Abel, Jacobi and the double homotopy fiber Domenico Fiorenza - PDF document

Abel, Jacobi and the double homotopy fiber Domenico Fiorenza Sapienza Universit` a di Roma March 5, 2014 Joint work with Marco Manetti, (hopefully) soon on arXiv Everything will be over the field C of complex numbers. Questions like “does this work over an arbitrary characteristic zero algebraically closed field K ?” are not allowed! (in any case the answer is “I guess so, but I don’t know”)

Let X be a smooth complex manifold and let Z ⊆ X be a complex codimension p smooth complex submanifold. Denote by Hilb X / Z the functor of infinitesimal deformations of Z inside X . T b 0 Hilb X / Z = H 0 ( Z ; N X / Z ) obs ( Hilb X / Z ) ⊆ H 1 ( Z ; N X / Z ) Actually one can control the obstructions better: � � i X → · · · → · · · Ω p − 1 H 1 ( Z ; N X / Z ) → H 2 p ( O X → Ω 1 obs ( Hilb X / Z ) ⊆ ker − ) X This has been originally shown by Bloch under a few additional hypothesis and recently by Iacono-Manetti and Pridham in full generality. The aim of this talk is to illustrate a bit of the (infinitesimal) geometry behind these proofs.

Idea: to exhibit a morphism of (derived) infinitesimal deformation functors AJ : Hilb X / Z → Jac 2 p X / Z where ◮ Jac 2 p X / Z is some deformation functor with obs ( Jac 2 p X / Z ) = 0 ◮ obs ( AJ ) is the restriction to obs ( Hilb X / Z ) of X → · · · → · · · Ω p − 1 i : H 1 ( Z ; N X / Z ) → H 2 p ( O X → Ω 1 ) X Infinitesimal deformation functors are “the same thing” as L ∞ -algebras. So the idea becomes: to exhibit a morphism of L ∞ -algebras ϕ : g → h such that: ◮ g � Hilb X / Z ◮ h is quasi-abelian (i.e. h is quasi-isomorphic to a cochain complex) H 2 ( ϕ ) ◮ the linear morphism H 2 ( g ) → H 2 ( h ) is naturally identified − − − with X → · · · → · · · Ω p − 1 i : H 1 ( Z ; N X / Z ) → H 2 p ( O X → Ω 1 ) X

� � � � Let χ : L → M a morphism of dglas. hofiber ( χ ) L χ � M 0 A convenient model for hofiber ( χ ) is the Thom-Whitney model M ⊗ Ω • (∆ 1 ) � � TW ( χ ) = { ( l , m ( t , dt )) ∈ L ⊕ | m (0) = 0 , m (1) = χ ( l ) } M ⊗ Ω • (∆ 1 ) � � It is a sub-dgla of L ⊕ . It is “big” even when L and M are small. However there is also another model which is just “as big as L and M ”.

cone ( χ ) = L ⊕ M [ − 1] , [( l , m )] 1 = ( dl , χ ( l ) − dm ) � � 2[ m 1 , χ ( l 2 )] + ( − 1) deg( l 1 ) [ l 1 , l 2 ] , 1 [( l 1 , m 1 ) , ( l 2 , m 2 )] 2 = [ χ ( l 1 ) , m 2 ] 2   B n − 1 �  , [( l 1 , m 1 ) , · · · , ( l n , m n )] n =  0 , ± [ m σ (1) , [ · · · , [ m σ ( n − 1) , χ ( l σ ( n ) )] · · · ]] ( n − 1)! σ ∈ S n for n ≥ 3 where the B n ’s are the Bernoulli numbers Why is this relevant for us? Let X be a complex manifold and let Z ⊆ X be a complex submanifold. Let A 0 , ∗ X (Θ X ) be the p = 0 Dolbeault dgla with coefficients in holomorphic vector fields on X and A 0 , ∗ X (Θ X )( − log Z ) = ker { A 0 , ∗ X (Θ X ) → A 0 , ∗ Z ( N X / Z ) } the sub-dgla of A 0 , ∗ X (Θ X ) of differential forms with coefficients vector fields tangent to Z . The deformation functor associated with � � A 0 , ∗ → A 0 , ∗ X (Θ X )( − log Z ) ֒ X (Θ X ) hofiber is Hilb X / Z .

Let L and M be two dglas, i : L → M [ − 1] a morphism of graded vector spaces . Let l : L → M a �→ l a = d i a + i da be the differential of i in the cochain complex Hom ( L , M ).The map i is called a Cartan homotopy for l if, for every a , b ∈ L , we have: i [ a , b ] = [ i a , l b ] , [ i a , i b ] = 0 . Note that i [ a , b ] = [ i a , l b ] implies that l is a morphism of differential graded Lie algebras: the Lie derivative associated with i . Example let X be a differential manifold, A 0 X ( T X ) be the Lie algebra of vector fields on X , and End ( A ∗ X ) be the dgla of endomorphisms of the de Rham complex of X . Then the contraction i : A 0 X ( T X ) → End ( A ∗ X )[ − 1] is a Cartan homotopy and its differential is the Lie derivative l = [ d , i ] =: A 0 X ( T X ) → End ( A ∗ X ) .

Example let X be a complex manifold, A 0 , ∗ X (Θ X ) be the p = 0 Dolbeault dgla with coefficients in holomorphic vector fields on X , and End ( A ∗ , ∗ X ) be the dgla of endomorphisms of the de Dolbeault complex of X . Then the contraction i : A 0 , ∗ X (Θ X ) → End ( A ∗ , ∗ X )[ − 1] is a Cartan homotopy and its differential is the holomorphic Lie derivative l = [ ∂, i ]: A 0 , ∗ X (Θ X ) → End ( A ∗ , ∗ X ) . Example let X be a complex manifold, A 0 , ∗ X (Θ X ) be the p = 0 Dolbeault dgla with coefficients in holomorphic vector fields on X , and End ( D X ) be the dgla of endomorphisms of the complex of smooth currents on X . Then the contraction ˆ i : A 0 , ∗ X (Θ X ) → End ( D X )[ − 1] is a Cartan homotopy and its differential is the holomorphic Lie derivative i ]: A 0 , ∗ ˆ l = [ˆ ∂, ˆ X (Θ X ) → End ( D X ) .

The composition of a Cartan homotopy with a morphism of DGLAs (on either sides) is a Cartan homotopy. The corresponding Lie derivative is the composition of the Lie derivative of i with the given dgla morphisms. Example ˆ i [2 p ]: A 0 , ∗ X (Θ X )( − log Z ) → End ( D X [2 p ])[ − 1] is a Cartan homotopy. Cartan homotopies are compatible with base change/extension of scalars: if i : L → M [ − 1] is a Cartan homotopy and Ω is a differential graded-commutative algebra, then its natural extension i ⊗ Id: L ⊗ Ω → ( M ⊗ Ω)[ − 1] , a ⊗ ω �→ i a ⊗ ω, is a Cartan homotopy.

� � � � � � Cartan homotopies and homotopy fibers Let now i : L → M [ − 1] be a Cartan homotopy with Lie derivative l , and assume the image of l is contained in the subdgla N of M L l N ι M Then we have a homotopy commutative diagram of dglas L l N i ι � M 0

� � � � � � � � � � � � And so, by the universal property of the homotopy fiber we get L l Φ hofiber ( ι ) N ι � M 0 When we choose cone ( ι ) as a model for the homotopy fiber we get a particularly simple expression for the L ∞ morpgism Φ : L → hofiber ( ι ): L l ( l , i ) cone ( ι ) N ι � M 0

� � � � A Cartan square is the following set of data: ◮ two morphisms of dglas ϕ L : L 1 → L 2 and ϕ M : M 1 → M 2 ; ◮ two Cartan homotopies i 1 : L 1 → M 1 [ − 1] and i 2 : L 2 → M 2 [ − 1] such that i 1 � M 1 [ − 1] L 1 ϕ L ϕ M [ − 1] i 2 � M 2 [ − 1] L 2 is a commutative diagram of graded vector spaces. A Cartan square induces a commutative diagram of dglas l 1 � M 1 , L 1 ϕ L ϕ M l 2 � M 2 L 2 where l 1 and l 2 are the Lie derivatives associated with i 1 and i 2 , respectively.

� � � � It also induces a Cartan homotopy ( i 1 , i 2 ) : TW ( L 1 → L 2 ) → TW ( M 1 → M 2 )[ − 1] whose Lie derivative is ( l 1 , l 2 ) : TW ( L 1 → L 2 ) → TW ( M 1 → M 2 ) . Now assume the commutative diagram of dglas associated with a Cartan square factors as ι 1 l 1 � N 1 L 1 M 1 ϕ L ϕ M ι 2 l 2 � N 2 � M 2 L 2 where ι 1 and ι 2 are inclusions of sub-dglas. Then we have a linear L ∞ morphism ( l 1 , l 2 , i 1 , i 2 ): TW ( L 1 → L 2 ) → cone ( TW ( N 1 → N 2 ) → TW ( M 1 → M 2 )) .

If moreover also ϕ L and ϕ M are inclusions, then in the (homotopy) category of cochain complexes the linear L ∞ -morphism ( l 1 , l 2 , i 1 , i 2 ) is equivalent to the span ∼ ( L 2 / L 1 )[ − 1] ← − cone ( L 1 → L 2 ) → ( M 2 / ( M 1 + N 2 ))[ − 2] , where the quasi isomorphism on the left is induced by the projection on the second factor, and the morphism on the right is ( a 1 , a 2 ) �→ i 2 , a 2 mod M 1 + N 2 . Hence, at the cohomology level, the morphism H n ( l 1 , l 2 , i 1 , i 2 ) is naturally identified with the morphism H n − 1 ( L 2 / L 1 ) → H n − 2 ( M 2 / ( M 1 + N 2 )) [ a ] �→ [ i 2 , ˜ a mod M 1 + N 2 ] , where ˜ a ∈ L 2 is an arbitrary representative of [ a ].

Where do we find Cartan squares? Let V be a chain complex, and let End ( V ) and aff ( V ) be the dgla of its linear endomorphisms and infinitesimal affine transformations, respectively. aff ( V ) = End ( V ) ⊕ V = { f ∈ End ( V ⊕ C , V ⊕ C ) | Im ( f ) ⊆ V } . [( f , v ) , ( g , w )] = ([ f , g ] , f ( w ) − ( − 1) f g g ( v )) d aff ( f , v ) = ( d End f , dv ) Every degree zero closed element v in V defines an embedding of dglas j v : End ( V ) → aff ( V ) f �→ ( f , − f ( v )) This is the identification of End ( V ) with the stabilizer of v under the action of aff ( V ) on V . In particular j 0 is the canonical embedding of End ( V ) into aff ( V ) given by f �→ ( f , 0).

Let i : L → End ( V )[ − 1] be a Cartan homotopy and let v be a degree zero closed element in V . Then i v : L → aff ( V )[ − 1] a �→ ( i a , − i a ( v )) is a Cartan homotopy. The corresponding Lie derivative is l v : L → aff ( V ) a �→ ( l a , − l a ( v )) Indeed, the linear map i v is the composition of the Cartan homotopy i with the dgla morphism j v , hence it is a Cartan homotopy. The corresponding Lie derivative is the composition of l with j v . So we have built a Cartan homotopy i v out of a Cartan homotopy i : L → End ( V )[ − 1] and of a closed element v in V . Let us now use the same ingredients to cook up a sub-dgla of L . L v = { a ∈ L such that i a ( v ) = 0 and l a ( v ) = 0 }