Abel, Jacobi and the double homotopy fiber

Transcription

1 Abel, Jacobi and the double homotopy fiber Domenico Fiorenza Sapienza Università 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 )

2 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 b0 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: { } obs(hilb X /Z ) ker H 1 (Z; N X /Z ) i H 2p (O X Ω 1 X Ωp 1 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.

3 Idea: to exhibit a morphism of (derived) infinitesimal deformation functors AJ : Hilb X /Z Jac 2p X /Z where Jac 2p X /Z is some deformation functor with obs(jac2p X /Z ) = 0 obs(aj) is the restriction to obs(hilb X /Z ) of i : H 1 (Z; N X /Z ) H 2p (O X Ω 1 X Ωp 1 X ) Infinitesimal deformation functors are the same thing as L -algebras. So the idea becomes: to exhibit a morphism of L -algebras such that: g Hilb X /Z ϕ : g h h is quasi-abelian (i.e. h is quasi-isomorphic to a cochain complex) the linear morphism H 2 (g) H2 (ϕ) H 2 (h) is naturally identified with i : H 1 (Z; N X /Z ) H 2p (O X Ω 1 X Ωp 1 X )

4 Let χ : L M a morphism of dglas. hofiber(χ) L 0 M χ A convenient model for hofiber(χ) is the Thom-Whitney model TW (χ) = {(l, m(t, dt)) L ( M Ω ( 1 ) ) m(0) = 0, m(1) = χ(l)} It is a sub-dgla of L ( M Ω ( 1 ) ). 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.

5 cone(χ) = L M[ 1], [(l, m)] 1 = (dl, χ(l) dm) ( ) [(l 1, m 1 ), (l 2, m 2 )] 2 = [l 1, l 2 ], 1 2 [m 1, χ(l 2 )] + ( 1)deg(l 1) [χ(l 1 ), m 2 ] 2 B n 1 [(l 1, m 1 ),, (l n, m n )] n = 0, ±[m (n 1)! σ(1), [, [m σ(n 1), χ(l σ(n) )] ]], 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 ( ) hofiber A 0, X (Θ X )( log Z) A 0, X (Θ X ) is Hilb X /Z.

6 Let L and M be two dglas, i : L M[ 1] a morphism of graded vector spaces. Let l: L M a l a = di 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 ).

7 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 î: A 0, X (Θ X ) End(D X )[ 1] is a Cartan homotopy and its differential is the holomorphic Lie derivative ˆl = [ ˆ, î]: A 0, X (Θ X ) End(D X ).

8 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 î[2p]: A 0, X (Θ X )( log Z) End(D X [2p])[ 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.

9 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 i N ι 0 M

10 And so, by the universal property of the homotopy fiber we get L Φ hofiber(ι) l N 0 M ι 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,i) l cone(ι) N 0 M ι

11 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 L 1 i 1 M 1 [ 1] L 2 ϕ L ϕ M [ 1] i 2 M 2 [ 1] is a commutative diagram of graded vector spaces. A Cartan square induces a commutative diagram of dglas L 1 l 1 M 1, ϕ L L 2 l 2 M 2 where l 1 and l 2 are the Lie derivatives associated with i 1 and i 2, respectively. ϕ M

12 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 L 1 l 1 N 1 ι 1 M 1 ϕ L L 2 l 2 N 2 ι 2 M 2 where ι 1 and ι 2 are inclusions of sub-dglas. ϕ M 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 )).

13 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,a2 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,ã mod M 1 + N 2 ], where ã L 2 is an arbitrary representative of [a].

14 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).

15 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}

16 For any sub-dgla L L v, the diagram L i L End(V )[ 1] L j 0 [ 1] i v aff(v )[ 1], where the left vertical arrow is the inclusion L L, is a Cartan square. Let now F be a subcomplex of V such that the dgla morphism l v : L aff(v ) takes its values in aff(v )( F ) = {(f, v) aff(v ) f (F ) F, v F }. Then we have a linear L -morphism L End(V )( F ) (l TW L,lv,i L,iv ) cone TW L aff(v )( F ) TW End(V ) aff(v ).

17 At the n-th cohomology level, this L -morphism gives the map H n 1 (L/ L) H n 2 (V /F ) [a] [iã(v) mod F ]. where ã is any representative of [a] in L. The L -algebra cone TW End(V )( F ) aff(v )( F ) TW End(V ) aff(v ) is a model for the double homotopy fiber of the commutative diagram End(V )( F ) End(V ) aff(v )( F ) aff(v )

18 cone TW TW End(V )( F ) End(V ) aff(v )( F ) aff(v ) But actually, due to the fact that we have sections 0 V aff(v ) End(V ) 0 F aff(v )( F ) End(V )( F ) there is a simpler model: 0 0 (V /F )[ 2] F [ 1] V [ 1] End(V )( F ) End(V ) aff(v )( F ) aff(v )

19 Let now X be a compact complex manifold and let Z X be a codimension p complex submanifold. Then integration over Z defines a closed (p, p)-current, which we will denote by the same symbol Z. By shifting the degrees, we can look at Z as a closed degree zero element v in the chain complex V = D(X )[2p]. Let F = (F p D(X ))[2p] be the sub-complex of V obtained by shifting the p-th term in the Hodge filtration on currents, F p D(X ) = i p D i, (X ). Finally, let L = A 0, X (Θ X ), let L = A 0, X (Θ X )( log Z) and let i : L End(V )[ 1] be the (shifted) contraction operator on currents: î[2p] : A 0, X (Θ X ) End(D(X )[2p], D(X )[2p])[ 1]. The 6-ple (L, L, V, F, v, i) defined this way satisfies the hypothesis of the slides above, so we get an L -morphims TW A 0, X (Θ X )( log Z) A 0, X (Θ X ) inducing in cohomology (D(X )/F p D(X ))[2p 2] H 0 (Z; N X /Z ) H 2p 1 (D(X )/F p D(X )) H 1 (Z; N X /Z ) H 2p (D(X )/F p D(X )) [x] [î xz mod F p D(X )] in degrees 1 and 2, where x is any representative of [x] in A 0, X (Θ X ).

20 Since H (D(X )/F p D(X )) = H (X ; O X Ω 1 X Ωp 1 X ), if we define Jac 2p X /Z to be the deformation functor associated to the abelian dgla (D(X )/F p D(X ))[2p 2] then we get from the L -morphis exhibited above a morphism of deformation functors AJ : Hilb X /Z Jac 2p X /Z with and daj : H 0 (Z; N X /Z ) i H 2p 1 (X ; O X Ω 1 X Ωp 1 X ) obs(aj) : H 1 (Z; N X /Z ) i H 2p (X ; O X Ω 1 X Ωp 1 X )