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”)
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 .
Tb0 HilbX /Z = H 0 (Z ; NX /Z )
obs(HilbX /Z ) ⊆ H 1 (Z ; NX /Z )
Actually one can control the obstructions better:
n
o
i
p−1
1
2p
1
obs(HilbX /Z ) ⊆ ker H (Z ; NX /Z ) →
− H (OX → ΩX → · · · → · · · ΩX )
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 → JacX2p/Z
where
I
JacX2p/Z is some deformation functor with obs(JacX2p/Z ) = 0
I
obs(AJ) is the restriction to obs(HilbX /Z ) of
i : H 1 (Z ; NX /Z ) → H2p (OX → Ω1X → · · · → · · · 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:
I
g ! HilbX /Z
I
h is quasi-abelian (i.e. h is quasi-isomorphic to a cochain
complex)
I
the linear morphism
with
H 2 (ϕ)
2
H (g) −−−→
H 2 (h) is naturally identified
i : H 1 (Z ; NX /Z ) → H2p (OX → Ω1X → · · · → · · · Ωp−1
X )
Let χ : L → M a morphism of dglas.
hofiber(χ)
0
s{
/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)}
•
1
It is a sub-dgla of L ⊕ M ⊗ Ω (∆ ) .
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 =
1
[l1 , l2 ], [m1 , χ(l2 )] +
2
(−1)deg(l1 )
2
!
[χ(l1 ), m2 ]

[(l1 , m1 ), · · · , (ln , mn )]n = 0,

Bn−1 X
±[mσ(1) , [· · · , [mσ(n−1) , χ(lσ(n) )] · · · ]],
(n − 1)!
σ∈Sn
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
0,∗
0,∗
A0,∗
X (ΘX )(− log Z ) = ker{AX (ΘX ) → AZ (NX /Z )}
the sub-dgla of A0,∗
X (ΘX ) of differential forms with coefficients
vector fields tangent to Z . The deformation functor associated
with
0,∗
0,∗
hofiber AX (ΘX )(− log Z ) ,→ AX (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 7→ 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, A0X (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 : A0X (TX ) → End(A∗X )[−1]
is a Cartan homotopy and its differential is the Lie derivative
l = [d, i] = : A0X (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(AX
) be the dgla of endomorphisms of the de Dolbeault
complex of X . Then the contraction
∗,∗
i : A0,∗
X (ΘX ) → End(AX )[−1]
is a Cartan homotopy and its differential is the holomorphic Lie
derivative
∗,∗
l = [∂, i] : A0,∗
X (ΘX ) → End(AX ).
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
î : A0,∗
X (ΘX ) → End(DX )[−1]
is a Cartan homotopy and its differential is the holomorphic Lie
derivative
ˆ î] : A0,∗ (ΘX ) → End(DX ).
l̂ = [∂,
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
0,∗
î[2p] : AX
(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],
is a Cartan homotopy.
a ⊗ ω 7→ ia ⊗ ω,
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
s{
i
N
"
ι
0
/M
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)
$ 0
!
/N
ι
/M
A Cartan square is the following set of data:
I
two morphisms of dglas ϕL : L1 → L2 and ϕM : M1 → M2 ;
I
two Cartan homotopies i1 : L1 → M1 [−1] and
i2 : L2 → M2 [−1]
such that
i1
L1
/ M1 [−1]
ϕL
i2
L2
ϕM [−1]
/ M2 [−1]
is a commutative diagram of graded vector spaces.
A Cartan square induces a commutative diagram of dglas
L1
l1
ϕL
L2
/ M1
,
ϕM
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
l1
ϕL
L2
l2
/ N1
/ N2
ι1
/ M1
ϕM
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 ) 7→ i2,a2 mod M1 + N2 .
Hence, at the cohomology level, the morphism H n (l1 , l2 , i1 , i2 ) is
naturally identified with the morphism
H n−1 (L2 /L1 ) → H n−2 (M2 /(M1 + N2 ))
[a] 7→ [i2,ã mod M1 + N2 ],
where ã ∈ 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
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 ))
daff (f , v ) = (dEnd f , dv )
Every degree zero closed element v in V defines an embedding of
dglas
jv : End(V ) → aff(V )
f 7→ (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 7→ (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 7→ (ia , −ia (v ))
is a Cartan homotopy. The corresponding Lie derivative is
lv : L → aff(V )
a 7→ (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 L̃ ⊆ Lv , the diagram
L̃
L
i
L̃
/ End(V )[−1]
j0 [−1]
iv
/ aff(V )[−1] ,
where the left vertical arrow is the inclusion L̃ ,→ 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



 
L̃

 End(V )(−F ) 
v v
  (l L̃ ,l ,i L̃ ,i )



 −−−−−−−→ cone TW 
 → TW
TW 



 



L
aff(V )(−F )


 End(V ) 


 .





aff(V )
At the n-th cohomology level, this L∞ -morphism gives the map
H n−1 (L/L̃) → H n−2 (V /F )
[a] 7→ −[iã (v ) mod F ].
where ã is any representative of [a] in L.
The L∞ -algebra




cone 
TW



 End(V )(−F ) 



 → TW




aff(V )(−F )

 End(V ) 








aff(V )
is a model for the double homotopy fiber of the commutative
diagram
/ End(V )
End(V )(−F )
aff(V )(−F )
/ aff(V )
cone
/ TW
/ TW
/ End(V )
End(V )(−F )
aff(V )(−F )
/ aff(V )
But actually, due to the fact that we have sections
0
0
v
/ aff(V )
/V
/F
t
/ aff(V )(−F )
/ End(V )
/ End(V )(−F )
there is a simpler model:
(V /F )[−2]
/ F [−1]
/ V [−1]
/ End(V )
End(V )(−F )
aff(V )(−F )
/ aff(V )
/0
/0
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,
M
p
F D(X ) =
D i,∗ (X ).
i≥p
0,∗
Finally, let L = A0,∗
X (ΘX ), let L̃ = AX (ΘX )(− log Z ) and let
i : L → End(V )[−1] be the (shifted) contraction operator on
currents:
î[2p] : A0,∗
X (ΘX ) → End(D(X )[2p], D(X )[2p])[−1].
The 6-ple (L, L̃, V , F , v , i) defined this way satisfies the hypothesis
of the slides above, so we get an L∞ -morphims


 A0,∗

 X (ΘX )(− log Z ) 


TW 
 → (D(X )/F p D(X ))[2p − 2]




0,∗
AX (ΘX )
inducing in cohomology
H 0 (Z ; NX /Z ) → H 2p−1 (D(X )/F p D(X ))
H 1 (Z ; NX /Z ) → H 2p (D(X )/F p D(X ))
[x] 7→ −[îx̃ Z mod F p D(X )]
in degrees 1 and 2, where x̃ is any representative of [x] in
A0,∗
X (ΘX ).
Since
H • (D(X )/F p D(X )) = H• (X ; OX → Ω1X → · · · Ωp−1
X ),
if we define JacX2p/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 : HilbX /Z → JacX2p/Z
with
i
dAJ : H 0 (Z ; NX /Z ) →
− H2p−1 (X ; OX → Ω1X → · · · Ωp−1
X )
and
i
obs(AJ) : H 1 (Z ; NX /Z ) →
− H2p (X ; OX → Ω1X → · · · Ωp−1
X )