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Connections Holomorphic connections Lie algebroid connections on a Lie algebroid

Atiyah classes of Lie Algebroids

Francesco Bottacin

University of Padova

ISAAC 2013 – Session 22

Cracow, 5–9 August 2013

Outline

1 ConnectionsMotivationLie algebroid connections

2 Holomorphic connectionsHolomorphic Lie algebroidsLie algebroid jets, Atiyah classes, . . .The cohomological Bianchi identity

3 Lie algebroid connections on a Lie algebroidThe (A , ])-Atiyah class of AThe Lie algebra structure

Outline

Fiber spaces and connections

Basic ingredients:

M a smooth (differentiable) manifold

π : E →M a smooth (complex) vector bundle over M (moregenerally, a smooth fiber bundle)

EP = π−1(P) the fiber over P ∈M

∇ : Γ (E)→ Γ (E)⊗Ω1M a connection on E

Using the connection, we can define the so-called “paralleltransport”Then, “parallel transport” provides isomorphisms betweenfibers of E over different points of M

Basic ingredients:

Using the connection, we can define the so-called “paralleltransport”

Then, “parallel transport” provides isomorphisms betweenfibers of E over different points of M

Basic ingredients:

Using the connection, we can define the so-called “paralleltransport”Then, “parallel transport” provides isomorphisms betweenfibers of E over different points of M

Then, if we have some kind of “structure” on E that iscompatible with the connection, properties that hold at somepoint of M must also hold at any other point of M .

The presence of a connection compatible with some geometricstructure prevents singular behaviour.

Example

Assume there exists a Poisson structure on M . Let E = ∧2TMand σ ∈ Γ (∧2TM) be the Poisson bivector.If there exists a connection compatible with σ , then σ musthave the same rank at all points of M (assume M is connected),hence M is a regular Poisson manifold.

Example

Let X be a complex manifold (or smooth algebraic variety).Let E be a DX -module that is coherent as an OX -module.The DX -module structure on E is equivalent to a flatconnection on E . Then, the presence of the flat connectionimplies that E is locally free.

IdeaModify the notion of connection in order to allow some kind ofsingular behaviour.

Example

Let X be a complex manifold (or smooth algebraic variety).Let E be a DX -module that is coherent as an OX -module.The DX -module structure on E is equivalent to a flatconnection on E . Then, the presence of the flat connectionimplies that E is locally free.

IdeaModify the notion of connection in order to allow some kind ofsingular behaviour.

Lie algebroids

DefinitionA Lie algebroid over a smooth manifold M is a vector bundleπ : A →M , together with a Lie algebra structure [·, ·] on thespace of sections Γ (A), and a morphism of vector bundles] : A → TM (the anchor) such that:

the induced map ] : Γ (A)→ Γ (TM) is a Lie algebrahomomorphism

for any sections α,β ∈ Γ (A) and any f ∈ C∞(M), we have

[α, fβ] = f [α,β] + ]α(f)β

The image of ] defines an integrable distribution in M . Theintegrable leaves are the orbits of A ; they form the orbitfoliation.

Lie algebroid connections

Let E be a complex vector bundle over M . Let π : A →M be aLie algebroid, let ] : A → TM be the anchor map. Let[ : T ∗M → A ∗ be the dual of the anchor map and letdA : C∞M → A ∗ be the C-derivation defined as the compositiondA = [ d , where d : C∞M → T ∗M is the usual differential.

Definition

An (A , ])-connection on E is a C-linear map

∇ : E → E ⊗A ∗

such that∇(fs) = f∇(s)+ s ⊗dA (f)

for any s ∈ Γ (E) and f ∈ Γ (C∞M ).

Let E be a complex vector bundle over M . Let π : A →M be aLie algebroid, let ] : A → TM be the anchor map. Let[ : T ∗M → A ∗ be the dual of the anchor map and letdA : C∞M → A ∗ be the C-derivation defined as the compositiondA = [ d , where d : C∞M → T ∗M is the usual differential.

Definition

An (A , ])-connection on E is a C-linear map

∇ : E → E ⊗A ∗

such that∇(fs) = f∇(s)+ s ⊗dA (f)

for any s ∈ Γ (E) and f ∈ Γ (C∞M ).

Theorem

Let E be a complex vector bundle over M. Let (A , ]) be a Liealgebroid over M. Then (A , ])-connections on E always exist.

Proof.Standard argument, using partitions of unity on M .

Most of the usual theory of connections can be extended to Liealgebroid connections.Given an (A , ])-connection ∇ : E → E ⊗A ∗ we can defineoperators

∇ : E ⊗∧pA ∗→ E ⊗∧p+1A ∗

Then we define the curvature of the (A , ])-connection

R = ∇∇ : E → E ⊗∧2A ∗

Theorem

Most of the usual theory of connections can be extended to Liealgebroid connections.

Given an (A , ])-connection ∇ : E → E ⊗A ∗ we can defineoperators

∇ : E ⊗∧pA ∗→ E ⊗∧p+1A ∗

R = ∇∇ : E → E ⊗∧2A ∗

Theorem

∇ : E ⊗∧pA ∗→ E ⊗∧p+1A ∗

R = ∇∇ : E → E ⊗∧2A ∗

Theorem

∇ : E ⊗∧pA ∗→ E ⊗∧p+1A ∗

R = ∇∇ : E → E ⊗∧2A ∗

The usual Chern–Weil theory of characteristic classes extendsto Lie algebroid connections.

Using the curvature R , which is a section of End(E)⊗∧2A ∗, wecan define (A , ])-Chern classes of E

c(A ,])i (E) ∈ H 2i (A) (Lie algebroid cohomology)

Def. of Lie algebroid cohomology

These classes are the image of the usual Chern classes underthe map H ∗dR(M)→ H ∗(A) induced by the dual of the anchormap.

Holomorphic connections

Holomorphic Lie algebroid connections

Holomorphic Lie algebroids

Let X be a complex manifold (or a smooth complex algebraicvariety), OX the sheaf of holomorphic functions on X .Let (A , ]) be a holomorphic Lie algebroid and E a holomorphicvector bundle over X (more generally, we can take aquasi-coherent OX -module E).

Definition

A (holomorphic) (A , ])-connection on E is a C-linearholomorphic map

∇ : E → E ⊗A ∗

such that∇(fs) = f∇(s)+ s ⊗dA (f),

for any sections s of E and f of OX

Holomorphic Lie algebroids

Let X be a complex manifold (or a smooth complex algebraicvariety), OX the sheaf of holomorphic functions on X .Let (A , ]) be a holomorphic Lie algebroid and E a holomorphicvector bundle over X (more generally, we can take aquasi-coherent OX -module E).

Definition

A (holomorphic) (A , ])-connection on E is a C-linearholomorphic map

∇ : E → E ⊗A ∗

such that∇(fs) = f∇(s)+ s ⊗dA (f),

for any sections s of E and f of OX

In the holomorphic (or algebraic) setting there are nopartitions of unity. It follows that there are obstructions to theexistence of holomorphic Lie algebroid connections.

For a holomorphic vector bundle E over X we define the bundleof first (A , ])-jets of E by setting

J 1(A ,])(E) = E ⊕ (E ⊗OX

A ∗)

as C-modules.The (right) OX -module structure on J 1

(A ,])(E) is defined by

(s ,σ) · f = (fs , fσ + s ⊗dA f)

for sections f ∈ Γ (OX ), s ∈ Γ (E) and σ ∈ Γ (E ⊗A ∗).

In the holomorphic (or algebraic) setting there are nopartitions of unity. It follows that there are obstructions to theexistence of holomorphic Lie algebroid connections.For a holomorphic vector bundle E over X we define the bundleof first (A , ])-jets of E by setting

J 1(A ,])(E) = E ⊕ (E ⊗OX

A ∗)

as C-modules.

The (right) OX -module structure on J 1(A ,])(E) is defined by

(s ,σ) · f = (fs , fσ + s ⊗dA f)

In the holomorphic (or algebraic) setting there are nopartitions of unity. It follows that there are obstructions to theexistence of holomorphic Lie algebroid connections.For a holomorphic vector bundle E over X we define the bundleof first (A , ])-jets of E by setting

J 1(A ,])(E) = E ⊕ (E ⊗OX

A ∗)

as C-modules.The (right) OX -module structure on J 1

(A ,])(E) is defined by

(s ,σ) · f = (fs , fσ + s ⊗dA f)

Jet bundle sequence

There is an exact sequence of OX -modules

0 −→ E ⊗A ∗ −→ J 1(A ,])(E) −→ E −→ 0

(which is split as a sequence of C-modules but not, in general,as a sequence of OX -modules).

LemmaA splitting of the above sequence is equivalent to aholomorphic (A , ])-connection on E.

Proof.

Trivial. Follows from the definition of the (right) OX -modulestructure of J 1

(A ,])(E).

Jet bundle sequence

0 −→ E ⊗A ∗ −→ J 1(A ,])(E) −→ E −→ 0

Proof.

(A ,])(E).

Jet bundle sequence

0 −→ E ⊗A ∗ −→ J 1(A ,])(E) −→ E −→ 0

Proof.

(A ,])(E).

Atiyah classes

The above exact sequence gives rise to an extension class

Definition

The (A , ])-Atiyah class of a vector bundle E is the class

a(A ,])(E) ∈ Ext1(E ,E ⊗A ∗) = H 1(M ,Hom(E ,E ⊗A ∗))

corresponding to the extension

0 −→ E ⊗A ∗ −→ J 1(A ,])(E) −→ E −→ 0

Corollary

A holomorphic (A , ])-connection on a vector bundle E exists ifand only if the (A , ])-Atiyah class of E vanishes.

Atiyah classes

Definition

0 −→ E ⊗A ∗ −→ J 1(A ,])(E) −→ E −→ 0

Corollary

Atiyah classes

Definition

0 −→ E ⊗A ∗ −→ J 1(A ,])(E) −→ E −→ 0

Corollary

Atiyah classes

The usual Atiyah class of E is the class a(E) ∈ Ext1(E ,E ⊗Ω1X )

that corresponds to the extension

0→ E ⊗Ω1X → J 1(E)→ E → 0

We have a morphism of extensions

0 // E ⊗Ω1X

J 1(E) //

0 // E ⊗A ∗ // J 1(A ,])(E)

// E // 0

that induces a morphism

Ext1(E ,E ⊗Ω1X )→ Ext1(E ,E ⊗A ∗)

The (A , ])-Atiyah class of E is the image of the usual Atiyahclass a(E) under the previous map.

Atiyah classes

0→ E ⊗Ω1X → J 1(E)→ E → 0

0 // E ⊗Ω1X

J 1(E) //

0 // E ⊗A ∗ // J 1(A ,])(E)

// E // 0

Ext1(E ,E ⊗Ω1X )→ Ext1(E ,E ⊗A ∗)

Atiyah classes

0→ E ⊗Ω1X → J 1(E)→ E → 0

0 // E ⊗Ω1X

J 1(E) //

0 // E ⊗A ∗ // J 1(A ,])(E)

// E // 0

Ext1(E ,E ⊗Ω1X )→ Ext1(E ,E ⊗A ∗)

Higher jet bundles

For any r ≥ 1 we can define inductively the sheaf of (A , ])-r-jetsof E by setting

J r(A ,])(E) = J r−1

(A ,])(E)⊕ (E ⊗Sr A ∗)

as C-modules (we set J 0(A ,])(E) = E ), where Sr A ∗ denotes the

symmetric r-th power of A ∗. We just have to define a suitable(right) OX -module structure of J r

(A ,])(E).

There is an exact sequence

0→ E ⊗Sr A ∗→ J r(A ,])(E)→ J r−1

(A ,])(E)→ 0

Higher jet bundles

For any r ≥ 1 we can define inductively the sheaf of (A , ])-r-jetsof E by setting

J r(A ,])(E) = J r−1

(A ,])(E)⊕ (E ⊗Sr A ∗)

as C-modules (we set J 0(A ,])(E) = E ), where Sr A ∗ denotes the

symmetric r-th power of A ∗. We just have to define a suitable(right) OX -module structure of J r

(A ,])(E).There is an exact sequence

0→ E ⊗Sr A ∗→ J r(A ,])(E)→ J r−1

(A ,])(E)→ 0

Differential operators

The sheaf of rings DX of finite-order (holomorphic) differentialoperators on X is generated, as an algebra, by OX and by TX .

We define D(A ,]) to be the algebra generated by OX and A , withthe commutation relations given by

af = ](a)(f)+ fa and a1a2 = a2a1 + [a1,a2]

for a , a1, a2 ∈ Γ (A) and f ∈ Γ (OX ).D(A ,]) is endowed with a filtration

0 ⊂ OX =D≤0(A ,]) ⊂ D

≤1(A ,]) ⊂ · · · ⊂ D

≤r(A ,]) ⊂ · · · ⊂ D(A ,])

where the OX -module D≤r(A ,]) is the dual of the sheaf of r-th

(A , ])-jets J r(A ,])(OX )

D≤r(A ,]) = HomOX

(J r(A ,])(OX ),OX )

The sheaf of rings DX of finite-order (holomorphic) differentialoperators on X is generated, as an algebra, by OX and by TX .We define D(A ,]) to be the algebra generated by OX and A , withthe commutation relations given by

for a , a1, a2 ∈ Γ (A) and f ∈ Γ (OX ).

D(A ,]) is endowed with a filtration

0 ⊂ OX =D≤0(A ,]) ⊂ D

≤1(A ,]) ⊂ · · · ⊂ D

≤r(A ,]) ⊂ · · · ⊂ D(A ,])

(A , ])-jets J r(A ,])(OX )

D≤r(A ,]) = HomOX

(J r(A ,])(OX ),OX )

The sheaf of rings DX of finite-order (holomorphic) differentialoperators on X is generated, as an algebra, by OX and by TX .We define D(A ,]) to be the algebra generated by OX and A , withthe commutation relations given by

for a , a1, a2 ∈ Γ (A) and f ∈ Γ (OX ).D(A ,]) is endowed with a filtration

0 ⊂ OX =D≤0(A ,]) ⊂ D

≤1(A ,]) ⊂ · · · ⊂ D

≤r(A ,]) ⊂ · · · ⊂ D(A ,])

(A , ])-jets J r(A ,])(OX )

D≤r(A ,]) = HomOX

(J r(A ,])(OX ),OX )

Example

The dual of the 1st jet sequence for E ∗

0 −→ E ∗ ⊗A ∗ −→ J 1(A ,])(E

∗) −→ E ∗ −→ 0

is the exact sequence

0 −→ E −→D≤1(A ,]) ⊗ E −→ E ⊗A −→ 0

that corresponds to the class

−a(A ,])(E) ∈ Ext1(E ⊗A ,E) = Ext1(E ,E ⊗A ∗)

The map σ :D≤r(A ,])→ Sr A , that associates to a

(A , ])-differential operator its highest order term, is welldefined and is called the principal symbol map.

For every r ≥ 0, there is an exact sequence

0→D≤r−1(A ,]) →D

≤r(A ,])→ Sr A → 0

which is the dual of

0→ Sr A ∗→ J r(A ,])(OX )→ J r−1

(A ,])(OX )→ 0

The associated graded ring of the filtered ring D(A ,]) isisomorphic to the symmetric algebra over A

gr(D(A ,])

) S·(A)

(A , ])-differential operator its highest order term, is welldefined and is called the principal symbol map.For every r ≥ 0, there is an exact sequence

0→D≤r−1(A ,]) →D

≤r(A ,])→ Sr A → 0

0→ Sr A ∗→ J r(A ,])(OX )→ J r−1

(A ,])(OX )→ 0

gr(D(A ,])

) S·(A)

(A , ])-differential operator its highest order term, is welldefined and is called the principal symbol map.For every r ≥ 0, there is an exact sequence

0→D≤r−1(A ,]) →D

≤r(A ,])→ Sr A → 0

0→ Sr A ∗→ J r(A ,])(OX )→ J r−1

(A ,])(OX )→ 0

gr(D(A ,])

) S·(A)

Notations

We introduce some notation in order to state the maintechnical result (the so-called Cohomological Bianchi Identity).

Let a , b ∈ H 1(X ,End(E)⊗A ∗)Their cup-product is

a ∪ b ∈ H 2(X ,End(E)⊗ End(E)⊗A ∗ ⊗A ∗)

Consider the map

End(E)⊗ End(E)⊗A ∗ ⊗A ∗→ End(E)⊗S2(A ∗)

φ⊗ψ ⊗α ⊗ β 7→ [φ,ψ]⊗ (α β)

We denote by [a ∪ b ] ∈ H 2(X ,End(E)⊗S2(A ∗)) the image ofa ∪ b under the induced map in cohomology.

Notations

a ∪ b ∈ H 2(X ,End(E)⊗ End(E)⊗A ∗ ⊗A ∗)

Consider the map

φ⊗ψ ⊗α ⊗ β 7→ [φ,ψ]⊗ (α β)

Notations

a ∪ b ∈ H 2(X ,End(E)⊗ End(E)⊗A ∗ ⊗A ∗)

Consider the map

φ⊗ψ ⊗α ⊗ β 7→ [φ,ψ]⊗ (α β)

Notations

a ∪ b ∈ H 2(X ,End(E)⊗ End(E)⊗A ∗ ⊗A ∗)

Consider the map

φ⊗ψ ⊗α ⊗ β 7→ [φ,ψ]⊗ (α β)

Notations

Let a ∈ H 1(X ,End(E)⊗A ∗) = Ext1(E ,E ⊗A ∗) = Ext1(A ⊗ E ,E)Let c ∈ Ext1(A ⊗A ,A)

Let us consider the composition

S2(A)⊗ E → A ⊗A ⊗ Ec⊗1−→ A ⊗ E [1]

a−→ E [2]

We denote by

a ∗ c ∈ Hom(S2(A)⊗ E ,E [2]) = Ext2(S2(A)⊗ E ,E)

= H 2(X ,End(E)⊗S2(A ∗))

the corresponding element.

Notations

S2(A)⊗ E → A ⊗A ⊗ Ec⊗1−→ A ⊗ E [1]

a−→ E [2]

We denote by

= H 2(X ,End(E)⊗S2(A ∗))

Notations

S2(A)⊗ E → A ⊗A ⊗ Ec⊗1−→ A ⊗ E [1]

a−→ E [2]

We denote by

= H 2(X ,End(E)⊗S2(A ∗))

The cohomological Bianchi identity

Theorem (Cohomological Bianchi identity)

Let a(A ,])(E) ∈ Ext1(E ,E ⊗A ∗) = H 1(M ,Hom(E ,E ⊗A ∗)) be the(A , ])-Atiyah class of a vector bundle E.

Let a(A ,])(A) ∈ Ext1(A ,A ⊗A ∗) = H 1(M ,Hom(A ,A ⊗A ∗)) be the(A , ])-Atiyah class of A.

Then we have the identity

2 [a(A ,])(E)∪a(A ,])(E)]+a(A ,])(E) ∗a(A ,])(A) = 0

in H 2(X ,End(E)⊗S2(A ∗)).

(the name comes from the Bianchi identity for the curvature ofa connection)

The cohomological Bianchi identity: proof

Lemma 1Let M be a D(A ,])-module with a good filtration Mi by vectorbundles. Let

µ : A ⊗Mi /Mi−1→Mi+1/Mi

be the multiplication map, induced by the D(A ,])-modulestructure on M . Let

φi ∈ Ext1(Mi+1

Mi−1

)be the extension class of the exact sequence

0 −→ Mi

Mi−1−→

Mi−→ 0

Then the class −a(A ,])(Mi /Mi−1) is the difference between thefollowing composition of morphisms:

A ⊗ Mi

Mi−1

µ−→

φi−→ Mi

Mi−1[1]

A ⊗ Mi

Mi−1

1⊗φi−1−→ A ⊗ Mi−1

Mi−2[1]

µ−→ Mi

Mi−1[1]

Proof of Lemma 1.Generalization of the proof of a similar result, due to Angénioland Lejeune-Jalabert: Le théorème de Riemann-Roch singulierpour les D-modules, Astérisque 130 (1985).

Then the class −a(A ,])(Mi /Mi−1) is the difference between thefollowing composition of morphisms:

A ⊗ Mi

Mi−1

µ−→

φi−→ Mi

Mi−1[1]

A ⊗ Mi

Mi−1

1⊗φi−1−→ A ⊗ Mi−1

Mi−2[1]

µ−→ Mi

Mi−1[1]

Proof of Lemma 1.Generalization of the proof of a similar result, due to Angénioland Lejeune-Jalabert: Le théorème de Riemann-Roch singulierpour les D-modules, Astérisque 130 (1985).

Let M =D(A ,]) ⊗ E with the filtration Mi =D≤i(A ,]) ⊗ E

The exact sequence

0 −→ M1

M0−→ M2

M1−→ 0

0 −→ A ⊗ E −→D≤2

(A ,]) ⊗ E

E−→ S2(A)⊗ E −→ 0

Let ξ ∈ Ext1(S2(A)⊗ E ,A ⊗ E) be the corresponding extensionclass (it is the φ1 of Lemma 1).Let σ : A ⊗A → S2(A) be the symmetrization.

Lemma 2

a(A ,])(A ⊗ E) = −ξ (σ ⊗1)−1⊗a(A ,])(E)

Let M =D(A ,]) ⊗ E with the filtration Mi =D≤i(A ,]) ⊗ E

The exact sequence

0 −→ M1

M0−→ M2

M1−→ 0

0 −→ A ⊗ E −→D≤2

(A ,]) ⊗ E

E−→ S2(A)⊗ E −→ 0

Let ξ ∈ Ext1(S2(A)⊗ E ,A ⊗ E) be the corresponding extensionclass (it is the φ1 of Lemma 1).Let σ : A ⊗A → S2(A) be the symmetrization.

Lemma 2

a(A ,])(A ⊗ E) = −ξ (σ ⊗1)−1⊗a(A ,])(E)

Proof of Lemma 2.Follows from Lemma 1.Since Mi =D≤i

(A ,]) ⊗ E , we have M1/M0 = A ⊗ E . Hence−a(A ,])(M1/M0) = −a(A ,])(A ⊗ E) is the difference between thefollowing composition of morphisms:

A ⊗A ⊗ Eσ⊗1−→ S2(A)⊗ E

ξ−→ A ⊗ E [1]

A ⊗A ⊗ E1⊗a(A ,])(E)−→ A ⊗ E [1]

− id−→ A ⊗ E [1]

It follows that −a(A ,])(A ⊗ E) = ξ (σ ⊗1)+1⊗a(A ,])(E)

Proof of cohom. Bianchi identity

Consider the filtration of M =D(A ,]) ⊗ E

0 ⊂M0 = E ⊂M1 =D≤1(A ,]) ⊗ E ⊂M2 =D≤2

(A ,]) ⊗ E ⊂ · · ·

The exact sequence 0→M0→M1→M1/M0→ 0 is

0→ E →D≤1(A ,]) ⊗ E → A ⊗ E → 0

whose extension class is −a(A ,])(E) ∈ Ext1(A ⊗ E ,A).The next exact seq. 0→M1/M0→M2/M0→M2/M1→ 0 is

0→ A ⊗ E →M2/M0→ S2(A)⊗ E → 0

whose extension class we have denoted by ξ.

0 ⊂M0 = E ⊂M1 =D≤1(A ,]) ⊗ E ⊂M2 =D≤2

(A ,]) ⊗ E ⊂ · · ·

0→ E →D≤1(A ,]) ⊗ E → A ⊗ E → 0

whose extension class is −a(A ,])(E) ∈ Ext1(A ⊗ E ,A).

The next exact seq. 0→M1/M0→M2/M0→M2/M1→ 0 is

0→ A ⊗ E →M2/M0→ S2(A)⊗ E → 0

0 ⊂M0 = E ⊂M1 =D≤1(A ,]) ⊗ E ⊂M2 =D≤2

(A ,]) ⊗ E ⊂ · · ·

0→ E →D≤1(A ,]) ⊗ E → A ⊗ E → 0

whose extension class is −a(A ,])(E) ∈ Ext1(A ⊗ E ,A).The next exact seq. 0→M1/M0→M2/M0→M2/M1→ 0 is

0→ A ⊗ E →M2/M0→ S2(A)⊗ E → 0

Standard results tell us that the composition (Yoneda product)of the two extensions is zero: a(A ,])(E) ξ = 0

S2(A)⊗ Eξ−→ A ⊗ E [1]

a(A ,])(E)−→ E [2]

From Lemma 2 we have

a(A ,])(A ⊗ E) = −ξ (σ ⊗1)−1⊗a(A ,])(E)

We can also write

a(A ,])(A ⊗ E) = a(A ,])(A)⊗1+1⊗a(A ,])(E)

2(1⊗a(A ,])(E)

)+a(A ,])(A)⊗1 = −ξ (σ ⊗1)

S2(A)⊗ Eξ−→ A ⊗ E [1]

a(A ,])(E)−→ E [2]

a(A ,])(A ⊗ E) = −ξ (σ ⊗1)−1⊗a(A ,])(E)

We can also write

a(A ,])(A ⊗ E) = a(A ,])(A)⊗1+1⊗a(A ,])(E)

2(1⊗a(A ,])(E)

)+a(A ,])(A)⊗1 = −ξ (σ ⊗1)

S2(A)⊗ Eξ−→ A ⊗ E [1]

a(A ,])(E)−→ E [2]

a(A ,])(A ⊗ E) = −ξ (σ ⊗1)−1⊗a(A ,])(E)

We can also write

a(A ,])(A ⊗ E) = a(A ,])(A)⊗1+1⊗a(A ,])(E)

2(1⊗a(A ,])(E)

)+a(A ,])(A)⊗1 = −ξ (σ ⊗1)

Now we take the Yoneda product of the previous expression

2(1⊗a(A ,])(E)

)+a(A ,])(A)⊗1 = −ξ (σ ⊗1)

with a(A ,])(E) (on the left), and we recall that a(A ,])(E) ξ = 0.We get

2 [a(A ,])(E)∪a(A ,])(E)]+a(A ,])(E) ∗a(A ,])(A) = 0

Lie algebroid connections on a Lie algebroid

The (A , ])-Atiyah class of A

Now we consider the special case E = A . The corresponding(A , ])-Atiyah class is

a(A ,])(A) ∈ Ext1(A ⊗A ,A) = H 1(X ,A ∗ ⊗A ∗ ⊗A)

Theorem

The (A , ])-Atiyah class of A is symmetric, i.e.,

a(A ,])(A) ∈ Ext1(S2(A),A) = H 1(X ,S2(A ∗)⊗A)

Proof.

Direct computation in local coordinates. or. . .

a(A ,])(A) ∈ Ext1(A ⊗A ,A) = H 1(X ,A ∗ ⊗A ∗ ⊗A)

Theorem

a(A ,])(A) ∈ Ext1(S2(A),A) = H 1(X ,S2(A ∗)⊗A)

Proof.

a(A ,])(A) ∈ Ext1(A ⊗A ,A) = H 1(X ,A ∗ ⊗A ∗ ⊗A)

Theorem

a(A ,])(A) ∈ Ext1(S2(A),A) = H 1(X ,S2(A ∗)⊗A)

Proof.

The Lie algebra structure

Let X and (A , ]) be as before. Let F be a quasi-coherent sheafof commutative OX -algebras. We consider the composition ofthe following maps: first we take the cup-product

H i (X ,A ⊗F )⊗H j (X ,A ⊗F )→ H i+j (X ,A ⊗A ⊗F ⊗F )

followed by the map

H i+j (X ,A ⊗A ⊗F ⊗F )→ H i+j (X ,A ⊗A ⊗F )

induced by the commutative multiplication F ⊗F → F .

Finally we take the composition (Yoneda product) witha(A ,])(A) ∈ H 1(X ,Hom(S2(A),A)):

H i+j (X ,A ⊗A ⊗F )→ H i+j+1(X ,A ⊗F )

Let X and (A , ]) be as before. Let F be a quasi-coherent sheafof commutative OX -algebras. We consider the composition ofthe following maps: first we take the cup-product

H i (X ,A ⊗F )⊗H j (X ,A ⊗F )→ H i+j (X ,A ⊗A ⊗F ⊗F )

followed by the map

H i+j (X ,A ⊗A ⊗F ⊗F )→ H i+j (X ,A ⊗A ⊗F )

induced by the commutative multiplication F ⊗F → F .Finally we take the composition (Yoneda product) witha(A ,])(A) ∈ H 1(X ,Hom(S2(A),A)):

H i+j (X ,A ⊗A ⊗F )→ H i+j+1(X ,A ⊗F )

So, for any i and j , we obtain maps

H i (X ,A ⊗F )⊗H j (X ,A ⊗F )→ H i+j+1(X ,A ⊗F )

Let us write gi = H i−1(X ,A ⊗F ). Then we have maps

gi ⊗ gj → gi+j

TheoremThe maps above define a graded Lie algebra structure on thegraded vector space g• =

⊕i gi

gi ⊗ gj → gi+j

⊕i gi

gi ⊗ gj → gi+j

⊕i gi

Sketch of proof

Let αi ∈ gi and αj ∈ gj .Let us denote the bracket by [αi ,αj ] ∈ gi+jThe bilinearity of the bracket is obvious.

The (graded) antisymmetry is

[αj ,αi ] = −(−1)ij [αi ,αj ]

This follows immediately from the graded commutativity of thecup product.It remains only to prove the (graded) Jacobi identity:

(−1)ik [αi , [αj ,αk ]] + (−1)ij [αj , [αk ,αi ]] + (−1)jk [αk , [αi ,αj ]] = 0

Sketch of proof

Let αi ∈ gi and αj ∈ gj .Let us denote the bracket by [αi ,αj ] ∈ gi+jThe bilinearity of the bracket is obvious.The (graded) antisymmetry is

[αj ,αi ] = −(−1)ij [αi ,αj ]

This follows immediately from the graded commutativity of thecup product.

It remains only to prove the (graded) Jacobi identity:

Sketch of proof

Let αi ∈ gi and αj ∈ gj .Let us denote the bracket by [αi ,αj ] ∈ gi+jThe bilinearity of the bracket is obvious.The (graded) antisymmetry is

[αj ,αi ] = −(−1)ij [αi ,αj ]

This follows immediately from the graded commutativity of thecup product.It remains only to prove the (graded) Jacobi identity:

Let us denote the left-hand side by θ(αi ,αj ,αk ). This defines anelement θ ∈ Hom(∧3g•,g•).

We can check that θ(αi ,αj ,αk ) is obtained by taking the cupproduct

αi ∪αj ∪αk ∈ H i+j+k−3(X ,A ⊗A ⊗A ⊗F )

followed by the Yoneda composition with an element ofH 2(X ,Hom(S3(A),A)).This element turns out to be the symmetrization of

[a(A ,])(A)∪a(A ,])(A)] ∈ H 2(X ,Hom(A ⊗S2(A),A))

Let us denote the left-hand side by θ(αi ,αj ,αk ). This defines anelement θ ∈ Hom(∧3g•,g•).We can check that θ(αi ,αj ,αk ) is obtained by taking the cupproduct

followed by the Yoneda composition with an element ofH 2(X ,Hom(S3(A),A)).

This element turns out to be the symmetrization of

Let us denote the left-hand side by θ(αi ,αj ,αk ). This defines anelement θ ∈ Hom(∧3g•,g•).We can check that θ(αi ,αj ,αk ) is obtained by taking the cupproduct

followed by the Yoneda composition with an element ofH 2(X ,Hom(S3(A),A)).This element turns out to be the symmetrization of

Now we use the cohomological Bianchi identity (for E = A ):

2[a(A ,])(A)∪a(A ,])(A)]+a(A ,])(A) ∗a(A ,])(A) = 0

From the definition, it follows that the symmetrization ofa(A ,])(A) ∗a(A ,])(A) is 0, hence the same is true for thesymmetrization of [a(A ,])(A)∪a(A ,])(A)].

This finally means that θ = 0, which proves the Jacobiidentity.

Now we use the cohomological Bianchi identity (for E = A ):

2[a(A ,])(A)∪a(A ,])(A)]+a(A ,])(A) ∗a(A ,])(A) = 0

From the definition, it follows that the symmetrization ofa(A ,])(A) ∗a(A ,])(A) is 0, hence the same is true for thesymmetrization of [a(A ,])(A)∪a(A ,])(A)].

This finally means that θ = 0, which proves the Jacobiidentity.

Modules over the Lie algebra

Let X , (A , ]) and F be as before. Let E be a holomorphic vectorbundle over X .We consider the composition of the following maps: first wetake the cup-product

H i (X ,A ⊗F )⊗H j (X ,E ⊗F )→ H i+j (X ,A ⊗ E ⊗F )

(we have used the multiplication F ⊗F → F , as before).

Then we take the composition (Yoneda product) witha(A ,])(E) ∈ H 1(X ,Hom(A ⊗ E ,E)):

H i+j (X ,A ⊗ E ⊗F )→ H i+j+1(X ,E ⊗F )

Let us write gi = H i−1(X ,A ⊗F ) and Vj = H j−1(X ,E ⊗F )

(we have used the multiplication F ⊗F → F , as before).Then we take the composition (Yoneda product) witha(A ,])(E) ∈ H 1(X ,Hom(A ⊗ E ,E)):

H i+j (X ,A ⊗ E ⊗F )→ H i+j+1(X ,E ⊗F )

(we have used the multiplication F ⊗F → F , as before).Then we take the composition (Yoneda product) witha(A ,])(E) ∈ H 1(X ,Hom(A ⊗ E ,E)):

H i+j (X ,A ⊗ E ⊗F )→ H i+j+1(X ,E ⊗F )

Then, for any i and j , we have maps gi ⊗Vj → Vi+j

TheoremThe maps above define a structure of graded module on thegraded vector space V• =

⊕j Vj , over the graded Lie algebra

Sketch of proof

Let αi ∈ gi , αj ∈ gj and vk ∈ Vk . We have to prove that

[αi ,αj ]vk −αi (αj vk )+ (−1)ijαj (αi vk ) = 0

The left-hand side defines an element φ ∈ Hom(∧2g• ⊗V•,V•)

Sketch of proof

We can check that φ is obtained by taking the cup product

αi ∪αj ∪ vk ∈ H i+j+k−3(X ,A ⊗A ⊗ E ⊗F )

followed by the Yoneda composition with an element ofH 2(X ,Hom(S2(A)⊗ E ,E)).

This element is precisely

2[a(A ,])(E)∪a(A ,])(E)]+a(A ,])(E) ∗a(A ,])(A)

which vanishes by the cohomological Bianchi identity.

We can check that φ is obtained by taking the cup product

αi ∪αj ∪ vk ∈ H i+j+k−3(X ,A ⊗A ⊗ E ⊗F )

followed by the Yoneda composition with an element ofH 2(X ,Hom(S2(A)⊗ E ,E)).

This element is precisely

2[a(A ,])(E)∪a(A ,])(E)]+a(A ,])(E) ∗a(A ,])(A)

which vanishes by the cohomological Bianchi identity.

References

Z. Chen, M. Stiénon, P. Xu, From Atiyah classes tohomotopy Leibniz algebras, arXiv:1204.1075v2.

R. Loja Fernandes, Lie algebroids, holonomy andcharacteristic classes, Adv. Math. 170 (2002), 119–179.

M. Kapranov, Rozansky–Witten invariants via Atiyahclasses, Compositio Math. 115 (1999), 71–113.

Thank you!

Appendix

Lie algebroid cohomology

Let (A , ]) be a Lie algebroid, let [ : T ∗M → A ∗ be the dual of theanchor map and let dA : Γ (C∞M )→ Γ (A ∗) be the C-derivationdefined as the composition dA = [ d , where d : Γ (C∞M )→Ω1

Mis the usual differential.

We can extend the definition of dA todA : Γ (∧pA ∗)→ Γ (∧p+1A ∗). Then we obtain a complex

· · · → ∧p−1A ∗→∧pA ∗→∧p+1A ∗→ ·· ·

whose cohomology is called the Lie algebroid cohomology ofA , denoted by H•(A).

Appendix

Let (A , ]) be a Lie algebroid, let [ : T ∗M → A ∗ be the dual of theanchor map and let dA : Γ (C∞M )→ Γ (A ∗) be the C-derivationdefined as the composition dA = [ d , where d : Γ (C∞M )→Ω1

Mis the usual differential.

We can extend the definition of dA todA : Γ (∧pA ∗)→ Γ (∧p+1A ∗). Then we obtain a complex

· · · → ∧p−1A ∗→∧pA ∗→∧p+1A ∗→ ·· ·

whose cohomology is called the Lie algebroid cohomology ofA , denoted by H•(A).

Appendix

The dual of the anchor map defines a homomorphism ofexterior algebras [ :Ω•(M)→ Γ (∧•A ∗).

This induces a ring homomorphism

[ : H•dR(M)→ H•(A)

Return

Appendix

Sheaves of torsors

Let Conn(A ,])(E) be the sheaf whose sections over U ⊂ X arethe holomorphic (A , ])-connections defined on E |U .

This is an affine space over Γ (U ,End(E)⊗A ∗).

Then Conn(A ,])(E) is a sheaf of torsors over End(E)⊗A ∗.

Sheaves of torsors over End(E)⊗A ∗ are classified by elementsof H 1(X ,End(E)⊗A ∗).

a(A ,])(E) is the element that classifies Conn(A ,])(E).

Appendix

Sheaves of torsors

Appendix

Sheaves of torsors

Appendix

Sheaves of torsors

Appendix

Sheaves of torsors

If we take E = A , then to any (A , ])-connection ∇ on A we canassociate its torsion T ∈ Hom(∧2A ,A)

T(a ,b) = ∇ab −∇b a − [a ,b ]

Let Conntf(A ,])(A) be the sheaf whose sections over U ⊂ X are

the torsion free (A , ])-connections on A |U .Then Conntf

(A ,])(A) is a sheaf of torsors over S2(A ∗)⊗A .Since the sheaf of torsors Conn(A ,])(A) is obtained fromConntf

(A ,])(A) by “change of scalars” (i.e., from S2(A ∗)⊗A toA ∗ ⊗A ∗ ⊗A ) it follows that the classifying elementa(A ,])(A) ∈ H 1(X ,A ∗ ⊗A ∗ ⊗A) actually belongs to the summandH 1(X ,S2(A ∗)⊗A)

Return

Appendix

Sheaves of torsors

T(a ,b) = ∇ab −∇b a − [a ,b ]

the torsion free (A , ])-connections on A |U .

Then Conntf(A ,])(A) is a sheaf of torsors over S2(A ∗)⊗A .

Since the sheaf of torsors Conn(A ,])(A) is obtained fromConntf

Return

Appendix

Sheaves of torsors

T(a ,b) = ∇ab −∇b a − [a ,b ]

(A ,])(A) is a sheaf of torsors over S2(A ∗)⊗A .

Since the sheaf of torsors Conn(A ,])(A) is obtained fromConntf

Return

Appendix

Sheaves of torsors

T(a ,b) = ∇ab −∇b a − [a ,b ]

(A ,])(A) is a sheaf of torsors over S2(A ∗)⊗A .Since the sheaf of torsors Conn(A ,])(A) is obtained fromConntf

Return

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