L2--cohomology groups of some singular complex spacestjh/FinalNDMarch2013.pdf · L2-@-cohomology groups of some singular complex spaces Sophia Vassiliadou Department of Mathematics

L2-∂-cohomology groups of some singularcomplex spaces

Sophia Vassiliadou

Department of Mathematics and StatisticsGeorgetown University

March 16, 2013

logo

Introduction: objects and problemsHistory-Methods and results before 2008

Results after 2008Another description of the H

p,q(2)

(X′r)Open questions

Non-isolated singularities

Outline

1 Introduction: objects and problems

2 History-Methods and results before 2008

3 Results after 2008

4 Another description of the Hp,q(2) (X ′r)

5 Open questions

6 Non-isolated singularities

Sophia Vassiliadou L2-∂-cohomology groups of some singular complex spaces

logo



p,q(2)



The ∂-operator has played an important role in complex geometry,from Dolbeault isomorphism theorem, to representation of ordinarycohomology in terms of forms, to Hodge decomposition for compactKahler manifolds i.e Hr(X, C) ∼= ⊕p+q=rHp,q(X), Hp,q = Hq,p.

What happens to some of these theorems if the space is no longercompact or if the space is singular?

In the 1980-1990’s people realized that singular projective varietiesdo not satisfy Hodge decomposition, Poincare duality or Lefschetzhyperplane theorems; a new topological theory emerged to deal withcertain stratified spaces, called intersection homology, and a questensued to answer the following question : Is there an analytic theoryto represent intersection cohomology (just like the de Rham theoryrepresents ordinary cohomology). Maybe, L2-cohomology.

Fundamental works by Cheeger, Goresky, MacPherson, Nagase,Ohsawa, Pardon and Stern, Saper, Zucker..... appeared.


logo



p,q(2)








logo



p,q(2)








logo



p,q(2)








logo



p,q(2)



Theorem ( Ohsawa 1981, 1987, Ohsawa-Takegoshi 1988, Demailly 1990)

Let (X,ω) be a Kahler manifold of dimension n. Suppose that X isweakly 1-convex (i.e it has a plurisubharmonic exhaustion function that isstrongly plurisubharmonic outside a compact subset). Then there is aHodge decomposition

HkDR(X, C) ∼= ⊕p+q=kHp,q(X); Hp,q(X) = Hq,p(X) k ≥ n+ 1

HkDR,c(X, C) ∼= ⊕p+q=kHp,q

c (X); Hp,qc (X) = Hq,p

c (X) k ≤ n− 1

where H ·,·c denotes cohomology with compact support. Moreover wehave a Lefschetz isomorphism

ωn−p−q ∧ • : Hp,qc (X)→ Hn−q,n−p(X), p+ q ≤ n− 1.


logo



p,q(2)



Set-up

Let X be a pure n-dimensional complex analytic set embedded insome CN with an isolated singularity at 0 (resp. a complex projectivevariety embedded in some CPN with an isolated singular point).

Let X ′ := X \ SingX be the set of smooth points of X. WhenX ⊂ CN , pure n-dimensional C-analytic, we let Xr := X ∩Br(0),be a small neighborhood of 0 and X ′r := X ′ ∩Br(0). We shallchoose an R > 0 small enough, so that bBr(0) intersects Xtransversally for all 0 < r < R.

The restriction on X ′ of the Euclidean metric in CN (resp. theFubini-Study metric in CPN ) induces an incomplete metric on X ′,which we call the ambient metric.

We consider smooth forms f on X ′ (resp. on X ′r) such that f, ∂fare square integrable on X ′ (resp. X ′r) with respect to the ambientmetric and let (Ap,.(2)(·), ∂) denote the corresponding Dolbeault

complex on X ′ (resp. X ′r) for each p ≥ 0.


logo



p,q(2)



Set-up







logo



p,q(2)



Set-up







logo



p,q(2)



Due to the incompleteness of the metric, there could be many (possibly)different closed extensions for the ∂-operator which might lead todifferent L2-∂-cohomology groups.Three closed extensions of ∂ will appear in this talk .

Let ∂min := L2-graph closure of ∂C∞0 (·) ( where C∞0 (·) denotes the

space of smooth compactly supported forms on · and where · is X ′

or X ′r)

∂1

:= L2-graph closure of ∂C∞0 (Xr\0)(where C∞0 (Xr \ 0)

denotes those smooth compactly supported forms on Xr that satisfya Dirichlet condition near the singularity).

∂max: the distributional ∂, i.e. u ∈ L2 ∩ Dom(∂) if ∂u ∈ L2 (in D′).

In general

Dom(∂min) ⊂ Dom(∂1) ⊂ Dom(∂max)


logo



p,q(2)






or X ′r)

∂1




In general



logo



p,q(2)






or X ′r)

∂1




In general



logo



p,q(2)






or X ′r)

∂1




In general



logo



p,q(2)



Formulation of the problems

A. Local Problem: Let p, q be integers with 0 ≤ p, q ≤ n. Givenf ∈ Lp,q(2)(X

′r), ∂max f = 0 on X ′r does there exist a u ∈ Lp,q−1(2) (X ′r)

such that ∂maxu = f on X ′r?

•1 If there are obstructions, i.e. we have non-trivial ∂max-cohomology groups Hp,q

(2) (X ′r), are they finite dimensional for any

p, q?

•2 Can we relate the dimensions of these cohomology groups tocertain invariants of the singularities?

B. Global problem: Given f ∈ Lp,q(2)(X′), ∂max f = 0 on X ′ does

there exist a u ∈ Lp,q−1(2) (X ′) such that ∂maxu = f on X ′?


(2) (X ′), are they finite dimensional for any

p, q? Can we identify them? In the case of projectives varieties withan isolated singularity, are these cohomology groups birational invariants?


logo



p,q(2)



Formulation of the problems

A. Local Problem: Let p, q be integers with 0 ≤ p, q ≤ n. Givenf ∈ Lp,q(2)(X

′r), ∂max f = 0 on X ′r does there exist a u ∈ Lp,q−1(2) (X ′r)

such that ∂maxu = f on X ′r?


(2) (X ′r), are they finite dimensional for any

p, q?

•2 Can we relate the dimensions of these cohomology groups tocertain invariants of the singularities?

B. Global problem: Given f ∈ Lp,q(2)(X′), ∂max f = 0 on X ′ does

there exist a u ∈ Lp,q−1(2) (X ′) such that ∂maxu = f on X ′?


(2) (X ′), are they finite dimensional for any

p, q? Can we identify them? In the case of projectives varieties withan isolated singularity, are these cohomology groups birational invariants?


logo



p,q(2)



Five methods have been developed to answer the local/globalproblem and related questions.

M1. Branched covering method (used to prove finite dimensionalityof certain L2-∂-cohomology groups)

• We can think of X as a finite branched cover over a suitable Cn.

• We can remove a hypersurface S from X such thatπ : X \ S → Cn is unbranched. On X \ S we can solve ∂u = fusing Hormander’s L2-theory.

• Then we can use a detailed geometric analysis of the singularspace to modify this u to obtain a solution with L2-estimates up toa finite dimensional set of obstructions.

Pro’s: This method works well for (p, 1)-forms and can be used toprove some results for irreducible projective varieties withnon-isolated singularities (Øvrelid-V. (2009)).

Con’s: Does not work well for general (p, q)-forms with q > 1.


logo



p,q(2)



This method was applied successfully by Fornæss in (1999) forhomogeneous surfaces X in the unit ball in C3 and later byDiederich-Fornæss-V. (2000), for a generic isolated surface singularityin CN , to show that dimH0,1

(2) (X ′1) <∞.

There were two projections πj : C3 → C2 for j = 1, 2 such that πj |Xis a finite branched covering. The ramification locii Σj intersect onlyat 0. Each Σj is a finite union of lines through the origin.

According to Hormander’s L2-theory for Stein Riemann domainsthere exist solutions to ∂uj = f on X \ Σj . Then the difference ofthese solutions is a meromorphic function with poles on Σ1 ∪ Σ2.

By analyzing its Laurent series expansion in conical neighborhoodsof the lines in Σ1, Fornæss showed that the vanishing of a certainfinite set of coefficients in this expansion is a necessary and sufficientcondition for the solvability of ∂u = f in L2(X ′). Moreover heproved that the dimensions of the local L2-(0, 1)-∂-cohomologygroups grow at most like d3 where d > 2 is the degree of thehomogeneous polynomial that defines the surface.


logo



p,q(2)




(2) (X ′1) <∞.





logo



p,q(2)




(2) (X ′1) <∞.





logo



p,q(2)



M2. Construction of complete Kahler metrics and appropriateweight functions, application of Andreotti-Vesentini orDonnelly-Fefferman vanishing theorems

This method was pioneered by Ohsawa (1987, 1991, 1999) andDemailly and subsequently used in Pardon-Stern (1991, 2001),Fornæss-Øvrelid-V (2005), Ruppenthal (2010, 2012).

Different techniques were employed to attack the local problemdepending on whether p+ q > n, p+ q < n.

Case p+ q > n. Ohsawa constructed a family of auxiliary completeKahler metrics ωε := i∂∂ψεε in a deleted ball X ′r around 0 suchthat the lengths of ∂ψε with respect to ωε were uniformly boundedfrom above.

Using the uniform vanishing of the Lp,q2,ωε-∂max-cohomology groups

when p+ q > n and a standard weak limit argument, he was able toobtain the existence of a square-integrable (with respect to theambient metric) solution u to the equation ∂maxu = f on X ′ρ withρ < r thus showing lim−→Hp,q

(2) (X ′s) = 0 when p+ q > n.


logo



p,q(2)









(2) (X ′s) = 0 when p+ q > n.


logo



p,q(2)









(2) (X ′s) = 0 when p+ q > n.


logo



p,q(2)









(2) (X ′s) = 0 when p+ q > n.


logo



p,q(2)



Case p+ q < n. Ohsawa constructed a complete Kahler metric on adeleted neighborhood of 0 in X with Kahler form ω := i∂∂φ (φ wasan appropriate strictly plurisubharmonic function on X ′r) and provedthat certain weighted Lp,q2, ω-∂max-cohomology groups of X ′r vanishwhen p+ q < n. As a consequence of this vanishing he obtained asolution u (compactly supported in Xr) to ∂maxu = f on X ′r,whenever f was a square-integrable (with respect to the ambientmetric), ∂max-closed form, of bidegree p+ q < n, compactlysupported on Xr.

However, he did not provide any information on Hp,q(2) (X ′r) when

p+ q < n.

Similarly, the local Lp,q(2)-∂-cohomology groups when p+ q = n and

p > 0, q > 0 remained a mystery.


logo



p,q(2)





p+ q < n.




logo



p,q(2)





p+ q < n.




logo



p,q(2)



Inspired by these previous works, we showed the following inFornæss-Øvrelid-V (2005)

dimHp,q(2) (X ′r) <∞ when p+ q < n, q > 0

Using a local L2-solvability result for ∂-closed forms with compactsupport in Xr, we reduced the question of local solvability for ∂ onX ′r to a local solvability result for ∂ on Xr \Bρ with ρ < r and thentried to understand the L2-∂-cohomology groups of Xr \Bρ.

Hp,q(2) (X ′r) = 0, when p+ q > n

We considered the natural inclusion j : Lp,q(2)(X′r)→ Lp,q2, loc(X

′r) and

studied the corresponding induced homomorphismj∗ : Hp,q

(2) (X ′r)→ Hq(X ′r,Ωp).

Theorem

Let j∗ : Hp,q(2) (X ′r)→ Hq(X ′r,Ω

p) be the obvious homomorphism induced

by the inclusion j : Lp,q(2)(X′r)→ Lp,q2, loc(X

′r). Then the map j∗ is injective

for p+ q ≤ n− 1 and q > 0 and bijective for p+ q ≤ n− 2 and q > 0.


logo



p,q(2)






Hp,q(2) (X ′r) = 0, when p+ q > n


′r) and


(2) (X ′r)→ Hq(X ′r,Ωp).

Theorem







logo



p,q(2)






Hp,q(2) (X ′r) = 0, when p+ q > n


′r) and


(2) (X ′r)→ Hq(X ′r,Ωp).

Theorem







logo



p,q(2)






Hp,q(2) (X ′r) = 0, when p+ q > n


′r) and


(2) (X ′r)→ Hq(X ′r,Ωp).

Theorem







logo



p,q(2)



When (X, 0) was a Cohen-Macauley singularity and 1 ≤ q ≤ n− 2,the previous theorem along with Scheja’s extension of cohomologyclasses theorem will yield that

Hq(Xr,O)Scheja’s isom Thm−→ Hq(X ′r,O)

∼→ H0,q(2) (X ′r)

But Xr is a Stein space so Hq(Xr,O) = 0 for all q ≥ 1.

(X, 0) Cohen-Macaulay and 1 ≤ q ≤ n− 2, H0,q(2) (X ′r) = 0 (n ≥ 3).

Actually, the previous theorem and Scheja’s extension theorem willyield that Hp,q

(2) (X ′r) = 0 when p+ q ≤ n− 2, q > 0 and 1 ≤ q ≤codh(ωp)0 − 2, where ωp is any coherent analytic sheaf extendingthe sheaf of holomorphic p-forms on X ′r and codh(F) is thehomological codimension of a coherent analytic sheaf. (If (X, 0) is acomplete-intesection isolated singularity such restrictions betweenp, q are possible with ωp = π∗(Ω

p

X) for π : X → X a

desingularization and Ωp=sheaf of holomorphic p-forms on themanifold X).Con’s: What happens if we are not in the above situation(s) or whenn = 2?


logo



p,q(2)





∼→ H0,q(2) (X ′r)





p




logo



p,q(2)





∼→ H0,q(2) (X ′r)





p




logo



p,q(2)





∼→ H0,q(2) (X ′r)





p




logo



p,q(2)



M3. Resolution of singularities and transfer of the problem on amanifold

Pioneering work of Haskell, Hsiang-Pati, Nagase, Ohsawa, Pardon,Saper, Stern, ........

Interest in relating L2-∂-(0, q)-cohomology groups on X ′ withcohomology groups on a desingularized space arose from thefollowing

Conjecture (MacPherson)

Let X be a projective variety, X ′ := regX (endowed with theFubini-Study metric) and π : X → X be a desingularization.Then

χ(2)(X) :=∑i

(−1)i dimH0,i(2)(X

′)?= χ(X, O)


logo



p,q(2)








χ(2)(X) :=∑i

(−1)i dimH0,i(2)(X

′)?= χ(X, O)


logo



p,q(2)








χ(2)(X) :=∑i

(−1)i dimH0,i(2)(X

′)?= χ(X, O)


logo



p,q(2)



MacPherson’s conjecture had to be modified

(MMC)∑i

(−1)i dimH0,iD (X ′)

?= χ(X, O)

where H0,iD (X ′) are the L2-∂min-cohomology groups.

Haskell proved (MMC) for projective curves and surfaces withisolated singularities

Pardon-Stern proved (MMC) for projective varieties with arbitrarysingularities

Ruppenthal used Pardon and Stern’s techniques and proved (MMC)for compact Hermitian complex spaces.

In mid 80’s Pardon, while studying MacPherson’s conjectureproposed an alternative one


logo



p,q(2)




(MMC)∑i


?= χ(X, O)







logo



p,q(2)




(MMC)∑i


?= χ(X, O)







logo



p,q(2)




(MMC)∑i


?= χ(X, O)







logo



p,q(2)




(MMC)∑i


?= χ(X, O)







logo



p,q(2)



Conjecture (Pardon)

χ(2)(X) :=∑i

(−1)i dimH0,i(2)(X

′)?= χ(X, O(Z − E))

where π : X → X is a desingularization, E := π−1(SingX) is a SNCD(reduced) and Z := π−1(SingX) is the unreduced exceptional divisor.

Pardon (1989) showed that the above conjecture holds for projectivecurves with isolated singularities and products of such curves withsmooth projective curves.

Pardon-Stern, (1991) proved the above conjecture for projectivesurfaces with isolated singularities, by showing that

H0,i(2)(X

′) ∼= Hi(X, O(Z − E))

for all i with 0 ≤ i ≤ 2 (the proof for i = 1 turned out not to becomplete, so groups of people started looking at these ideas again)


logo



p,q(2)



Conjecture (Pardon)

χ(2)(X) :=∑i

(−1)i dimH0,i(2)(X

′)?= χ(X, O(Z − E))

where π : X → X is a desingularization, E := π−1(SingX) is a SNCD(reduced) and Z := π−1(SingX) is the unreduced exceptional divisor.

Pardon (1989) showed that the above conjecture holds for projectivecurves with isolated singularities and products of such curves withsmooth projective curves.

Pardon-Stern, (1991) proved the above conjecture for projectivesurfaces with isolated singularities, by showing that

H0,i(2)(X

′) ∼= Hi(X, O(Z − E))

for all i with 0 ≤ i ≤ 2 (the proof for i = 1 turned out not to becomplete, so groups of people started looking at these ideas again)


logo



p,q(2)



M4. Other methods

Berndtsson and Sibony (2001) developed a theory for solving ∂ oncurrents (since then this theory found applications in complex dynamicsand lamination theory)

Andersson and Samuelsson, Wulcan,.... developed a theory for solving ∂using residue theory.


logo



p,q(2)



Theorem (Ruppenthal, 2008)

Let X be the affine cone over a smooh projective variety Y n−1 ⊂ CPNand let X := Tot(OY (−1)) and let N denote the normal bundle of Y inX. Then(i)

H0,q(2) (X ′r) → ⊕

µ≥q−nHq(Y,O(N−µ))

(ii)⊕

µ≥1+q−nHq(Y,O(N−µ)) → H0,q

(2) (X ′r)

(iii)H0,q

(2) (X ′r)∼= Hq(Xr,O) ∼= Hq(X, O), 1 ≤ q ≤ n− 2.

H0,1(2) (X ′r) = 0 if Y = z ∈ CP2 : z2 = xy, H0,1

(2) (X ′r) 6= 0, if Y is an

elliptic curve.

Remark: To construct these maps he relied on a quite technical companion paper where he studied

a Dolbeault complex with weights according to SNCD.


logo



p,q(2)



Theorem (Ruppenthal, 2010)

Let (X,h) be a Hermitian complex space of pure dimension n, D ⊂⊂ Xsmoothly bounded strongly pseudoconvex domain such thatbD ∩ SingX = ∅. Let π : X → X be a desingularization such thatE := π−1(SingX) is a SNCD and let Z := π−1(SingX) be theunreduced exceptional divisor.

If L−Z+E (the line bundle associated to the divisor Z − E) is locallysemi-positive with respect to X, then

H0,q(2) (D) ∼= Hq(X, O(Z − E))

for all q with 0 ≤ q ≤ n.

Remark: The requirement that L−Z+E is locally semi-positive w.r.t. X isalways satisfied if E is irreducible or Z has the same multiplicity alongeach irreducible component.


logo



p,q(2)



Theorem (Øvrelid-V, (2011))

Let X be a complex analytic subset of CN of pure dimension n ≥ 2 withan isolated singularity at 0, and let π : X → X be a desingularization.Then, there exists a well-defined, linear mapping

φ∗ : Hq(Xr, O)→ H0,q(2) (X ′r)

sending[g]→ [(π−1)∗gXr\E ]

such that

(i) φ∗ is bijective if 1 ≤ q ≤ n− 2

(ii) φ∗ is injective if q = n− 1.

Remark: A completely analogous global theorem is true, in the case ofcompact Hermitian complex spaces with isolated singularities.


logo



p,q(2)



The proof of the previous theorem is based on the following observations

Since Xr is a smoothly bounded strongly pseudoconvex manifoldH0,q

(2) (Xr) ∼= Hq(Xr, O) (here the H0,q(2) (Xr) cohomology groups are

taken w.r.t. a non-degenerate Hermitian metric on X).

If g ∈ L0,q(2)(Xr), then (π−1)∗ g ∈ L0,q

(2)(X′r) since

‖π−1)∗ g‖L0,q(2)

(X′r)= ‖g‖Lp,q2, γ(Xr)

≤ C‖g‖Lp,q(2)

(Xr)

where γ := π∗(ambient metric).

Remark: One of the unfortunate features of the desingularizationmethod is that there is no easy way to compare norms of(p, q)-forms on X with respect to the pull-back of the ambientmetric and some non-degenerate metric on X, unless(p, q) = (n, q) or (p, q) = (0, q).

The exceptional set E of the desingularization is an exceptional setin the sense of Grauert (i.e. it has arbitrarily small stronglypseudoconvex neighborhoods).


logo



p,q(2)








(2)(X′r) since

‖π−1)∗ g‖L0,q(2)


≤ C‖g‖Lp,q(2)

(Xr)





logo



p,q(2)








(2)(X′r) since

‖π−1)∗ g‖L0,q(2)


≤ C‖g‖Lp,q(2)

(Xr)





logo



p,q(2)



Karras proved in 1986 that if E is an exceptional set in Xr

HqE(Xr, O) = 0 for all q < n

The above result followed from the following interesting theorem

Theorem (Karras)

Let X be a reduced complex space and E an exceptional subset of X. IfF is a coherent analytic sheaf on X such that depthx F ≥ d forx ∈M \E where M is a smooth strongly pseudoconvex neighborhood ofE then

γi : HiE(X, F)→ Hi

c(M, F)

is an isomorphism for i < d.

Remark: If M is a strongly psdcx mfld containing E, F := O then

HiE(X,O) = Hi

E(M,O) ∼= Hic(M,O) ∼= (Hn−i(M,KM ))′ = 0

since by Grauert-Riemenschneider’s vanishing theorem Riπ∗KM = 0for all i > 0.


logo



p,q(2)



Karras proved in 1986 that if E is an exceptional set in Xr

HqE(Xr, O) = 0 for all q < n

The above result followed from the following interesting theorem

Theorem (Karras)

Let X be a reduced complex space and E an exceptional subset of X. IfF is a coherent analytic sheaf on X such that depthx F ≥ d forx ∈M \E where M is a smooth strongly pseudoconvex neighborhood ofE then

γi : HiE(X, F)→ Hi

c(M, F)

is an isomorphism for i < d.

Remark: If M is a strongly psdcx mfld containing E, F := O then

HiE(X,O) = Hi

E(M,O) ∼= Hic(M,O) ∼= (Hn−i(M,KM ))′ = 0

since by Grauert-Riemenschneider’s vanishing theorem Riπ∗KM = 0for all i > 0.


logo



p,q(2)



We have a commutative diagram

Hq(Xr, O)

r∗

++H0,q

(2) (Xr) //

∼=

ff

φ∗

Hq(Xr \ E, O)

∼=

H0,q(2) (X ′r)

j∗ // Hq(X ′r, O).

and a long exact sequence on cohomology

....→ HiE(Xr, O)→ Hi(Xr, O)

r∗→ Hi(Xr \ E, O)→ Hi+1E (Xr, O)→

Since for 1 ≤ q ≤ n− 2 r∗ and j∗ are isomorphisms, the commutativityof the above diagram will imply that for 1 ≤ q ≤ n− 2 φ∗ is anisomorphism. On the other hand for q = n− 1, the maps r∗ and j∗ are only injective,

hence φ∗ is an injective map.


logo



p,q(2)



Suppose for the moment that instead of the sheaf O in the previous slide,we had the sheaf of holomorphic p-forms on Xr. We still have the longexact sequence

HqE(Xr, Ωp)→ Hq(Xr, Ωp)

r∗→ Hq(Xr\E, Ωp)→ Hq+1E (Xr, Ωp)→ ....

By Karras’ result we know that HqE(Xr,Ω

p)→ Hqc (Xr, Ωp) is an

isomorphism for all q < n.

By Ohsawa’s and Demailly’s Hard-Lefschetz theorem for 1-convex, Kahlermanifolds (Xr, ω) we know that

ωn−p−q∧ : Hqc (Xr,Ω

p)→ Hn−p(Xr,Ωn−q)

is injective when p+ q ≤ n− 1.

But the above map factors through Hq(Xr, Ωp); hence we can deducethat the natural morphism Hq

c (Xr, Ωp)→ Hq(Xr,Ωp) is injective when

p+ q ≤ n− 1 and thus HqE(Xr,Ω

p)→ Hq(Xr,Ωp) is injective for the

same range of p, q.Sophia Vassiliadou L2-∂-cohomology groups of some singular complex spaces

logo



p,q(2)



Putting these together one obtains


There exists a surjective linear map T : Hq(Xr, Ωp)→ Hp,q(2) (X ′r) whose

kernel is naturally isomorphic to HqE(Xr, Ωp) when p+ q ≤ n− 2, q > 0.

Proof:

Hq(Xr, Ωp)r∗ //

T

Hq(Xr \ E,Ωp)

∼=(π−1)∗

Hp,q

(2) (X ′r)j∗

∼=// Hq(X ′r, Ωp).

Since r∗ is surjective we can set T := (j∗)−1 (π−1)∗ r∗. Then,

kernT = kern(r∗) ∼= HqE(Xr, Ωp) (by previous slide)


logo



p,q(2)



The case q = n− 1 turned out to be more delicate.

The cokernel of φ∗ : Hn−1(Xr, O)→ H0,n−1(2) (X ′r) will play a role

in computing H0,n−1(2) (X ′r).

In [FOV] from (2005) we had shown that if

∫X′r

f ∧ ∂ χ ∧ ψ = 0 ∀ ψ ∈ Hn,0(2) (X ′r)(:= Ln,0(2) (X ′r) ∩ kern(∂)

where χ ∈ C∞0 (Xr), χ = 1 near 0, then the following system issolvable

∂w = π∗(∂χ ∧ f)

w ∈ L0,n−1(2) (X ′r)

suppw ⊂⊂ Xr


logo



p,q(2)



The case q = n− 1 turned out to be more delicate.

The cokernel of φ∗ : Hn−1(Xr, O)→ H0,n−1(2) (X ′r) will play a role

in computing H0,n−1(2) (X ′r).

In [FOV] from (2005) we had shown that if

∫X′r

f ∧ ∂ χ ∧ ψ = 0 ∀ ψ ∈ Hn,0(2) (X ′r)(:= Ln,0(2) (X ′r) ∩ kern(∂)

where χ ∈ C∞0 (Xr), χ = 1 near 0, then the following system issolvable

∂w = π∗(∂χ ∧ f)

w ∈ L0,n−1(2) (X ′r)

suppw ⊂⊂ Xr


logo



p,q(2)



For such f ’s

f = χf − φ(w)︸︷︷︸f1

+ φ(w) + (1− χf)︸︷︷︸f2

where

•f1 is ∂-closed, square-integrable and compactly supported in Xr

(thus f1 = ∂u for some u ∈ L2(X ′r) and compactly supported inXr, from earlier work).

•f2 := φ(w+π∗((1−χ)f)) ∈ Z0,n−1(2) (X ′r) := kern(∂)∩L0,n−1

(2) (X ′r).

[f ] = φ∗([w + π∗(1− χ) f ])

Recap: For f ’s that satisfy∫X′rf ∧ ∂ χ ∧ ψ = 0 ∀ ψ ∈ Hn,0

(2) (X ′r)(:= Ln,0(2) (X ′r) ∩ kern(∂)

we have [f ] = φ∗([w + π∗(1− χ) f ]).


logo



p,q(2)



For such f ’s

f = χf − φ(w)︸︷︷︸f1

+ φ(w) + (1− χf)︸︷︷︸f2

where




(2) (X ′r).

[f ] = φ∗([w + π∗(1− χ) f ])


(2) (X ′r)(:= Ln,0(2) (X ′r) ∩ kern(∂)

we have [f ] = φ∗([w + π∗(1− χ) f ]).


logo



p,q(2)



For such f ’s

f = χf − φ(w)︸︷︷︸f1

+ φ(w) + (1− χf)︸︷︷︸f2

where




(2) (X ′r).

[f ] = φ∗([w + π∗(1− χ) f ])


(2) (X ′r)(:= Ln,0(2) (X ′r) ∩ kern(∂)

we have [f ] = φ∗([w + π∗(1− χ) f ]).


logo



p,q(2)



Properties of ∂1-operator

f ∈ Dom(∂1) iff ξ f ∈ Dom(∂min) for any ξ cut-off function (compt.

supported in Xr).

g ∈ L0,n−1(2) (X ′r) ∩ Dom(∂) then φ(g) ∈ Dom(∂

1).

If f ∈ kern(∂1) then ∀ ψ ∈ Hn,0

(2) (X ′r) we have∫X′r

f ∧ ∂χ ∧ ψ = 0

Putting all these together one sees that if [f ] = φ∗([g]) then∫X′r

f ∧ ∂χ ∧ ψ = 0 ∀ ψ ∈ Hn,0(2) (X ′r)


logo



p,q(2)





supported in Xr).


1).



f ∧ ∂χ ∧ ψ = 0


f ∧ ∂χ ∧ ψ = 0 ∀ ψ ∈ Hn,0(2) (X ′r)


logo



p,q(2)





supported in Xr).


1).



f ∧ ∂χ ∧ ψ = 0


f ∧ ∂χ ∧ ψ = 0 ∀ ψ ∈ Hn,0(2) (X ′r)


logo



p,q(2)





supported in Xr).


1).



f ∧ ∂χ ∧ ψ = 0


f ∧ ∂χ ∧ ψ = 0 ∀ ψ ∈ Hn,0(2) (X ′r)


logo



p,q(2)



A non-degenerate pairing

H0,n−1(2) (X ′r)

φ∗(H0,n−1(Xr))× kern(∂)n,0

kern(∂1)n,0→ C

Define a pairing

< [f ], [ψ] >:=

∫X′r

f ∧ ∂χ ∧ ψ

when f ∈ Z0,n−1(2) (X ′r), ψ ∈ Hn,0

(2) (X ′r).∫X′rf ∧ ∂χ ∧ ψ = 0 for all ψ ∈ Hn,0

(2) (X ′r) iff [f ] ∈ Imφ∗∫X′rf ∧ ∂χ ∧ ψ = 0 for all f ∈ Z0,n−1

(2) (X ′r) iff ψ ∈ kern(∂1).

These two properties show that < , > is a a non-degenerate pairing.


logo



p,q(2)




H0,n−1(2) (X ′r)

φ∗(H0,n−1(Xr))× kern(∂)n,0

kern(∂1)n,0→ C

Define a pairing

< [f ], [ψ] >:=

∫X′r

f ∧ ∂χ ∧ ψ

when f ∈ Z0,n−1(2) (X ′r), ψ ∈ Hn,0



(2) (X ′r) iff ψ ∈ kern(∂1).



logo



p,q(2)




H0,n−1(2) (X ′r)

φ∗(H0,n−1(Xr))× kern(∂)n,0

kern(∂1)n,0→ C

Define a pairing

< [f ], [ψ] >:=

∫X′r

f ∧ ∂χ ∧ ψ

when f ∈ Z0,n−1(2) (X ′r), ψ ∈ Hn,0



(2) (X ′r) iff ψ ∈ kern(∂1).



logo



p,q(2)




H0,n−1(2) (X ′r)

φ∗(H0,n−1(Xr))× kern(∂)n,0

kern(∂1)n,0→ C

Define a pairing

< [f ], [ψ] >:=

∫X′r

f ∧ ∂χ ∧ ψ

when f ∈ Z0,n−1(2) (X ′r), ψ ∈ Hn,0



(2) (X ′r) iff ψ ∈ kern(∂1).



logo



p,q(2)




H0,n−1(2) (X ′r)

φ∗(H0,n−1(Xr))× kern(∂)n,0

kern(∂1)n,0→ C

Define a pairing

< [f ], [ψ] >:=

∫X′r

f ∧ ∂χ ∧ ψ

when f ∈ Z0,n−1(2) (X ′r), ψ ∈ Hn,0



(2) (X ′r) iff ψ ∈ kern(∂1).



logo



p,q(2)




Let X be a complex analytic subset of CN of pure dimension n ≥ 2 withan isolated singularity at 0. Let π : X → X be a desingularization suchthat the exceptional locus E of π is a simple, normal crossings divisor.Let Z = π−1(SingX) be the unreduced exceptional divisor of theresolution, let the support of Z be denoted by |Z| := E and letD := Z − |Z|. Then, there exists a natural surjective linear map

T : Hn−1(Xr, O(D))→ H0,n−1(2) (X ′r)

whose kernel is naturally isomorphic to Hn−1E (Xr, O(D)).

Ruppenthal in (2012) gave another proof of the above theorem.


logo



p,q(2)



Corollary 1: If L−Z+E is locally semi-positive with respect to X,then by Serre duality, Karras’ result and Silva-Takegoshi’s twistedrelative vanishing theorem one obtains Hn−1

E (Xr, O(D)) = 0.

Corollary 2: If dimX = 2, then the above map T is always anisomorphism (this completes the proof of Pardon’s conjecture in thecase of projective surfaces with isolated singularities).

This followed from the fact that the natural mapH1E(X, O(Z − E))→ H1

E(X, O(Z)) is injective, which in its turnwas proven using the fact that for a surface isolated singularity theintersection matrix (Ei · Ej) is negative definite.

When dimX > 2 then

Hn−1E (X, O(D)) = 0⇐⇒ Hn−2(E, OX(Z)E ) = 0.

An analogous global theorem is also true for compact puren-dimensional complex spaces with isolated singularities.


logo



p,q(2)




E (Xr, O(D)) = 0.




When dimX > 2 then

Hn−1E (X, O(D)) = 0⇐⇒ Hn−2(E, OX(Z)E ) = 0.



logo



p,q(2)




E (Xr, O(D)) = 0.




When dimX > 2 then

Hn−1E (X, O(D)) = 0⇐⇒ Hn−2(E, OX(Z)E ) = 0.



logo



p,q(2)



Idea of the proof

Recall the map j∗ : H0, n−1(2) (X ′r)→ Hn−1(X ′r, O) is injective.

There exists a natural map `∗ : Hn−1(Xr, O(D))→ Hn−1(X ′r, O)sending [g]→ (π−1)∗[g].

If we could show that Im j∗ = Im `∗, then the map T := j∗−1 `∗

will be the desired map of the theorem. So we need to characterizeImj∗.

Lemma

Let f ∈ L0,n−12, loc (X ′r) ∩ kern ∂. Then,

i) If [f ] ∈ Im j∗, then < f,ψ >:=∫X′rf ∧ ∂χ ∧ ψ = 0 when

ψ ∈ Kern(∂1)n,0.

ii) On the other hand, if∫X′r|f |2 ‖z‖B dV <∞ for some B > 0 large

enough and < f, ψ >= 0 when ψ ∈ kern(∂1)n,0, then [f ] ∈ Im j∗.


logo



p,q(2)



Idea of the proof





Lemma



ψ ∈ Kern(∂1)n,0.




logo



p,q(2)



Idea of the proof





Lemma



ψ ∈ Kern(∂1)n,0.




logo



p,q(2)



Idea of the proof





Lemma



ψ ∈ Kern(∂1)n,0.




logo



p,q(2)



A twisted version of an L2-Cauchy problem will show thatIm j∗ ⊂ Im `∗ and will help in the construction of a mapS : H0,n−1

(2) (X ′r)→ Hn−1(Xr,O(D)).

To prove Im`∗ ⊂ Imj∗ we will use part ii) of the previous lemma; keyingredient here will be an observation by Ruppenthal that for

ψ ∈ kern(∂1)n,0

π∗ψ ∈ Ln,0(2) (Xr) ∩ Γ(Xr, K(−D))

The composition j∗−1 `∗ will be the desired map T and

T S = Id.

What happens to H0,n(2) (X ′r)?

Well, H0,n(2) (X ′r) = 0 since the map φ∗ : Hn(Xr, O)→ H0,n

(2) (X ′r) is

surjective and by a result of Malgrange Hn(Xr, O) = 0.


logo



p,q(2)




(2) (X ′r)→ Hn−1(Xr,O(D)).



π∗ψ ∈ Ln,0(2) (Xr) ∩ Γ(Xr, K(−D))


T S = Id.



(2) (X ′r) is



logo



p,q(2)




(2) (X ′r)→ Hn−1(Xr,O(D)).



π∗ψ ∈ Ln,0(2) (Xr) ∩ Γ(Xr, K(−D))


T S = Id.



(2) (X ′r) is



logo



p,q(2)




(2) (X ′r)→ Hn−1(Xr,O(D)).



π∗ψ ∈ Ln,0(2) (Xr) ∩ Γ(Xr, K(−D))


T S = Id.



(2) (X ′r) is



logo



p,q(2)




(2) (X ′r)→ Hn−1(Xr,O(D)).



π∗ψ ∈ Ln,0(2) (Xr) ∩ Γ(Xr, K(−D))


T S = Id.



(2) (X ′r) is



logo



p,q(2)



Global H0,n(2) (X

′)

Consider the map φn∗ : Hn(X, O)→ H0,n(2) (X

′) defined by

φn∗ ([f ]) = [ (π−1)∗ f ]. It is easily seen to be surjective.

Let in∗ : Hn(X, O)→ Hn(X, O(D)) be the map on cohomology induced bythe sheaf inclusion i : O → O(D).

Corollary

With X, X,D, φn∗ , in∗ as above we have kern(φn∗ ) = kern(in∗ ) and

H0,n(2) (X

′) ∼= Hn(X, O(D)).

Hn(X, O)

in∗

φ∗n // H0,n(2) (X

′)

Hn(X, O(D))

88

As the maps φn∗ , in∗ are surjective and kern(in∗ ) = kern(φn∗ ) the dotted map

will be an isomorphism.Sophia Vassiliadou L2-∂-cohomology groups of some singular complex spaces

logo



p,q(2)



Pro’s: The kernel of in∗ can be computed using standard long exactsequences on cohomology and cohomology with support on E.

Corollary allows us to describe the kernel of φn∗ which in some sensemeasures when X is compact the difference between the minimalH0,nD -cohomology group on X ′ (which is isomorphic to Hn(X, O)), and

the corresponding cohomology group using the ∂max-operator (i.e.H0,n

(2) (X ′)).

For example if,

dimCHn−1(U ,O(D))

Hn−1(U ,O)6= dimC

Hn−1(X,O(D))

Hn−1(X,O),

then the kernel of in∗ would be non-trivial (and where U a stronglypseudoconvex neighborhood of E).


logo



p,q(2)



Another description of the Hp,q(2)(X

′r)

Choose ρ < r such that Σ := X ∩ ‖z‖ = ρ is a C∞-compact stronglypseudoconvex hypersurface in X ′.

In [ FOV] (2005) paper, we showed that there exists a map

θ : Hp,q(2) (X ′r)→ Hp,q

b (Σ)

such that

• θ is bijective if p+ q ≤ n− 2, q > 0

• θ is injective if p+ q = n− 1, q > 0

Stephen Yau studied the Hp,qb (Σ) in the case of a hypersurface with an

isolated singularity by showing that Hp,qb (Σ) ∼= Hq+1

0 (Xr, Ωp) if q ≥ 1

(the dimensions of the latter cohomology are the Brieskorn invariants ofthe singularity) and then using commutative algebra to calculate thedimensions of these cohomology groups.


logo



p,q(2)



Open questions

Can we identify Hp,q(2) (X ′r) or Hp,q

(2) (X ′) when p+ q = n− 1 and

p > 0 with sheaf cohomology groups on a desingularization?

What happens to Hp,q(2) (X ′r) or Hp,q

(2) (X ′) when p+ q = n and p > 0

and q > 0?

What happens if the singular point is on the boundary of a smallStein set U? What can one say about Hp,q

(2) (U ′)


logo



p,q(2)



Open questions


(2) (X ′) when p+ q = n− 1 and



(2) (X ′) when p+ q = n and p > 0

and q > 0?


(2) (U ′)


logo



p,q(2)



Open questions


(2) (X ′) when p+ q = n− 1 and



(2) (X ′) when p+ q = n and p > 0

and q > 0?


(2) (U ′)


logo



p,q(2)




How about non-isolated singularities? Very few results there

Pardon-Stern showed in (1991) that Hn,q(2) (X ′) ∼= Hn,q(X) where

π : X → X is a desingularization and X a projective variety witharbitrary singularities. (Ruppenthal extended P-S’ theorem to compactcomplex Hermitian spaces X).

Berndtsson-Sibony obtained in (2001) parts of Pardon-Stern’s results asa special case of a more general theorem for solving ∂ for currents.

Øvrelid-V obtained in (2006) some results on product singularities (of thetype Y := X ′r ×∆ where X ′r a deleted neighborhood of an isolatedsingualarity and ∆ = |z| < 1.)Øvrelid-V in (2009) proved that Hp,1

(2) (X ′) <∞ when X irreducible

projective variety in CN .

Thank you.


Documents

L2--cohomology groups of some singular complex spacestjh/FinalNDMarch2013.pdf · L2-@-cohomology groups of some singular complex spaces Sophia Vassiliadou Department of Mathematics