODi(−D0 ∩Di).

In the first case (1),

cDi1 (Gr(Di, k;E ′)) = cDi1 (Gr(Di, k;E)).

In the second case (2),

cDi1 (Gr(Di, k;E ′)) = cDi1 (Gr(Di, k;E))− [D0 ∩Di].

Applying (ξi)∗ gives the following proposition.

Proposition 4.5.2. Suppose E ′ is an elementary transformation of E. Then there exista unique invariant degloc that satisfy the following properties:

degloc(Dj, k; E

):= 0,

degloc(Dj, k; E(m.Pi)

):= m,

and for divisor components Di intersecting D0 transversally, the change in Chern classof the associated-graded pieces is

degloc (Gr(Dj, k;E ′), Pi) := degloc (Gr(Dj, k;E), Pi)− τ(E,E ′; k)

where τ(E,E ′; k) = 0 or 1 in cases (1) or (2) respectively.

99

Definition 4.5.3. Let S and S are two sheaves of finite length with support at points Pi.Suppose We have bundles Gr(Dj, k, E) and Gr(Dj, k, E) over Dj respectively Dj . LetF be the intersection of subsheaves Gr(Dj, k, E) and Gr(Dj, k, E). Define lg to be thelength, and let lg(S, Pi) be the length of the part supported set-theoretically at Pi. Thus

lg(S) =∑i

lg(S, Pi)

and similarly for S. Consider the sequences:

F −→ Gr(Dj, k, E) −→ S −→ 0

F −→ Gr(Dj, k, E) −→ S −→ 0.

Definedegloc (Gr(Dj, k;E,Pi)) := lg(S, Pi)− lg(S, Pi).

Thendeg (Gr(Dj, k, E)) = deg(F ) + lg(S)

deg(Gr(Dj, k, E)

)= deg(F ) + lg(S),

therefore

lg(S)− lg(S) =∑Pi

[lg(S, Pi)− lg(S, Pi)

]=∑Pi

degloc (Gr(Dj, k;E,Pi)) .

Suppose i 6= 0, if Dj are non-exceptional divisor. Then

deg (Gr(Dj, k;E) = deg(Gr(Dj, k; E

)+∑Pi

degloc (Gr(Dj, k, E), Pi)) .

This completes the proof of the proposition.

4.6 The local parabolic invariantLet E be a bundle, with β(D0, k) = 0, ∀k. Then we would like to define the terms inthe following equation:

∆Par(E) = ∆Par(E) +∑Pi

∆Parloc (E,Pi).

Assume that D is a union of smooth divisors meeting in some multiple points. Thedivisor D is obtained by blowing up the points Pu of multiplicity ≥ 3. Let

ϕ : X → X

100

be the birational transformation. We use the previous formula to break down ∆Par(E)into a global contribution which depends only on E, plus a sum of local contributionsdepending on the choice of extension of the parabolic structure across Pu.

Let S denote the set of divisor components in D (before blowing-up) and define theglobal term ∆Par(E) by the formula

∆Par(E) := ∆V b(E)

+∑i∈S

r∑k=1

β(Di, k).deg(Gr(Di, k)

)

− 1

2

∑i∈S

r∑k=1

β(Di, k)2.[Di]2

− 1

2

∑i 6=j

r∑k=1

∑y∈Irr(Di∩Dj)

β(Di, k).β(Dj, σ(y, i, j)(k)).[y].

This formula imitates the formula for ∆Par by considering only pairwise intersectionsof divisor components even though several different pairwise intersections could occurat the same point. Recall that [Di]

2 = [Di]2−m where m is the number of points on Di

which are blown up to pass to Di.To define the local terms, suppose at least one of the divisors, say D0 = P , is the

exceptional locus for a birational transformation blowing up the point P . We define a lo-cal contribution ∆Par

loc (E,P ) to ∆Par by isolating the local contributions in the previousformula.

Notice first of all that for any Di meeting P transversally, we have defined abovedegloc (Gr(E;Di, k), P ), the local contribution at P , in such a way that

deg (Gr(E;Di, k)) = deg(Gr(E;Di, k)

)+∑Pu

degloc (Gr(E;Di, k), Pu)

where the sum is over the exceptional divisors Pu meeting Di, which correspond to thepoints Pu ∈ Di which are blown up.

Let S(P ) denote the set of divisor components which meet P but not includingP = D0 itself. Define

101

∆Parloc (E,P ) := ∆V b

loc(E,P )

+r∑

k=1

β(P, k).deg (Gr(E;P, k))

+∑i∈S(P )

r∑k=1

β(Di, k).degloc (Gr(E;Di, k), P )

+1

2

r∑k=1

β(P, k)2

+1

2

∑i∈S(P )

r∑k=1

β(Di, k)2

−∑i∈S(P )

r∑k=1

β(Di, k).β(P, σ(i, P )(k)).[y]

+1

2

∑i 6=j,P∈Di∩Dj

r∑k=1

β(Di, k).β(Dj, σ(P , i, j)(k)).[P ]

In the next to last term, σ(i, P ) := σ(y, i, v) where P = Dv and y is the uniqueintersection point of P = Dv and Di. The factor of 1/2 disappears because we areimplicitly choosing an ordering of the indices i, j = 0 which occur here. The last termis put in to cancel with the corresponding term in the global expression for E above,and [P ] designates any lifting of the point P to a point on P .

Theorem 4.6.1. With the above definitions, we have

∆Par(E) = ∆Par(E) +∑Pu

∆Parloc (E,Pu),

where the sum is over the exceptional divisors.

Proof. This follows by comparing the above definitions with the formula of Proposition4.2.12.

102

Let ϕ∗E denote the parabolic bundle on X given by using the trivial extension ϕ∗Eas underlying vector bundle, and setting β(Pu, k) := 0 for all exceptional divisor com-ponents Pu. Note that ∆V b

loc(ϕ∗E,P ) = 0. Then

∆Parloc (ϕ∗E, P ) =

1

2

∑i∈S(P )

r∑k=1

β(Di, k)2

+1

2

∑i 6=j,P∈Di∩Dj

r∑k=1

β(Di, k).β(Dj, σ(P , i, j)(k)).[P ],

and

∆Parloc (E,P )−∆Par

loc (ϕ∗E, P ) = ∆V bloc(E,P )

+r∑

k=1

β(P, k).deg (Gr(E;P, k))

+∑i∈S(P )

r∑k=1

β(Di, k).degloc (Gr(E;Di, k), P )

+1

2

r∑k=1

β(P, k)2

−∑i∈S(P )

r∑k=1

β(Di, k).β(P, σ(i, P )(k)).[y].

A different local-global decomposition may be obtained by noting that

∆Par(E) = ∆Par(ϕ∗E) +∑u

(∆Parloc (E,Pu)−∆Par

loc (ϕ∗E, Pu))

with the local terms (∆Parloc (E,Pu) −∆Par

loc (ϕ∗E, Pu)) being given by the previous for-mula.

103

4.7 Normalization via standard elementary transforma-tions

There is another modification of parabolic structures due to elementary transformations.This may also be viewed as a shift of the parabolic structures in the viewpoint of acollection of sheaves. If E· = Eα1,...,αn is a parabolic sheaf, then we can shift thefiltration at the i-th place defined by

(CiθE)α1,...,αn := Eα1,...,αi−θ,...,αn .

This may also be viewed as tensoring with a parabolic line bundle

CiE = E ⊗O(θDi).

The weights of the parabolic structure CiθE along Di are of the form αi + θ for αi

weights of E.In the point of view of a vector bundle with filtration, it may correspond to doing an

elementary transformation. Suppose 0 < θ < 1. Then

(CiθE)0 = E0,...,0,−θ,0,...,0

and we have an exact sequence

0→ (CiθE)0 → E0 → (E0/F

i−θE0)→ 0.

Therefore (CiθE)0 is obtained by elementary transformation of E0 along one of the

elements of the parabolic filtration on the divisor Di.This is specially useful when the rank is 2. Suppose rk(E) = 2. There is a single

choice for the elementary transformation. If the weights of E at Di are αtoti − βi andαtoti +βi then the weights of the elementary transformation will be θ+αtoti +βi− 1 andθ+αtoti − βi. The shift θ should be chosen so that these lie in (−1, 0]. The new weightsmay be written as

(αtoti − βi, αtoti + βi

withαtoti := (θ + αtoti −

1

2)

which is the new average value, and

βi :=1

2− βi.

Corollary 4.7.1. In the case rk(E) = 2, by replacing E with its shift CiθE if necessary,

we may assume that

0 ≤ βi ≤1

4.

104

Proof. If βi >1

4then do the shift which corresponds to an elementary transformation;

for the new parabolic structure βi =1

2− βi and 0 ≤ βi ≤

1

4.

4.8 The rank two caseIn order to simplify the further constructions and computations, we now restrict to thecase when E has rank 2. The parabolic structures along Di are rank one subbundlesF i ⊂ E|Di . The associated graded pieces are Gr(Di, 1) = F i and Gr(Di, 2) =E|Di/F i. The normalized weights may be written as

β(Di, 1) = −βi, β(Di, 2) = βi

with 0 ≤ βi <1

2, and by Corollary 4.7.1 we may furthermore suppose 0 ≤ βi ≤

1

4.

Define degδ(EDi , Fi) :=

(deg(EDi/F

i)− deg(F i)). This has a local version as

discussed in Definition 4.5.3,

degδloc(EDi , Fi, P ) :=

(degloc(EDi/F

i, P )− degloc(Fi, P )

)whenever Di meets P transversally.

The main formula may now be rewritten:

Proposition 4.8.1. ∆Par(E) = ∆V b(E)

+∑i∈S

βi degδ(EDi , Fi)

−∑i∈S

β2i .[Di]

2

−∑i 6=j

∑y∈Irr(Di∩Dj)

τ(y, i, j)βiβj.[y]

where τ(y, i, j) = 1 if F i(y) = F j(y) and τ(y, i, j) = −1 if F i(y) 6= F j(y) as sub-spaces of E(y).

Similarly for the local parabolic invariants, denoting P = D0 we have

∆Parloc (E,P )−∆Par

loc (ϕ∗E, P ) = ∆V bloc(E,P )

105

+ β0. degδ(ED0 , F0)

+∑i∈S(P )

βi. degδloc(EDi , Fi, P )

+ β20

− 2∑i∈S(P )

τ(i, P )βiβ0.[y].

Example 4.8.2. LetE be a non pure rank two bundle, we haveE |P= O(m1)⊕O(m2),

then ∆V bloc(E,P ) ≥ 1

2(m2 −m1 −

1

2).

Example 4.8.3. Let E be a rank 2 bundle with E|P = O ⊕ O(1). The reduction byelementary transformation is E|P = O ⊕O(1) E ′. We get an exact sequence

0 −→ O(1) −→ E ′|P −→ O(1) −→ 0

then E ′ is pure, E ′ = µ?E(−P ).

Suppose we start with the bundle E, by doing the sequence of elementary transfor-mation we get E(m.Pi). Number the sequence in opposite direction, we get a sequenceof bundles of the form:

E(m.Pi) = E(0), E(1), E(2), ..., E(g) = E

where g is the number of steps, and E(j − 1) = (E(j))′ for j = 1, ..., g. we recall thatif E|P ∼= O(m1)⊕O(m2) with m1 ≤ m2 then µ(E) = m2 −m1. Also µ(E) = 0 =⇒E = E(miPi). Furthermore if m1 < m2 then µ(E ′) < µ(E). We see that

0 = µ(E0) < µ(E1) < µ(E2) < ... < µ(Eg−1).

To calculate ∆(E,P )loc we use the proposition 4.4.13 applied to each E(j):

∆V bloc(E(j)′, P0) = ∆V b

loc(E(j), P0)− 1

2A,

where

A =k∑i=2

(m2 −m1 −1

2)r1ri = m2(E(j))−m1(E(j))− 1

2= µ(E(j))− 1

2.

106

Therefore∆V bloc(E(j − 1)) = ∆V b

loc(E(j))− 1

2(µ(E(j))− 1

2),

and putting them all together,

∆V bloc(E(0)) = 0; ∆V b

loc(E(g)) =1

2

g∑i=1

(µ(E(j))− 1

2).

We haveµ(E(j − 1)) < µ((E(j)), (4.8.1)

so µ(E(j)) ≥ j.

Now we divide the work in two parts, first term µ(E(g)) − 1

2, then the sum of the

others. For each µ(E(j)) is at least one greater than the previous one, this gives that the

sum of the other terms is at least equal to (1 + 2 + 3 + ...+ (g − 1))− (g − 1)

2. Then

∆V bloc(E(g)) ≥ 1

2(1 + 2 + 3 + ...+ (g − 1))− g

4+

1

2µ(E(g))

=g(g − 1)

4− g

4+

1

2µ(E).

We have therefore proven the following:

Proposition 4.8.4. If E is a bundle which is brought to pure form in g ≥ 1 steps ofelementary transformation, and µ(E) = m2(E|P )−m1(E|P ), then we have the lowerbound

∆V bloc(E) = ∆V b

loc(E(g)) ≥ g2 − 2g

4+

1

2µ(E).

We have for each 1 ≤ k ≤ g,∣∣degδloc(E(k), F i, P )− degδloc(E(k − 1), F i, P )∣∣ ≤ 1,

but also E(0) = ϕ∗(E) and degδloc(E(0), F i, P ) = 0, so∣∣degδloc(E(g), F i, P )∣∣ ≤ g.

Also along P we have EP = O(m1) ⊕ O(m2) with m1 ≤ m2. For any subbundleF 0 ⊂ EP we have deg(F 0) ≤ m2 and deg(EP/F

0) ≥ m1 so

degδ(EP , F0) ≥ m1 −m2 = −µ(E).

Then

107

∆Parloc (E,P )−∆Par

loc (ϕ∗E, P ) = ∆V bloc(E,P )

+ β0. degδ(ED0 , F0)

+∑i∈S(P )

βi. degδloc(EDi , Fi, P )

+ β20

− 2∑i∈S(P )

τ(i, P )βiβ0.[y]

≥ g2 − 2g

4+

1

2µ(E)

− β0µ(E)

−∑i∈S(P )

βig

+ β20

− 2∑i∈S(P )

βiβ0.

But we know that | βi |≤1

2. Then we get the following theorem.

Theorem 4.8.5.

∆Parloc (E;P ) ≥ ∆Par

loc (ϕ∗E, P ) +g2 − 2g

4− g + 1

2.κ.

Where κ = #S(P ) is the number of divisors of Di meeting P .

Theorem 4.8.6. If E is a vector bundle of rank 2 on X with parabolic structures onthe components Di, then on X obtained by blowing up the multiple points of D, theparabolic invariant ∆Par

loc (E,P ) attains a minimum for some extension of the bundle Eand some parabolic structures on the exceptional loci.

108

Proof. From the above theorem, the number of elementary transformations g needed toget to any E with ∆Par

loc (E;P ) ≤ ∆Parloc (ϕ∗E, P ), is bounded. Furthermore the number

of numerical possibilities for the degrees degδloc(EDi , Fi, P ) and degδ(EP , F

0, P ) lead-ing to such a minimum, is finite. The parabolic weight β0 may be chosen to lie in the

closed interval [0,1

4], so the set of possible numerical values lies in a compact subset;

hence a minimum is attained.

Denote the parabolic extension which achieves the minimum by Emin. There mightbe several possibilities, although we conjecture that usually it is unique. Thus

∆Parloc (Emin, P ) = minE

(∆Parloc (E,P )

).

With the minimum taken over all parabolic extensions E of E|U across the exceptionaldivisor P .

The minimal Emin exists at each exceptional divisor and they fit together to give aglobal parabolic bundle. Define

∆Parmin(E) := ∆Par(Emin)

= ∆Par(E) +∑Pu

∆Parloc (Emin, Pu).

4.8.1 Panov differentiationD. Panov in his thesis [Pa] used the idea of differentiation with respect to the parabolicweight. A version of this technique allows us to gain more precise information on theminimum.

Lemma 4.8.7. Let E = Emin be the parabolic bundle extending E|U which achievesthe minimum value ∆Par

loc (Emin, P ). By making an elementary transformation we may

assume 0 ≤ β0 ≤1

4. Denote also by E the underlying vector bundle. Then for any

subbundle F ′ ⊂ E|P we have

deg(E|P/F ′)− deg(F ′) ≥ −κ.

Thus if E|P = OP (m1)⊕OP (m2) then

|m2 −m1| ≤ κ.

Where κ = #S(P ) is the number of divisors of Di meeting P .

109

Proof. We show that deg(E|P/F ′) − deg(F ′) ≥ −κ − 1

4which implies the stated

inequality since the left side and κ are integers. Let F = F 0 ⊂ E|P be the subbundlecorresponding to the parabolic structure Emin. Consider two cases:(i) if F 0 is the destabilizing bundle of E|P and β0 > 0; or(ii) if F 0 is not the destabilizing bundle of E|P , or else β0 = 0.

In case (i) note that β0 may be allowed to range in the full interval [0,1

2) so the

invariant ∆Parloc (E,P ) is a local minimum considered as a function of β0 ∈ (0,

1

2). Then

d

dβ0

∆Parloc (E,P ) = 0.

This gives the formula

degδ(ED0 , F0) = 2 β0 + 2

∑i∈S(P )

τ(F i, F 0)βi, so

degδ(ED0 , F0) ≥ −κ

2.

Since F 0 is the destabilizing bundle it implies that

degδ(ED0 , F′) ≥ −κ

2for any other subbundle F ′ also, which is stronger than the desired inequality in thiscase.

In case (ii) we have β0. degδ(ED0 , F0) ≥ 0 because in the contrary case that would

imply that F 0 is the destabilizing subbundle. Suppose F ′ ⊂ E|P is a possibly differentsubbundle such that

deg(E|P/F ′)− deg(F ′) < −1

4(1 + 4κ).

Then make a new parabolic structure E ′ using F ′ instead of F , with parabolic weight

β′0 =1

4. We have

∆Parloc (E ′, P )−∆Par

loc (E,P ) =

1

4degδ(ED0 , F

′)

110

− β0. degδ(ED0 , F0)

+1

16

− β20

+ 2∑i∈S(P )

τ(F, F i)βiβ0.[y]

− 1

2

∑i∈S(P )

τ(F ′, F i)βi.[y]

≤ 1

4degδ(ED0 , F

′) +1

16(1 + 4κ)

< 0.This contradicts minimality of Emin, which shows the desired inequality.

Corollary 4.8.8. In the case of 3 divisor components κ = 3 and the minimal extensionEmin satisfies |m2 − m1| ≤ 3. It is connected to ϕ∗(E) by at most three elementarytransformations.

This should permit an explicit description of all possible cases for κ = 3, we starton this below.

4.8.2 The Bogomolov-Gieseker inequalitySuppose C ⊂ X is an ample curve meeting D transversally. Then E|C is a parabolicbundle on C.

Proposition 4.8.9. Suppose E|C is a stable parabolic bundle. Then for any extensionE to a parabolic bundle over X , there exists an ample divisor H on X such that Eis H-stable. Hence ∆Par(E) ≥ 0. In particular ∆Par(Emin) ≥ 0. If E comes froman irreducible unitary representation of π1(X − D) then the parabolic extension on Xcorresponding to the same unitary representation must be some choice of Emin.

Proof. Fix an ample divisorH ′. Then any divisor of the formH = nC+H ′ is ample onX , and for n sufficiently large E will be H-stable. The Bogomolov-Gieseker inequalityfor parabolic bundles says that ∆Par(E) ≥ 0 with equality if and only if E comes froma unitary representation. However, ∆Par(E) ≥ ∆Par(Emin) ≥ 0 and if E comes froma unitary representation then ∆Par(E) = ∆Par(Emin) = 0. It follows in this case thatE is one of the choices of Emin.

111

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