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 HilbX/Z the functor of infinitesimal deformations of Z inside X. Tb0HilbX/Z = H0(Z; NX/Z)

• bs(HilbX/Z) ⊆ H1(Z; NX/Z)

Actually one can control the obstructions better:

• bs(HilbX/Z) ⊆ ker
• H1(Z; NX/Z)

i

− → H2p(OX → Ω1

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.

Idea: to exhibit a morphism of (derived) infinitesimal deformation functors AJ : HilbX/Z → Jac2p

X/Z

where

◮ Jac2p X/Z is some deformation functor with obs(Jac2p X/Z) = 0 ◮ obs(AJ) is the restriction to obs(HilbX/Z) of

i : H1(Z; NX/Z) → H2p(OX → Ω1

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 ϕ : g → h such that:

◮ g HilbX/Z ◮ h is quasi-abelian (i.e. h is quasi-isomorphic to a cochain

complex)

◮ the linear morphism H2(g) H2(ϕ)

− − − → H2(h) is naturally identified with i : H1(Z; NX/Z) → H2p(OX → Ω1

X → · · · → · · · Ωp−1 X

)

Let χ : L → M a morphism of dglas. hofiber(χ)

• L

χ

• M
• 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”.

cone(χ) = L ⊕ M[−1], [(l, m)]1 = (dl, χ(l) − dm) [(l1, m1), (l2, m2)]2 =

• [l1, l2], 1

2[m1, χ(l2)] + (−1)deg(l1) 2 [χ(l1), m2]

• [(l1, m1), · · · , (ln, mn)]n =

 0, Bn−1 (n − 1)!

• σ∈Sn

±[mσ(1), [· · · , [mσ(n−1), χ(lσ(n))] · · · ]]  ,

for n ≥ 3 where the Bn’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 A0,∗

X (ΘX) be the p = 0 Dolbeault dgla with

coefficients in holomorphic vector fields on X and A0,∗

X (ΘX)(− log Z) = ker{A0,∗ X (ΘX) → A0,∗ Z (NX/Z)}

the sub-dgla of A0,∗

X (ΘX) of differential forms with coefficients

vector fields tangent to Z. The deformation functor associated with hofiber

• A0,∗

X (ΘX)(− log Z) ֒

→ A0,∗

X (ΘX)

• is HilbX/Z.

Let L and M be two dglas, i : L → M[−1] a morphism of graded vector spaces. Let l: L → M a → la = dia + ida 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] = [ia, lb], [ia, ib] = 0. Note that i[a,b] = [ia, lb] implies that l is a morphism of differential graded Lie algebras: the Lie derivative associated with i.

Example

let X be a differential manifold, A0

X(TX) 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: A0

X(TX) → End(A∗ X)[−1]

is a Cartan homotopy and its differential is the Lie derivative l = [d, i] =: A0

X(TX) → End(A∗ X).

Example

let X be a complex manifold, A0,∗

X (ΘX) 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: A0,∗

X (ΘX) → End(A∗,∗ X )[−1]

is a Cartan homotopy and its differential is the holomorphic Lie derivative l = [∂, i]: A0,∗

X (ΘX) → End(A∗,∗ X ).

Example

let X be a complex manifold, A0,∗

X (ΘX) be the p = 0 Dolbeault

dgla with coefficients in holomorphic vector fields on X, and End(DX) be the dgla of endomorphisms of the complex of smooth currents on X. Then the contraction ˆ i: A0,∗

X (ΘX) → End(DX)[−1]

is a Cartan homotopy and its differential is the holomorphic Lie derivative ˆ l = [ˆ ∂,ˆ i]: A0,∗

X (ΘX) → End(DX).

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[2p]: A0,∗

X (ΘX)(− log Z) → End(DX[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 → ia ⊗ 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

ι

• M

i

And so, by the universal property of the homotopy fiber we get L

l

• Φ
• hofiber(ι)
• N

ι

• M

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

A Cartan square is the following set of data:

◮ two morphisms of dglas ϕL : L1 → L2 and ϕM : M1 → M2; ◮ two Cartan homotopies i1 : L1 → M1[−1] and

i2 : L2 → M2[−1] such that L1

ϕL

• i1 M1[−1]

ϕM[−1]

• L2

i2 M2[−1]

is a commutative diagram of graded vector spaces. A Cartan square induces a commutative diagram of dglas L1

ϕL

• l1

M1

ϕM

• L2

l2

M2

, where l1 and l2 are the Lie derivatives associated with i1 and i2, respectively.

It also induces a Cartan homotopy (i1, i2) : TW (L1 → L2) → TW (M1 → M2)[−1] whose Lie derivative is (l1, l2) : TW (L1 → L2) → TW (M1 → M2). Now assume the commutative diagram of dglas associated with a Cartan square factors as L1

ϕL

• l1

N1

ι1

• M1

ϕM

• L2

l2

N2

ι2

M2

where ι1 and ι2 are inclusions of sub-dglas. Then we have a linear L∞ morphism (l1, l2, i1, i2):TW (L1 → L2) → cone (TW (N1 → N2) → TW (M1 → M2)) .

If moreover also ϕL and ϕM are inclusions, then in the (homotopy) category of cochain complexes the linear L∞-morphism (l1, l2, i1, i2) is equivalent to the span (L2/L1)[−1]

∼

← − cone(L1 → L2) → (M2/(M1 + N2))[−2], where the quasi isomorphism on the left is induced by the projection on the second factor, and the morphism on the right is (a1, a2) → i2,a2 mod M1 + N2. Hence, at the cohomology level, the morphism Hn(l1, l2, i1, i2) is naturally identified with the morphism Hn−1(L2/L1) → Hn−2(M2/(M1 + N2)) [a] → [i2,˜

a mod M1 + N2],

where ˜ a ∈ L2 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

• f 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 gg(v)) daff(f , v) = (dEndf , dv) Every degree zero closed element v in V defines an embedding of dglas jv : 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 j0 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 iv : L → aff(V )[−1] a → (ia, −ia(v)) is a Cartan homotopy. The corresponding Lie derivative is lv : L → aff(V ) a → (la, −la(v)) Indeed, the linear map iv is the composition of the Cartan homotopy i with the dgla morphism jv, hence it is a Cartan

• homotopy. The corresponding Lie derivative is the composition of l

with jv. So we have built a Cartan homotopy iv 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. Lv = {a ∈ L such that ia(v) = 0 and la(v) = 0}

For any sub-dgla ˜ L ⊆ Lv, the diagram ˜ L

• i
• ˜

L End(V )[−1]

j0[−1]

• L

iv

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 lv : 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 TW     ˜ L

• L

   

(l

• ˜

L,lv,i

• ˜

L,iv)

− − − − − − − → cone       TW       End(V )(−F)

• aff(V )(−F)

      → TW       End(V )

• aff(V )

            .

At the n-th cohomology level, this L∞-morphism gives the map Hn−1(L/˜ L) → Hn−2(V /F) [a] → −[i˜

a(v) mod F].

where ˜ a 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 )

cone

TW

• TW
• End(V )(−F)
• End(V )
• aff(V )(−F)

aff(V )

But actually, due to the fact that we have sections

V aff(V ) End(V )

• F

aff(V )(−F) End(V )(−F)

• there is a simpler model:

(V /F)[−2]

F[−1]

• V [−1]
• End(V )(−F)
• End(V )
• aff(V )(−F)

aff(V )

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 pD(X))[2p] be the sub-complex of V obtained by shifting the p-th term in the Hodge filtration on currents, F pD(X) =

• i≥p

Di,∗(X). Finally, let L = A0,∗

X (ΘX), let ˜

L = A0,∗

X (ΘX)(− log Z) and let

i : L → End(V )[−1] be the (shifted) contraction operator on currents: ˆ i[2p] : A0,∗

X (ΘX) → End(D(X)[2p], D(X)[2p])[−1].

The 6-ple (L, ˜ L, V , F, v, i) defined this way satisfies the hypothesis

• f the slides above, so we get an L∞-morphims

TW        A0,∗

X (ΘX)(− log Z)

• A0,∗

X (ΘX)

       → (D(X)/F pD(X))[2p − 2] inducing in cohomology H0(Z; NX/Z) → H2p−1(D(X)/F pD(X)) H1(Z; NX/Z) → H2p(D(X)/F pD(X)) [x] → −[ˆ i˜

xZ mod F pD(X)]

in degrees 1 and 2, where ˜ x is any representative of [x] in A0,∗

X (ΘX).

Since H•(D(X)/F pD(X)) = H•(X; OX → Ω1

X → · · · Ωp−1 X

), if we define Jac2p

X/Z to be the deformation functor associated to

the abelian dgla (D(X)/F pD(X))[2p − 2] then we get from the L∞-morphis exhibited above a morphism of deformation functors AJ : HilbX/Z → Jac2p

X/Z

with dAJ : H0(Z; NX/Z)

i

− → H2p−1(X; OX → Ω1

X → · · · Ωp−1 X

) and

• bs(AJ) : H1(Z; NX/Z)

i

− → H2p(X; OX → Ω1

X → · · · Ωp−1 X

)