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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
Connections Holomorphic connections Lie algebroid connections on a Lie algebroid
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
Connections Holomorphic connections Lie algebroid connections on a Lie algebroid
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
Connections Holomorphic connections Lie algebroid connections on a Lie algebroid
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
Connections Holomorphic connections Lie algebroid connections on a Lie algebroid
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
Connections Holomorphic connections Lie algebroid connections on a Lie algebroid
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
Connections Holomorphic connections Lie algebroid connections on a Lie algebroid
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
Connections Holomorphic connections Lie algebroid connections on a Lie algebroid
M
γ
γ
EP
•x
•P
EQ
• y
•Q
Connections Holomorphic connections Lie algebroid connections on a Lie algebroid
M
γ
γ
EP
•x
•P
EQ
• y
•Q
Connections Holomorphic connections Lie algebroid connections on a Lie algebroid
M
γ
γ
EP
•x
•P
EQ
• y
•Q
Connections Holomorphic connections Lie algebroid connections on a Lie algebroid
M
γ
γEP
•x
•P
EQ
• y
•Q
Connections Holomorphic connections Lie algebroid connections on a Lie algebroid
Fiber spaces and connections
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.
Connections Holomorphic connections Lie algebroid connections on a Lie algebroid
Fiber spaces and connections
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.
Connections Holomorphic connections Lie algebroid connections on a Lie algebroid
Fiber spaces and connections
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.
Connections Holomorphic connections Lie algebroid connections on a Lie algebroid
Fiber spaces and connections
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.
Connections Holomorphic connections Lie algebroid connections on a Lie algebroid
Fiber spaces and connections
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.
Connections Holomorphic connections Lie algebroid connections on a Lie algebroid
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.
Connections Holomorphic connections Lie algebroid connections on a Lie algebroid
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 ).
Connections Holomorphic connections Lie algebroid connections on a Lie algebroid
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 ).
Connections Holomorphic connections Lie algebroid connections on a Lie algebroid
Lie algebroid connections
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 ∗
Connections Holomorphic connections Lie algebroid connections on a Lie algebroid
Lie algebroid connections
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 ∗
Connections Holomorphic connections Lie algebroid connections on a Lie algebroid
Lie algebroid connections
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 ∗
Connections Holomorphic connections Lie algebroid connections on a Lie algebroid
Lie algebroid connections
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 ∗
Connections Holomorphic connections Lie algebroid connections on a Lie algebroid
Lie algebroid connections
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.
Connections Holomorphic connections Lie algebroid connections on a Lie algebroid
Lie algebroid connections
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.
Connections Holomorphic connections Lie algebroid connections on a Lie algebroid
Lie algebroid connections
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.
Connections Holomorphic connections Lie algebroid connections on a Lie algebroid
Holomorphic connections
Holomorphic Lie algebroid connections
Connections Holomorphic connections Lie algebroid connections on a Lie algebroid
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
Connections Holomorphic connections Lie algebroid connections on a Lie algebroid
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
Connections Holomorphic connections Lie algebroid connections on a Lie algebroid
Holomorphic Lie algebroid connections
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 ∗).
Connections Holomorphic connections Lie algebroid connections on a Lie algebroid
Holomorphic Lie algebroid connections
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 ∗).
Connections Holomorphic connections Lie algebroid connections on a Lie algebroid
Holomorphic Lie algebroid connections
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 ∗).
Connections Holomorphic connections Lie algebroid connections on a Lie algebroid
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).
Connections Holomorphic connections Lie algebroid connections on a Lie algebroid
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).
Connections Holomorphic connections Lie algebroid connections on a Lie algebroid
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).
Connections Holomorphic connections Lie algebroid connections on a Lie algebroid
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.
Connections Holomorphic connections Lie algebroid connections on a Lie algebroid
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.
Connections Holomorphic connections Lie algebroid connections on a Lie algebroid
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.
Connections Holomorphic connections Lie algebroid connections on a Lie algebroid
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) //
E //
0
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.
Connections Holomorphic connections Lie algebroid connections on a Lie algebroid
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) //
E //
0
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.
Connections Holomorphic connections Lie algebroid connections on a Lie algebroid
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) //
E //
0
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.
Connections Holomorphic connections Lie algebroid connections on a Lie algebroid
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
(which is split as a sequence of C-modules but not, in general,as a sequence of OX -modules).
Connections Holomorphic connections Lie algebroid connections on a Lie algebroid
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
(which is split as a sequence of C-modules but not, in general,as a sequence of OX -modules).
Connections Holomorphic connections Lie algebroid connections on a Lie algebroid
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 )
Connections Holomorphic connections Lie algebroid connections on a Lie algebroid
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 )
Connections Holomorphic connections Lie algebroid connections on a Lie algebroid
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 )
Connections Holomorphic connections Lie algebroid connections on a Lie algebroid
Differential operators
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 ∗)
Connections Holomorphic connections Lie algebroid connections on a Lie algebroid
Differential operators
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)
Connections Holomorphic connections Lie algebroid connections on a Lie algebroid
Differential operators
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)
Connections Holomorphic connections Lie algebroid connections on a Lie algebroid
Differential operators
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)
Connections Holomorphic connections Lie algebroid connections on a Lie algebroid
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.
Connections Holomorphic connections Lie algebroid connections on a Lie algebroid
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.
Connections Holomorphic connections Lie algebroid connections on a Lie algebroid
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.
Connections Holomorphic connections Lie algebroid connections on a Lie algebroid
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.
Connections Holomorphic connections Lie algebroid connections on a Lie algebroid
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.
Connections Holomorphic connections Lie algebroid connections on a Lie algebroid
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.
Connections Holomorphic connections Lie algebroid connections on a Lie algebroid
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.
Connections Holomorphic connections Lie algebroid connections on a Lie algebroid
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)
Connections Holomorphic connections Lie algebroid connections on a Lie algebroid
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,
Mi
Mi−1
)be the extension class of the exact sequence
0 −→ Mi
Mi−1−→
Mi+1
Mi−1−→
Mi+1
Mi−→ 0
Connections Holomorphic connections Lie algebroid connections on a Lie algebroid
The cohomological Bianchi identity: proof
Then the class −a(A ,])(Mi /Mi−1) is the difference between thefollowing composition of morphisms:
A ⊗ Mi
Mi−1
µ−→
Mi+1
Mi
φi−→ Mi
Mi−1[1]
and
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).
Connections Holomorphic connections Lie algebroid connections on a Lie algebroid
The cohomological Bianchi identity: proof
Then the class −a(A ,])(Mi /Mi−1) is the difference between thefollowing composition of morphisms:
A ⊗ Mi
Mi−1
µ−→
Mi+1
Mi
φi−→ Mi
Mi−1[1]
and
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).
Connections Holomorphic connections Lie algebroid connections on a Lie algebroid
The cohomological Bianchi identity: proof
Let M =D(A ,]) ⊗ E with the filtration Mi =D≤i(A ,]) ⊗ E
The exact sequence
0 −→ M1
M0−→ M2
M0−→ M2
M1−→ 0
is
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)
Connections Holomorphic connections Lie algebroid connections on a Lie algebroid
The cohomological Bianchi identity: proof
Let M =D(A ,]) ⊗ E with the filtration Mi =D≤i(A ,]) ⊗ E
The exact sequence
0 −→ M1
M0−→ M2
M0−→ M2
M1−→ 0
is
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)
Connections Holomorphic connections Lie algebroid connections on a Lie algebroid
The cohomological Bianchi identity: proof
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]
and
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)
Connections Holomorphic connections Lie algebroid connections on a Lie algebroid
The cohomological Bianchi identity: proof
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.
Connections Holomorphic connections Lie algebroid connections on a Lie algebroid
The cohomological Bianchi identity: proof
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.
Connections Holomorphic connections Lie algebroid connections on a Lie algebroid
The cohomological Bianchi identity: proof
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.
Connections Holomorphic connections Lie algebroid connections on a Lie algebroid
The cohomological Bianchi identity: proof
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)
hence
2(1⊗a(A ,])(E)
)+a(A ,])(A)⊗1 = −ξ (σ ⊗1)
Connections Holomorphic connections Lie algebroid connections on a Lie algebroid
The cohomological Bianchi identity: proof
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)
hence
2(1⊗a(A ,])(E)
)+a(A ,])(A)⊗1 = −ξ (σ ⊗1)
Connections Holomorphic connections Lie algebroid connections on a Lie algebroid
The cohomological Bianchi identity: proof
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)
hence
2(1⊗a(A ,])(E)
)+a(A ,])(A)⊗1 = −ξ (σ ⊗1)
Connections Holomorphic connections Lie algebroid connections on a Lie algebroid
The cohomological Bianchi identity: proof
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
Connections Holomorphic connections Lie algebroid connections on a Lie algebroid
Lie algebroid connections on a Lie algebroid
Lie algebroid connections on a Lie algebroid
Connections Holomorphic connections 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. . .
Connections Holomorphic connections 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. . .
Connections Holomorphic connections 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. . .
Connections Holomorphic connections Lie algebroid connections on a Lie algebroid
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 )
Connections Holomorphic connections Lie algebroid connections on a Lie algebroid
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 )
Connections Holomorphic connections Lie algebroid connections on a Lie algebroid
The Lie algebra structure
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
Connections Holomorphic connections Lie algebroid connections on a Lie algebroid
The Lie algebra structure
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
Connections Holomorphic connections Lie algebroid connections on a Lie algebroid
The Lie algebra structure
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
Connections Holomorphic connections Lie algebroid connections on a Lie algebroid
The Lie algebra structure
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
Connections Holomorphic connections Lie algebroid connections on a Lie algebroid
The Lie algebra structure
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
Connections Holomorphic connections Lie algebroid connections on a Lie algebroid
The Lie algebra structure
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
Connections Holomorphic connections Lie algebroid connections on a Lie algebroid
The Lie algebra structure
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))
Connections Holomorphic connections Lie algebroid connections on a Lie algebroid
The Lie algebra structure
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))
Connections Holomorphic connections Lie algebroid connections on a Lie algebroid
The Lie algebra structure
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))
Connections Holomorphic connections Lie algebroid connections on a Lie algebroid
The Lie algebra structure
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.
Connections Holomorphic connections Lie algebroid connections on a Lie algebroid
The Lie algebra structure
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.
Connections Holomorphic connections Lie algebroid connections on a Lie algebroid
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 )
Connections Holomorphic connections Lie algebroid connections on a Lie algebroid
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 )
Connections Holomorphic connections Lie algebroid connections on a Lie algebroid
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 )
Connections Holomorphic connections Lie algebroid connections on a Lie algebroid
Modules over the Lie algebra
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
g•.
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•)
Connections Holomorphic connections Lie algebroid connections on a Lie algebroid
Modules over the Lie algebra
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
g•.
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•)
Connections Holomorphic connections Lie algebroid connections on a Lie algebroid
Modules over the Lie algebra
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
g•.
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•)
Connections Holomorphic connections Lie algebroid connections on a Lie algebroid
Modules over the Lie algebra
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
g•.
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•)
Connections Holomorphic connections Lie algebroid connections on a Lie algebroid
Modules over the Lie algebra
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.
Connections Holomorphic connections Lie algebroid connections on a Lie algebroid
Modules over the Lie algebra
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.
Connections Holomorphic connections Lie algebroid connections on a Lie algebroid
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.
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
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
Lie algebroid cohomology
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
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
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
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
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
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
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
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