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IntroductionDolbeault DGA of Formal NeighborhoodGeometric Description of Dolbeault DGA
Cohesive Modules
Coherent Analytic Sheaves over Formal Neighborhoods
Shilin Yu
Department of Mathematics, Penn State University
Algebraic Geometry Seminar
U. Wisconsin, Madison
Mar 9, 2012
Shilin Yu Coherent Analytic Sheaves over Formal Neighborhoods
psulogo
IntroductionDolbeault DGA of Formal NeighborhoodGeometric Description of Dolbeault DGA
Cohesive Modules
Outline
1 Introduction
2 Dolbeault DGA of Formal Neighborhood
Review of Formal Neighborhood
Dolbeault DGA of Formal Neighborhood
3 Geometric Description of Dolbeault DGA
Goal
Kapranov’s Result
The Case of General Embeddings
4 Cohesive Modules
Review of DG-categories
Cohesive Modules over a DGA
Application to Complex Analytical Geometry
Shilin Yu Coherent Analytic Sheaves over Formal Neighborhoods
psulogo
IntroductionDolbeault DGA of Formal NeighborhoodGeometric Description of Dolbeault DGA
Cohesive Modules
Motivations
The infinitesimal geometry of closed embeddings i : X ↪→ Y of complex
manifolds is encoded by the formal neighborhood X(∞)
Y .
Eventual Goals: understand
• The Yoneda algebra
Ext•OY(i∗OX , i∗OX) ' Ext•O
X(∞)Y
(i∗OX , i∗OX)
and its relation with Lie theory. (Kontsevich, Caldararu,
Calaque-Caldararu-Tu, etc.)
• The derived self-intersection X×RY X. (Arinkin-Caldararu)
• Infinitesimal deformations of X inside Y. (Deligne, Drinfeld, Feigin,
Kontsevich-Soibelman, Manetti, etc.)
Shilin Yu Coherent Analytic Sheaves over Formal Neighborhoods
psulogo
IntroductionDolbeault DGA of Formal NeighborhoodGeometric Description of Dolbeault DGA
Cohesive Modules
Review of Formal NeighborhoodDolbeault DGA of Formal Neighborhood
Formal Neighborhood
Let Y be a complex manifold and let X be a closed submanifold. Denote by
i : X ↪→ Y the closed embedding.
The rth formal neighborhood X(r)Y of X in Y is X equipped with the structure
sheaf
OX(r)
Y:= OY/I
r+1 ,
where I is the ideal sheaf of holomorphic functions vanishing on X.
Similarly, the (complete) formal neighborhood X(∞)
Y is defined as X equipped
with the structure sheaf
OX(∞)
Y:= lim←−
rOY/I
r+1 .
Shilin Yu Coherent Analytic Sheaves over Formal Neighborhoods
psulogo
IntroductionDolbeault DGA of Formal NeighborhoodGeometric Description of Dolbeault DGA
Cohesive Modules
Review of Formal NeighborhoodDolbeault DGA of Formal Neighborhood
Examples
• Let Y = Cn ×Cm = {(z, w)}, X = 0×Cm = {(0, w)}. Then
OX(∞)
Y' OXJz1 , · · · , znK.
• Suppose E→ X is a holomorphic vector bundle and let Y be its total
space. X is embedded in Y as the zero section. Then
OX(∞)
Y' S(E∨) = ∏
i≥0SiE∨ ,
which is the complete symmetric algebra generated by E∨
• Consider the diagonal embedding ∆ : X ↪→ X× X. Denote by
πi : X× X → X, i = 1, 2, the projections. Then OX(∞)
X×X, or rather
π1∗OX(∞)X×X
, is the holomorphic jet bundle J∞X of X.
Shilin Yu Coherent Analytic Sheaves over Formal Neighborhoods
psulogo
IntroductionDolbeault DGA of Formal NeighborhoodGeometric Description of Dolbeault DGA
Cohesive Modules
Review of Formal NeighborhoodDolbeault DGA of Formal Neighborhood
Dolbeault DGA
Take k = C.
• A differential graded algebra (= dga) (A• , d) is a Z≥0-graded associativealgebra A• over k with a derivation d : A• → A•+1, s.t.
• Leibniz’s rule: d(a · b) = d(a) · b + (−1)|a|a · d(b), a, b ∈ A•;• Integrability: d2 = 0.
• The Dolbeault dga (A•(Z), ∂) of a complex manifold Z:
• A•(Z) is the graded algebra of all C∞ (0, q)-forms on Z with the wedge
product.• ∂ is the (0, 1)-part of the de Rham differential.
• It is well known that the sheafy version A •Z of the Dolbeault dga gives a
soft, flat (Malgrange) resolution of OZ.
Shilin Yu Coherent Analytic Sheaves over Formal Neighborhoods
psulogo
IntroductionDolbeault DGA of Formal NeighborhoodGeometric Description of Dolbeault DGA
Cohesive Modules
Review of Formal NeighborhoodDolbeault DGA of Formal Neighborhood
• Question: Is there a notion of Dolbeault dga for formal neighborhoods?
• First attempt: Take the Dolbeault resolution of OX(r)
Yas an OY-module:
A •X(r)
Y= A •Y ⊗OY O
X(r)Y
= A •Y /(I r+1 ·A •Y )
and
A •X(∞)
Y= lim←−
rA •
X(r)Y
• In the case of Y = Cm+n = {(z, w)} and X = Cm = {(0, w)} ⊂ Y, such
defined A•X(∞)
Y
looks like
A•X(∞)
Y= A(X)Jzi , ziK⊗C ∧•(Cdz j ⊕Cdz j)
Too big for our applications...
Shilin Yu Coherent Analytic Sheaves over Formal Neighborhoods
psulogo
IntroductionDolbeault DGA of Formal NeighborhoodGeometric Description of Dolbeault DGA
Cohesive Modules
Review of Formal NeighborhoodDolbeault DGA of Formal Neighborhood
We want to build an ’economical’ Dolbeault dga A•(X(∞)
Y ) or its sheafy
version A •X(∞)
Y
for an arbitrary embedding X ↪→ Y, s.t.
• Locally it should look like A•(X ∩U)Jz1 , · · · , znK, no zi ’s or dzi ’s.
• If Y is the total space of a holomorphic vector bundle E→ X and X ↪→ Y
the zero section, then we should have
A•(X(∞)
Y ) ' A0,•X (S(E∨)),
where the RHS is the Dolbeault resolution of the (infinite-dimensional)
vector bundle S(E∨).
• In the case of diagonal embedding, A•(X(∞)
X×X) should be naturally
identified with A0,•X (J∞
X ), the Dolbeault resolution of the jet bundle J∞X .
Shilin Yu Coherent Analytic Sheaves over Formal Neighborhoods
psulogo
IntroductionDolbeault DGA of Formal NeighborhoodGeometric Description of Dolbeault DGA
Cohesive Modules
Review of Formal NeighborhoodDolbeault DGA of Formal Neighborhood
Review of Jet Bundles
• At each point p ∈ X, the fiber of the rth order jet bundle J rX is the space
of r-jets [ f ]rp of holomorphic functions defined about p:
J rp := OX,p/m
r+1p ,
with mp being the maximal ideal of OX,p.
• Holomorphic (resp. C∞-)sections of J rX are locally of the form (for
simplicity, assume dim X = 1):
s(p) = [a0(p) + a1(p)(z− p) + · · ·+ ar(p)(z− p)r]p ∈ J rp ,
for any p ∈ X, where ai are holomorphic (resp. C∞-)functions over U.
Shilin Yu Coherent Analytic Sheaves over Formal Neighborhoods
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IntroductionDolbeault DGA of Formal NeighborhoodGeometric Description of Dolbeault DGA
Cohesive Modules
Review of Formal NeighborhoodDolbeault DGA of Formal Neighborhood
Review of Jet Bundles (cont’d)
• Given a local C∞-section
s(p) = [a0(p) + a1(p)(z− p) + · · ·+ ar(p)(z− p)r]p ∈ J rp ,
the ∂-connection on J rX can be written locally as:
∂s = ∂a0 ⊗ [1] + ∂a1 ⊗ [z− p] + · · ·+ ∂ar ⊗ [(z− p)r] ∈ A0,1X (J r
X).
• C∞-sections of J rX can be thought of as (equiv. classes of) functions
defined near the diagonal X∆ ⊂ X× X, that are smooth in the first factor
of X× X and holomorphic in the second.
• However, such a perspective uses the special features of X× X and is
not available in the general case of an embedding i : X ↪→ Y.
Shilin Yu Coherent Analytic Sheaves over Formal Neighborhoods
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IntroductionDolbeault DGA of Formal NeighborhoodGeometric Description of Dolbeault DGA
Cohesive Modules
Review of Formal NeighborhoodDolbeault DGA of Formal Neighborhood
• To overcome this, we realize A0X(J
rX) as a quotient of the algebra
A0(X× X) by some equiv. relation.
Define a0r to be the ideal of A0(X× X) consisting of all functions f on
X× X s.t.
V1V2 · · ·Vk f |X∆= 0, 0 ≤ k ≤ r,
where Vj are smooth (1, 0)-vector fields in the direction to the second on
X× X. (The condition when k = 0 meas f |X∆= 0.)
Then there is an isomorphism σ : A0(X× X)/a0r'−→ AX(J r
X) defined by
[ f ] 7→ s, s(p) = [ f |{p}×X ]rp , ∀ p ∈ X.
• Observation: the Vj’s in the condition above can be chosen as being of
any direction.
Shilin Yu Coherent Analytic Sheaves over Formal Neighborhoods
psulogo
IntroductionDolbeault DGA of Formal NeighborhoodGeometric Description of Dolbeault DGA
Cohesive Modules
Review of Formal NeighborhoodDolbeault DGA of Formal Neighborhood
Dolbeault DGA of X(r)Y
For a closed embedding i : X ↪→ Y, define aqr (q ≥ 0), to be the subset of
A0,q(Y) consisting of (0,q)-formsω satisfying:
i∗(LV1LV2 · · · LVkω) = 0, 0 ≤ k ≤ r,
where Vj are any smooth (1, 0)-vector fields on Y and LVj denote the Lie
derivatives.
Lemma
a•r is a dg-ideal of (A0,•(Y), ∂).
Definition (Yu)
The dga A•(X(r)Y ) := A0,•(Y)/a•r with the inherited differential ∂ is called the
Dolbeault dga of the rth formal neighborhood X(r)Y .
Shilin Yu Coherent Analytic Sheaves over Formal Neighborhoods
psulogo
IntroductionDolbeault DGA of Formal NeighborhoodGeometric Description of Dolbeault DGA
Cohesive Modules
Review of Formal NeighborhoodDolbeault DGA of Formal Neighborhood
Dolbeault DGA of X(r)Y
We have inclusions of ideals a•r+1 ⊂ a•r and hence a projective system of
dgas A•(X(r+1)
Y )→ A•(X(r)Y ).
Definition (Yu)
The Dolbeault dga of the complete formal neighborhood X(∞)
Y is defined to be
A•(X(∞)
Y ) := lim←−rA•(X(r)
Y ).
Remark: In fact, we have a natural isomorphism
A0,•(Y)/ ⋂
r≥0a•r'−→ A•(X(∞)
Y ),
based on the fact that any formal power series is the Taylor series of some
C∞-function (E. Borel).
Shilin Yu Coherent Analytic Sheaves over Formal Neighborhoods
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IntroductionDolbeault DGA of Formal NeighborhoodGeometric Description of Dolbeault DGA
Cohesive Modules
Review of Formal NeighborhoodDolbeault DGA of Formal Neighborhood
Making A•(X(∞)
Y ) into a Sheaf
To make A•(X(∞)
Y ) into a sheaf of dgas A •X(∞)
Y
(over either Y or X), we define
over each open subset U ⊆ Y,
A •X(∞)
Y(U) := A•((U ∩Y)(∞)
U ).
In local charts AX(∞)
Ylooks like
A •X(∞)
Y(U) ' A•(U ∩Y)Jz1 , · · · , znK
We then obtain an exact sequence of sheaves
0→ OX(∞)
Y→ A 0
X(∞)Y
∂−→ A 1X(∞)
Y
∂−→ · · · ∂−→ A mX(∞)
Y→ 0,
where m = dim X. In other words, A •X(∞)
Y
is a soft resolution of OX(∞)
Y.
Shilin Yu Coherent Analytic Sheaves over Formal Neighborhoods
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IntroductionDolbeault DGA of Formal NeighborhoodGeometric Description of Dolbeault DGA
Cohesive Modules
Review of Formal NeighborhoodDolbeault DGA of Formal Neighborhood
Also we have the following:
Proposition
Let ∆ : X ↪→ X× X be the diagonal embedding. There are natural
isomorphisms of dgas
(A•(X(r)X×X), ∂)
'−→ (A0,•X (J r
X), ∂)
and
(A•(X(∞)
X×X), ∂)'−→ (A0,•
X (J∞X ), ∂).
Shilin Yu Coherent Analytic Sheaves over Formal Neighborhoods
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IntroductionDolbeault DGA of Formal NeighborhoodGeometric Description of Dolbeault DGA
Cohesive Modules
GoalKapranov’s ResultThe Case of General Embeddings
Geometric Description of Dolbeault DGA
• Question: How to describe A•(X(∞)
Y ) in terms of the differential
geometry of the embedding?
• Answer: We want to construct an isomorphism of dgas
(A•(X(∞)
Y ), ∂) ' (A0,•X (S(N∨)),D)
where N∨ = N∨X/Y is the conormal bundle of X, and write down the
differential D = ∂+ ? using connections, curvatures, etc.
• The answer in the case of diagonal embedding is due to M. Kapranov.
Shilin Yu Coherent Analytic Sheaves over Formal Neighborhoods
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IntroductionDolbeault DGA of Formal NeighborhoodGeometric Description of Dolbeault DGA
Cohesive Modules
GoalKapranov’s ResultThe Case of General Embeddings
Kapranov’s Result
Assume X is Kähler. Then the (1, 0)-part ∇ of the Levi-Civita connection on
TX is
• torsion-free: ∇XY−∇YX = [X, Y], ∀ X, Y ∈ Γ(TX).
• flat: [∇,∇] = 0
The curvature of the full Levi-Civita connection ∇ = ∇+ ∂ is
R = [∂,∇] ∈ A1,1(End(TX)) = A0,1(Hom(TX⊗ TX, TX)),
such that
• In fact, R ∈ A0,1(Hom(S2TX, TX) by torsion-freeness of ∇.
• ∂R = 0 and the corresponding class [R] ∈ Ext1OX
(S2TX, TX) is the
Atiyah class of TX.
Shilin Yu Coherent Analytic Sheaves over Formal Neighborhoods
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IntroductionDolbeault DGA of Formal NeighborhoodGeometric Description of Dolbeault DGA
Cohesive Modules
GoalKapranov’s ResultThe Case of General Embeddings
Kapranov’s Result
Define
R2 := R, Rn := ∇i−2R ∈ A0,1(Hom(TX⊗n , TX)), ∀ n > 2.
In fact, by flatness of ∇ we have
Rn ∈ A0,1(Hom(SnTX, TX)), ∀ n ≥ 2.
Now take the transpose of Rn:
R∗n ∈ A0,1(Hom(T∗X, SnT∗X))
and extend R∗n to derivations R∗n of degree +1 on A0,•(S(T∗X)) and set
D = ∂ + ∑n≥2
R∗n .
Shilin Yu Coherent Analytic Sheaves over Formal Neighborhoods
psulogo
IntroductionDolbeault DGA of Formal NeighborhoodGeometric Description of Dolbeault DGA
Cohesive Modules
GoalKapranov’s ResultThe Case of General Embeddings
Kapranov’s Result
Theorem (Kapranov)
Let ∆ : X ↪→ X× X. There is an isomorphism of dgas
(A•(X(∞)
X×X), ∂) ' (A0,•X (S(T∗X)),D)
where D = ∂ + ∑n≥2 R∗n. In particular, D2 = 0.
Originally in his paper, Kapranov constructed a ‘formal exponential map’
exp : X(∞)
TX → X(∞)
X×X
using the Kähler structure. The isomorphism above can be regarded as the
pullback of functions along exp∗.
Shilin Yu Coherent Analytic Sheaves over Formal Neighborhoods
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IntroductionDolbeault DGA of Formal NeighborhoodGeometric Description of Dolbeault DGA
Cohesive Modules
GoalKapranov’s ResultThe Case of General Embeddings
Sketch of Proof
We reformulate the proof by constructing the pullback map exp∗ directly.
Consider ∇ as the connection on T∗X:
∇ : A(T∗X)→ A(T∗X⊗ T∗X)
and lift it to a constant family of (1, 0)-connections on X× X in the direction
of the second component.
By the torsion-freeness and flatness of ∇, set
exp∗([ω]) = (∆∗ω, ∆∗∇ω, ∆∗∇2ω, · · · , ∆∗∇nω, · · · ) ∈ A0,•(S(T∗X))
for any [ω] ∈ A•(X(∞)
X×X), where ∇ω = ∂ω and ∇nω = ∇n−1∂ω.
Then check that
exp∗([∂ω])− ∂ exp∗([ω]) = ∑n≥2
R∗n(exp∗([ω]))
Q.E.D.
Shilin Yu Coherent Analytic Sheaves over Formal Neighborhoods
psulogo
IntroductionDolbeault DGA of Formal NeighborhoodGeometric Description of Dolbeault DGA
Cohesive Modules
GoalKapranov’s ResultThe Case of General Embeddings
The Case of General Embeddings
Theorem (Yu)
Let Y be a Kähler manifold and let X be a submanifold. Then there exists an
isomorphism of dgas
(A•(X(∞)
Y ), ∂) ' (A0,•X (S(N∨)),D)
in which D can be explicitly expressed in geometric terms.
Shilin Yu Coherent Analytic Sheaves over Formal Neighborhoods
psulogo
IntroductionDolbeault DGA of Formal NeighborhoodGeometric Description of Dolbeault DGA
Cohesive Modules
GoalKapranov’s ResultThe Case of General Embeddings
Some Differential Geometry...
• Choose a C∞-splitting τ : i∗TY → TX and ρ : TX → i∗TY of the ex. seq.
0→ TX → i∗TY → N → 0
and define
β := ∂τ ∈ A0,1X (Hom(N, TX)) = A0,1
X (Hom(T∗X, N∨)).
• The shape operator A : TX⊗ N → TX or T∗X → T∗X⊗ N∨ is defined
by
Aµ(V) = −τ(∇Vρ(µ)), ∀ µ ∈ C∞(N), V ∈ C∞(TX).
• Decompose the curvature R ∈ A0,1Y (Hom(T∗Y, S2T∗Y)) of Y over X to
get
R⊥ ∈ A0,1X (Hom(N∨ , S2 N∨)) and R> ∈ A0,1
X (Hom(T∗X, S2 N∨)).
Shilin Yu Coherent Analytic Sheaves over Formal Neighborhoods
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IntroductionDolbeault DGA of Formal NeighborhoodGeometric Description of Dolbeault DGA
Cohesive Modules
GoalKapranov’s ResultThe Case of General Embeddings
Low-Order Terms of D
• D acts on f ∈ A0,0X (S0 N∨) = A0,0(X) by
D f = ∂ f +β ◦ ∂ f + (β · A + R>) ◦ ∂ f + · · · .
• D acts on µ ∈ A0,0X (S1 N∨) = A0,0
X (N∨) by
Dµ = ∂µ + (β · ∇Nµ + R⊥ ◦µ) + · · · ,
where ∇N is the induced connection on N∨.
• If you try to show D2 = 0 for these terms by hand, e.g.,
∂((β · ∇N + R⊥) ◦µ) + (β · ∇N + R⊥) ◦ ∂µ = 0,
then you will see the Ricci equation!
• If you try harder, you will get all the Gauss-Codazzi-Ricci equations.
Shilin Yu Coherent Analytic Sheaves over Formal Neighborhoods
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IntroductionDolbeault DGA of Formal NeighborhoodGeometric Description of Dolbeault DGA
Cohesive Modules
Review of DG-categoriesCohesive Modules over a DGAApplication to Complex Analytical Geometry
DG-Categories
Definition
• A differential graded (dg-)category C is a category in which the set of
morphisms C(X, Y) between two objects has the structure of a complex,
in such a way that the composition maps
C(Y, Z)⊗k C(X, Y)→ C(X, Z), X, Y, Z ∈ Obj(C)
are chain maps of complexes
• A dg functor F : C1 → C2 is given by chain maps of complexes
F : C1(X, Y)→ C2(F(X), F(Y)), X, Y ∈ Obj(C)• The homotopy category Ho C of the dg category C has
• objects: same as Obj C• morphisms: Ho C(X, Y) := H0(C(X, Y))
Shilin Yu Coherent Analytic Sheaves over Formal Neighborhoods
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IntroductionDolbeault DGA of Formal NeighborhoodGeometric Description of Dolbeault DGA
Cohesive Modules
Review of DG-categoriesCohesive Modules over a DGAApplication to Complex Analytical Geometry
Example: DG Category of Complexes
Let R be a k-algebra. Define Cdg(R) to be the dg-category of complexes of
(left) R-modules, with
• Objects: complexes (M• , dM) of R-modules
• Morphisms: Homk(M• , N•) := ∏i∈Z HomR(Mi , Ni+k)
· · · dM> Mi dM
> Mi+1 dM> · · ·
· · ·dN> Ni+n
dN>
f>
Ni+n+1dN>
f>
· · ·
with the differential
dHom( f ) = dN ◦ f − (−1)n f ◦ dM
Then Ho Cdg(R) ' K(R) = the category of complexes of R-modules with
homotopy classes of chain maps as morphisms.
Shilin Yu Coherent Analytic Sheaves over Formal Neighborhoods
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IntroductionDolbeault DGA of Formal NeighborhoodGeometric Description of Dolbeault DGA
Cohesive Modules
Review of DG-categoriesCohesive Modules over a DGAApplication to Complex Analytical Geometry
Cohesive Modules over a DGA
Definition (J. Block)
Let A = (A• , d) be a dga. A cohesive module E = (E• ,E) over A is:
1 a bounded, finitely generated projective Z-graded (left) module E• over
A = A0, with
2 a superconnection E : A• ⊗A E• → A• ⊗A E• of total degree 1,satisfying
• Leibniz rule: E(ω · e) = dω · e + (−1)|ω|ω ·E(1⊗ e)• Integrability: E ◦E = 0.
By the Leibniz rule, E is determined by its values on E• ⊆ A• ⊗ E•. Thus
E = E0 + E1 + E2 + · · · ,
where the kth component is
Ek : E• → Ak ⊗A E•−k+1 .
Shilin Yu Coherent Analytic Sheaves over Formal Neighborhoods
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IntroductionDolbeault DGA of Formal NeighborhoodGeometric Description of Dolbeault DGA
Cohesive Modules
Review of DG-categoriesCohesive Modules over a DGAApplication to Complex Analytical Geometry
Cohesive Modules over a DGA
All Ek are A-linear except for E1, which satisfies the Leibniz rule.
Under this decomposition, the integrability condition E ◦ E = 0 can be written
as:
E0 ◦ E0 = 0 (E• ,E0) is a chain complex
E0E1 + E1E0 = 0 E0 (anti-)commutes with E1
E0E2 + E1E1 + E2E0 = 0 E1 is flat up to a homotopy E2
· · · ‘higher homotopies’
Shilin Yu Coherent Analytic Sheaves over Formal Neighborhoods
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IntroductionDolbeault DGA of Formal NeighborhoodGeometric Description of Dolbeault DGA
Cohesive Modules
Review of DG-categoriesCohesive Modules over a DGAApplication to Complex Analytical Geometry
Cohesive Modules over the Dolbeault dga
Let A = (A0,•(X), ∂) be the Dolbeault dga of a (compact) complex manifold.
Given a cohesive module (E• ,E) over (A0,•(X), ∂), then:
• Each Ek can be regarded as a C∞-vector bundle over X.
• (E• ,E0) is a complex of C∞-vector bundles with (non-flat)
(0, 1)-connections.
• If Ek = 0, ∀ k ≥ 2, which forces E1 ◦ E1 = 0, then we have a complex of
holomorphic vector bundles.
• In general, (E• ,E) is sort of ‘twisted’ complex of holomorphic vector
bundles.
Shilin Yu Coherent Analytic Sheaves over Formal Neighborhoods
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IntroductionDolbeault DGA of Formal NeighborhoodGeometric Description of Dolbeault DGA
Cohesive Modules
Review of DG-categoriesCohesive Modules over a DGAApplication to Complex Analytical Geometry
Perfect Category of Cohesive Modules
Definition (J. Block)
The perfect category of cohesive modules PA over a dga A = (A• , d) is the
dg-category that consists of
• Objects: cohesive modules over (A• , d)
• Morphisms:
P kA(E1 , E2) =
φ : A• ⊗A E•1 → A• ⊗A E•2
∣∣∣∣∣∣∣∣deg(φ) = k,
φ(a · e) = a ·φ(e),∀ a ∈ A• .
equipped with a differential d of degree 1 s.t.
d(φ)(e) = E2(φ(e))− (−1)|φ|φ(E1(e))
Shilin Yu Coherent Analytic Sheaves over Formal Neighborhoods
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IntroductionDolbeault DGA of Formal NeighborhoodGeometric Description of Dolbeault DGA
Cohesive Modules
Review of DG-categoriesCohesive Modules over a DGAApplication to Complex Analytical Geometry
Advantages
Why cohesive modules?
• On a smooth projective variety, every coherent sheaf can be resolved by
vector bundles. This is no longer true in complex analytical geometry...
But we have
Theorem (J. Block)
Let A = (A0,•(X), ∂) be the Dolbeault dga of a compact complex manifold X.
Then HoPA is equivalent to Dbcoh(X), the bounded derived category of
OX-modules with coherent cohomology.
• It also works in noncommutative geometry, where local constructions are
unavailable. (e.g., noncommutative T-duality, J. Block & C. Daenzer)
• ∞-version of Riemann-Hilbert correspondence (J. Block & A. Smith)
Shilin Yu Coherent Analytic Sheaves over Formal Neighborhoods
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IntroductionDolbeault DGA of Formal NeighborhoodGeometric Description of Dolbeault DGA
Cohesive Modules
Review of DG-categoriesCohesive Modules over a DGAApplication to Complex Analytical Geometry
Sketch of Proof
Part 1: Making a sheaf: For E = (E• ,E) ∈ PA, define the sheaves E p,q by
E p,q(U) = A0,q(U)⊗A(X) Ep
Define a fully faithful functor
α : HoPA −→ Dbcoh(X)
(E• ,E) 7−→ (E • ,E) = (∑p+q=• E p,q ,E)
Then it can be shown that (E • ,E) has coherent cohomology and
ExtkOX
(E •1 , E •2 ) ' Hk(PA(E1 , E2)).
Shilin Yu Coherent Analytic Sheaves over Formal Neighborhoods
psulogo
IntroductionDolbeault DGA of Formal NeighborhoodGeometric Description of Dolbeault DGA
Cohesive Modules
Review of DG-categoriesCohesive Modules over a DGAApplication to Complex Analytical Geometry
Sketch of Proof (cont’d)
Part 2: Given (E • , d) ∈ Dbcoh(X), we need to find a cohesive module
E = (E• ,E) s.t. there is quasi-isomorphism α(E) '−→ (E• , d).
Try: set E •∞ = E • ⊗OX AX, then we have a quasi-isomorphism
(E • , d) '−→ (A •X ⊗OX E • , ∂⊗ 1 + 1⊗ d).
Want to set (E• ,E) = (Γ(X, E •∞), ∂⊗ 1 + 1⊗ d).
But... it is only a quasi-cohesive module over A (i.e., without finiteness and
projectivity conditions).
Shilin Yu Coherent Analytic Sheaves over Formal Neighborhoods
psulogo
IntroductionDolbeault DGA of Formal NeighborhoodGeometric Description of Dolbeault DGA
Cohesive Modules
Review of DG-categoriesCohesive Modules over a DGAApplication to Complex Analytical Geometry
Sketch of Proof (cont’d)
However, there is a (strictly) perfect complex (E• ,E0) of A(X)-modules and a
quasi-isomorphism
e0 : (E• ,E0)→ (Γ(X, E •∞), 1⊗ d)
since (E •∞ , 1⊗ d) is a perfect complex of AX-modules (Illusie).
Block showed that E0 can be extended to a Z-connection E and e0 can be
extended at the same time to a closed morphism
e : (E• ,E)→ (Γ(X, E •∞), d⊗ 1 + 1⊗ ∂)
which induces a quasi-isomorphism on the level of sheaves. Thus the
cohesive module (E• ,E) serves our purpose. Q.E.D.
Shilin Yu Coherent Analytic Sheaves over Formal Neighborhoods
psulogo
IntroductionDolbeault DGA of Formal NeighborhoodGeometric Description of Dolbeault DGA
Cohesive Modules
Review of DG-categoriesCohesive Modules over a DGAApplication to Complex Analytical Geometry
Dolbeault Resolutions over Formal Neighborhood
It is not clear if AX(∞)
Yis flat over O
X(∞)Y
. However, we can still show the
following:
Theorem (Yu)
The functor F 7→ AX(∞)
Y⊗O
X(∞)Y
F from Coh(X(∞)
Y ) to the category of
AX(∞)
Y-modules is exact.
Theorem (Yu)
Suppose F ∈ Coh(X(∞)
Y ), then we have an exact sequence of
OX(∞)
Y-modules:
0→ F → A 0X(∞)
Y⊗O
X(∞)Y
F∂−→ A 1
X(∞)Y⊗O
X(∞)Y
F∂−→ · · · ∂−→ A m
X(∞)Y⊗O
X(∞)Y
F → 0,
where m = dim X.
Shilin Yu Coherent Analytic Sheaves over Formal Neighborhoods
psulogo
IntroductionDolbeault DGA of Formal NeighborhoodGeometric Description of Dolbeault DGA
Cohesive Modules
Review of DG-categoriesCohesive Modules over a DGAApplication to Complex Analytical Geometry
Cohesive Modules over Formal Neighborhood
Theorem (Yu)
A holomorphic vector bundle E over X(∞)
Y (i.e., a locally free OX(∞)
Y-module of
finite type) is equivalent to a locally free AX(∞)
Y-module of finite type E∞
equipped with a ∂-connection ∂E : E∞ → A 1X(∞)
Y
⊗AX(∞)
Y
E∞ s.t. ∂2E = 0.
Combining these results and follow the same argument of Block, we get
Theorem (hopefully...)
Suppose X is a closed compact submanifold of a complex manifold Y. Let
A = (A•(X(∞)
Y ), ∂) be the Dolbeault dga of the formal neighborhood X(∞)
Y and
PA be the perfect category of cohesive modules over A. Then
HoPA ' Dbcoh(X(∞)
Y ).
Shilin Yu Coherent Analytic Sheaves over Formal Neighborhoods
psulogo
IntroductionDolbeault DGA of Formal NeighborhoodGeometric Description of Dolbeault DGA
Cohesive Modules
Review of DG-categoriesCohesive Modules over a DGAApplication to Complex Analytical Geometry
The End
Thank You!
Shilin Yu Coherent Analytic Sheaves over Formal Neighborhoods