# Domenico Fiorenza The periods map of a complex projective manifold. An oo-category perspective

Notes from the talk given at the Quaterly seminar on topology and geometry, Utrecht University, March 11, 2011

#### Abstract

After recalling the classical Kodaira-Spencer and Newlander-Nierenberg approaches to deformations of complex manifolds, we will show how these apparently very different descriptions are just two models for the same derived object: the derived global sections of the holomorphic tangent sheaf of the manifold. With this in mind we will see how the periods map of a projective manifold has a very natural and simple local description in terms of oo-groupoids: its classical description is then recovered by globalization and truncation.

The talk will be completely elementary, and no prior knowledge of deformation theory or higher category theory will be assumed.

# Contents

## A review of classical deformation theory

Given a smooth complex manifold $X$, a deformation of $X$ over a pointed basis $(B,b)$ is a pullback diagram

$\array{ X &\to & \mathcal{X} \\ \downarrow&&\downarrow^{\mathrlap{\pi}} \\ * &\stackrel{b}{\to}& B }$

Here the basis $B$ is a (possibly singular) complex manifold, and $\pi:\mathcal{X}\to B$ is a proper holomorphic function. A morphism of deformations is a morphism of pullback diagrams.

We denote by $Def_X(B)$ the set of isomorphism classes of deformations of $X$ parametrized by the pointed base $(B,b)$

If $\mathcal{X}\to B$ is a deformation of $X$, and $(C,c)\to (B,b)$ is a morphism of pointed complex, then the pullback diagram

$\array{ X &\to& \mathcal{X}\times_B C &\to& \mathcal{X} \\ \downarrow &&\downarrow&&\downarrow\pi \\ *&\stackrel{c}{\to}& C&\to&B }$

exhibits $\mathcal{X}\times_B C\to C$ as a deformation of $X$ over $C$. Therefore $Def_X$ is a functor

$Def_X: \{pointed complex manifolds\}^{op}\to Sets.$

We are actually interested in the local behaviour of deformations in a neighborhood of the distinguished point $b$. So $Def_X$ is actually a functor

$Def_X: \{germs of pointed complex manifolds\}^{op}\to Sets.$

on germs of pointed complex manifolds.

We pass to infinitesimal deformations by going from germs of pointed complex manifolds to formal pointed manifolds, i.e., to spectra of local $\mathbb{C}$-Artin algebras. Since $Spec$ is a contravariant functor, we finally have the functor of infinitesimal deformations of $X$

$Def_X : Art \to Sets$

where $Art$ is the category of local Artin algebras with residue field $\mathbb{C}$. For an object $A$ in $Art$ we will denote by $\mathfrak{m}_A$ its maxial ideal.

### Newlander-Nierenberg approach

The Newlander-Nierenberg approach to $Def_X$ consists in noticing that from a differential point of view, the family $\mathcal{X}\to B$ is trivial (Ehresmann’s theorem), so we can think of it as a fixed smooth manifold on which we are changing the complex structure. And since the datum of a complex manifold structure on a smooth manifold is the datum of the subsheaf $\mathcal{O}_X$ of holomorphic functions inside the sheaf $C^\infty_X$ of smooth complex valued functions, what we are interested in is in deforming this subsheaf.

We have

$\mathcal{O}_X=ker\{\overline{\partial} C^\infty_X\to \mathcal{A}^{0,1}_X\}$

where $\mathcal{A}^{p,q}$ is the sheaf of smooth differential forms of type $(p,q)$, and

$\overline{\partial}:\mathcal{A}_X^{p,q}\to \mathcal{A}_X^{p,q+1}$

is the Dolbeault differential. So the idea is: to deform $\mathcal{O}_X$, deform $\overline{\partial}$.

This is achieved as follows: for any $\xi$ in the vector space $A^{0,k}(X,T_X^{1,0})$ of sections of the sheaf of $(0,k)$-forms with coefficients in the holomorphic tangent sheaf? of $X$, consider the holomorphic Lie derivative

$\boldsymbol{l}_\xi=[\partial,\boldsymbol{i}_\xi]: \mathcal{A}^{p,q}_X\to \mathcal{A}^{p,q+k},$

where

$\boldsymbol{i}_\xi: \mathcal{A}^{p,q}_X\to \mathcal{A}^{p-1,q+k}_X$

is the contraction of differential forms with the vector field (with coefficients in $\mathcal{A}^{0,k}_X$) $\xi$.

If $\xi\in A^{0,1}(T_X^{1,0})$, then $\boldsymbol{l}_\xi$ is a degree $(0,1)$ derivation of $\mathcal{A}^{*,*}_X$ and so we can use it to perturb $\overline{\partial}$. Since we want this perturbation to be infinitesimal, we actually fix an Artin algebra $(A,\mathfrak{m}_A)$ and pick $\xi$ in $A^{0,1}(X,T_X)\otimes \mathfrak{m}_A$. Then

$\partial+\boldsymbol{l}_\xi$

is our perturbed Dolbeault operator. By construction it is a derivation of $\mathcal{A}^{*,*}_X$, but it is in general not a differential. We have to impose the condition $(\overline{\partial}+\boldsymbol{l}_\xi)^2=0$, i.e.,

$[\overline{\partial},\boldsymbol{l}_\xi]+\frac{1}{2}[\boldsymbol{l}_\xi,\boldsymbol{l}_\xi]=0,$

where the bracket is the graded commutator in the graded associative algebra $End^*(A^{*,*}_X)$. The corresponding graded Lie algebra has a natural differential graded Lie algebra structure, with differential given by the adjoint of the Dolbeault operator, so the above equation reads

$d_{End}\boldsymbol{l}_\xi+\frac{1}{2}[\boldsymbol{l}_\xi,\boldsymbol{l}_\xi]=0,$

i.e. is the Maurer-Cartan equation in the dgla $End^*(A^{*,*}_X)$.

Also the graded vector space $A^{0,*}(X,T_X)$ of differential forms of type $(0,*)$ with coefficients in $T_X$ has a natural dgla structure, and the holomorphic Lie derivative

$\boldsymbol{l}: A^{0,*}(X,T_X)\to End^*(\mathcal{A}^{*,*}_X)$

is an morphism of dglas. Therefore the above condition on $\boldsymbol{l}_\xi$ becomes

$\boldsymbol{l}\left(d\xi+\frac{1}{2}[\xi,\xi]\right)=0,$

where $d$ is the differential in the dgla $A^{0,*}(X,T_X)$. Moreover, since $\boldsymbol{l}$ in injective, this is equivalent to

$d\xi+\frac{1}{2}[\xi,\xi]=0,$

i.e., we can naturally associate a deformation of the complex structure of $X$ to a Maurer-Cartan element in the dgla $A^{0,*}(X,T_X)$. To obtain the set $Def_X$ we still have to quotient out isomorphic copies of the same complex structure. Two Maurer-Cartan elements $\xi_0$ and $\xi_1$ give isomorphic complex structures if there exist a diffeomorphism of $X$ of the form $e^{\boldsymbol{l}_a}$ with $a$ in $A^{0,0}(X;T_X)$ such that the following diagram commutes

$\array{ \mathcal{A}^{0,*}_X&\stackrel{\overline{\partial}+\boldsymbol{l}_{\xi_0}}{\longrightarrow}& \mathcal{A}^{0,*}_X\\ e^{\boldsymbol{l}_a}\downarrow\phantom{mm}&&\downarrow e^{\boldsymbol{l}_a}\\ \mathcal{A}^{0,*}_X&\stackrel{\overline{\partial}+\boldsymbol{l}_{\xi_1}} {\longrightarrow}& \mathcal{A}^{0,*}_X }$

In other words

$Def_X = MC(A^{0,*}(X;T_X))/exp(A^{0,0}(X;T_X)$

### Kodaira-Spencer approach

The Newlander-Nierenberg approach to deformations of the sheaf $\mathcal{O}_X$ we described above was based on the remark that $\mathcal{O}_X$ is the kernel of a differenial operator globally defined on $X$. The Kodaira-Spencer approach is instead a local approach. Fix an open cover $\mathcal{U}$ of $X$, so that $\mathcal{O}_U=\mathcal{O}_X\vert_U$ for any $U$ in $\mathcal{U}$. Then the global sheaf $\mathcal{O}_X$ is obatined by glueing toghether the local sheaves $\mathcal{O}_U$ via the trivial glueing maps

$\mathcal{O}_U\vert_{U\cap V}\stackrel{Id}{\longrightarrow} \mathcal{O}_U\vert_{U\cap V}$

The Kodaira-Spencer’s idea is then that to deform $\mathcal{O}_X$ we have to deform the local sheaves $\mathcal{O}_U$ and the glueing maps between the restrictions $\mathcal{O}_U|_{U\cap V}$ and $\mathcal{O}_V\vert_{U\cap V}$. So, one takes an open cover of $X$ such that the “pieces” $(U,\mathcal{O}_U)$ have no nontrivial deformations (this happens, e.g., if $(U,\mathcal{O}_U)$ is isomorphic as a ringed space to a polidisk in $\mathbb{C}^n$ with its sheaf of holomorphic functions), then the datum of a deformation of $X$ is entirely encoded in the datum of a deformation of the glueing maps. Since we are working infinitesimally, an isomorphism

$\mathcal{O}_U\vert_{U\cap V}\stackrel{\varphi_{UV}}{\longrightarrow} \mathcal{O}_U\vert_{U\cap V}$

will be of the form $\varphi_{UV}=exp(\alpha)$, with $\alpha$ an holomorphic vector field on $U\cap V$. In order to define a global sheaf, the glueings $\varphi_{UV}$ have to satisfy the cocycle condition

$\varphi_{UV}\varphi_{VW}\varphi_{WU}=Id\qquad on U\cap V\cap W; \qquad \qquad \varphi_{UU}=Id\qquad on U.$

Two cocycles $\{\varphi_{UV}\}$ and $\{\psi_{UV}\}$ describe isomorphic deformations if and only if there are isomorphisms $\theta_U:\mathcal{O}_U\to\mathcal{O}_U$ such that the diagram

$\array{ \mathcal{O}_U\vert_{U\cap V}& \stackrel{\phi_{UV}}{\longrightarrow}& \mathcal{O}_V\vert_{U\cap V}\\ \theta_U\vert_{U\cap V}\downarrow\phantom{mmmm} && \phantom{mmm}\downarrow\theta_V\vert_{U\cap V}\\ \mathcal{O}_U\vert_{U\cap V}& \stackrel{\psi_{UV}}{\longrightarrow}& \mathcal{O}_V\vert_{U\cap V}\\ }$

commutes. And again, since we are working on an infinitesimal base, $\theta_U$ will be of the form $\theta_U=exp(\beta_U)$ with $\beta_U$ an holomorphic vector field on $U$. Therefore we see that deformations of $X$ are given by cohomology classes in nonabelian cohomology

$H^1(X;exp(\mathcal{T}_X)$

where $\mathcal{T}_X$ is the holomorphic tangent sheaf of $X$. This can be seen as the set of connetced components of the hom-space $\mathbf{H}(X,\mathbf{B}exp(\mathcal{T})$, by looking both at the manifold $X$ and to $\mathbf{B}exp(\mathcal{T})$ as simplicial presheaves over a suitable site of local models for complex geometry. Here $X$ is the presheaf represented by $X$, and $\mathcal{T}$ is the presheaf

$U\mapsto \{holomorphic vector fields on U\}.$

Finally, $\mathbf{B}exp(\mathcal{T})$ is the presheaf given by the delooping of

$U\mapsto exp(\mathcal{T}(U)).$

## From dglas to $\infty$-groupoids

Given an dgla $\mathfrak{g}$, we can functorially associate with any Artin algebra $(A,\mathfrak{m}_A)$ a nilpotent dgla by $(A,\mathfrak{m}_A)\mapsto \mathfrak{g}\otimes \mathfrak{m}_A$. With this in mind we will look at all dglas we will meet in what follows as functors $Art \to nilpotent dgla$.

### The Deligne groupoid

For $\mathfrak{g}=\bigoplus_{i}\mathfrak{g}^i$ a dgla, let $MC(\mathfrak{g})$ be the set of Maurer-Cartan elements, i.e.,

$MC(\mathfrak{g})=\{x\in \mathfrak{g}^1 such that dx+\frac{1}{2}[x,x]=0\}$

One thinks element in this set as flat $\mathfrak{g}$-connections: indeed

$x\in MC(\mathfrak{g}) \Leftrightarrow (d+[x,-])^2=0.$

The subspace $\mathfrak{g}^0$ of $\mathfrak{g}$ is a Lie algebra; the group

$exp(\mathfrak{g}^0)$

acts on as a group of gauge transformations on the set of $\mathfrak{g}$-connections (by conjugation), and this action preserves the subset of flat connections. Hence we have a gauge action of $exp(\mathfrak{g}^0)$ on $MC(\mathfrak{g})$:

$e^{a}(d+[x,-])e^{-a}=d+[e^a*x,-].$

Explicitely,

$e^a*x=x+\sum_{n=0}^\infty \frac{([a,-])^n}{(n+1)!}([a,x]-da)$

The Deligne groupoid $Del(\mathfrak{g})$ of the dgla $\mathfrak{g}$ is the action groupoid

$Del(\mathfrak{g})=MC(\mathfrak{g})// exp(\mathfrak{g}^0)$

Note that for $\mathfrak{g}$ a Lie algebra this is the delooping groupoid $\mathbf{B} exp(\mathfrak{g})=*//exp(\mathfrak{g})$.

We denote by $Def(\mathfrak{g}$ the decategorification of $Del(\mathfrak{g})$, i.e., the set of the isomorphism classes of its objects. Also, in what follows we will identify the Deligne groupoid with the simplicial set given by its nerve. For instance, we will write

$Def(\mathfrak{g})=\pi_{0}Del(\mathfrak{g}).$

### Gauge vs. homotopy equivalent Maurer-Cartan elements

Consider now two gauge equivalent Maurer-Cartan elements $x_0$ and $x_1$ for a dgla $\mathfrak{g}$. By definition, this means that there exists an element $\gamma$ in $\mathfrak{g}^0$ sucht that $e^\gamma*x_0=x_1$. We can make this way of going from $x_0$ to $x_1$ a “continuous path” simply by going from the dgla $\mathfrak{g}$ to the dgla $\mathfrak{g}\otimes \Omega_1$, where $\Omega^1$ is the commutative differential graded algebra of differential forms of the geometric realization of the 1-simplex. Indeed, $\gamma\,t$ is a degree zero element in $\mathfrak{g}\otimes \Omega_1$, and $x_0$ is an element in $MC(\mathfrak{g}\otimes\Omega_1)$, and so $x(t)=e^{\gamma\, t}*x_0$ is a Maurer-Cartan element in $\mathfrak{g}\orimes\Omega_1$, with $x(0)=x_0$ and $x(1)=x_1$.

Definition Two Maurer-Cartan elements $x_0$ ad $x_1$ are homotopy equivalent if there exists a maurer-Cartan element $x(t)$ in $mathfrak{g}\otimes\Omega^1$ such that $x(0)=x_0$ and $x(1)=x_1$.

At this level is not even clear that this is indeed an equivalence relation. This is an important point and we will come back to this in a minute.

By the above consideratins it is clear that

$gauge equivalence \Rightarrow homotopy equivalence$

Remarkably, also the converse is true (which, by the way shows that homotopy equivalence is an equivalence relation). Indeed, one can show that any Maurer-Cartan element for the dgla $\mathfrak{g}\otimes\Omega_1$ is of the form $x(t)=e^{\gamma(t)}*x_0$, for some $\gamma(t)$ in $\mathfrak{g}^0[t]$ and some $x_0$ in $MC(\mathfrak{g})$. So if $x(t)=e^{\gamma(t)}*x_0$ is a homotopy equivalence between $x_0$ and $x_1$, then $e^{\gamma(1)}$ is a gauge equivalence between them, i.e.

$homotopy equivalence \Rightarrow gauge equivalence$

### The $\infty$-groupoid of a dgla

With any nilpotent dgla $\mathfrak{g}$ is therefore naturally associated the simplicial set

$Def_\bullet(\mathfrak{g})=MC(\mathfrak{g}\otimes \Omega_\bullet),$

where $\Omega_\bullet$ is the simplicial differential graded commutative associative algebra of polynomial (or piecewise smooth, smooth with sitting instnts, etc.) differential forms on the standard $n$-simplexes, for $n\geq0$. The crucial property of the simplicial set $Def_\bullet(\mathfrak{g})$ is that it is a Kan complex, or (in maybe more evocative terms) an oo-groupoid. In particular this gives a much deeper explanation for homotopy equivalence of Maurer-Cartan elements beeing an equivalence relation: this is the horn filling condition for the 2-horn $\Lambda_1[2]$.

If the dgla $\mathfrak{g}$ is concentrated in nonnegative degrees, then parallel transport induces a natural morphism of simplicial sets

$\Def_\bullet(\mathfrak{g})\to Del(\mathfrak{g}).$

Indeed, a Maurer-Cartan element in $\mathfrak{g}\otimes \Omega_n$ is of the form $g+A$ with $g$ a $\mathfrak{g}^1$-valued 0-form on $\Delta[n]$, and $A$ a $\mathfrak{g}^0$-valued 1-form on $\Delta[n]$. The 2-form component of the Maurer-Cartan equation for $g+A$ is $d_\Omega A+\frac{1}{2}[A,A]=0$, where $d_\Omega$ is the differential on differential forms on the simplex. So $A$ is a flat $\mathfrak{g}^0$-connection on the simplex, and parallel transport of $A$ from the vertex 0 of the simplex to the other vertices will define a map from $Def(\mathfrak{g})_n$ to the $n$-simplices of the (nerve of) the Deligne groupoid. Here we are using the fact that the connection $A$ is flat and the $n$-simplex is simply connected.

It is a remarkable result by Hinich that the morphism $\Def_\bullet(\mathfrak{g})\to Del(\mathfrak{g})$ is an acyclic fibration. In particular $\Def_\bullet(\mathfrak{g})$ and $Del(\mathfrak{g})$ are (weakly) homotopy equivalent. An immediate corollary of this is that $\pi_{0}Def_\bullet(\mathfrak{g})=Def(\mathfrak{g})$, but note that by the

$gauge equivalence = homotopy equivalence$

mentioned above, we have

$\pi_{0}Def_\bullet(\mathfrak{g})=Def(\mathfrak{g}).$

for any dgla $\mathfrak{g}$, i.e., even for dglas non concentrated in nonegative degrees. On the other hand, when $\mathfrak{g}$ is not concentrated in nonegative degrees, we do not have, in general, a morphism $\Def_\bullet(\mathfrak{g})\to Del(\mathfrak{g})$ induced by parallel transport. Let us see why. If a degree $-1$ component $\mathfrak{g}^{-1}$ is present, then a Maurer-Cartan element in $\mathfrak{g}\otimes \Omega_n$ is of the form $g+A+h$ with $g$ a $\mathfrak{g}^1$-valued 0-form on $\Delta[n]$, $A$ a $\mathfrak{g}^0$-valued 1-form on $\Delta[n]$, and $h$ a $\mathfrak{g}^{-1}$-valued 2-form on $\Delta[n]$. The 2-form component of the Maurer-Cartan equation for $g+A+h$ is $d_\Omega A+\frac{1}{2}[A,A]+d_{\mathfrak{g}}h+[h,g]=0$, and we see that $A$ is not flat anymore, and that the obstruction to flatness is given by the term $d_{\mathfrak{g}}h+[h,g]$. The parallel transport along the boundary of a 2-simplex then does not gives the identity but an element in the irrelevant stabilizer of the starting vertex, where the irrelevant stabilizer of a Maurer-Cartan element $x$ is defined as

$Stab_{irr}(x)= \{d h+[x,h] \mid h \in \mathfrak{g}^{-1}\}$

(note that $Stab_{irr}(x)$ is actually a subgroup of the stabilizer $\{ a \in \mathfrak{g}^0\otimes \mathfrak{m}_A \mid e^a*x=x \}$). So we see that the groupoid $\pi_{\leq 1}Def_\bullet(\mathfrak{g})$ fails to be equivalent to $Del(\mathfrak{g})$ precisely because of nontrivial irrelevant stabilizers. And so we also see that we do have a morphism of simlicial sets $Def_\bullet(\mathfrak{g})\to Del(\mathfrak{g})$ as soon as irrelevant stabilizers are trivial, and in this case we have

$\pi_{\leq 1}Def_\bullet(\mathfrak{g})\simeq Del(\mathfrak{g}).$

A particular case of this is the case in which $\mathfrak{g}$ is concentrated in nonnegative degrees we mentioned above.

A very good reason for working with $\infty$-groupoids valued deformation functors rather than with their apparently handier set-valued or groupoid-valued versions is the following folk statement, which allows one to move homotopy constructions back and forth between dglas and (homotopy types of) “nice” topological spaces.

Theorem The functor $\Def_\bullet: DGLA \to [Art, \infty{-Grpd}]$ is an equivalence of (∞,1)-categories.

Here the $(\infty,1)$-category structures involved are the most natural ones, and they are both induced by standard model category structures. Namely, on the model structure on dg-Lie algebras has surjective morphisms as fibrations and quasi-isomorphisms as weak equivalences, just as in the case of differential complexes, whereas the model category structure on the right hand side is the model structure on simplicial presheaves induced by the standard model structure on simplicial sets.

A sketchy proof of the above equivalence can be found in (lurie-moduli); see also (Pridham).

We will refer to a functor $[Art,\infty{-Grpd}]$ as to a formal $\infty$-groupoid. In what follows, we will just say “$\infty$-groupoid” to mean “formal $\infty$-groupoid”.

### Back to classical deformations

We are now in position to directly relate the Newlander-Nierenberg to the Kodaira-Spencer approach to the infinitesimal deformation theory of a complex manifold $X$. Consider the holomorphic tangent sheaf $\mathcal{T}_X$ of $X$; it is a sheaf of Lie algebras. Then for any open subset $U$ of $X$ we can consider the oo-groupoid $Def_\bullet(\mathcal{T}_X(U))$. This defines a presheaf of oo-groupoids on the site of open subsets of $X$: the oo-presheaf of local deformations of $X$. To pass to global deformations we just have to take (derived) global sections, i.e., the homotopy limit homotopy limit of its local sections over a good open cover $\mathcal{U}$ of $X$. Since $Def_\bullet$ is an equivalence of $(\infty,1)$-categories, we have

$holim(Def_\bullet(\mathcal{T}_X(\mathcal{U}))\simeq Def_\bullet(holim \mathcal{T}_X(\mathcal{U}))$

On the left-hand side, since $\mathcal{T}$ is a sheaf of Lie algebras, and so $\mathcal{T}_X(U)$ is concentrated in nonnegative degree as a dgla, we have natural homotopy equivalences $Def_\bullet(\mathcal{T}_X(U))\simeq Del(\mathcal{T}_X(U))$, and so

$holim(Def_\bullet(\mathcal{T}_X(\mathcal{U}))\simeq holim(Del(\mathcal{T}_X(\mathcal{U}))=holim \mathbf{B} exp(\mathcal{T}_X(\mathcal{U})),$

and we find the Kodaira-Spencer description of the deformations of $X$. On the right-hand side, a way of computing the homotopy limit holim $\mathcal{T}_X(\mathcal{U})$, i.e., to compute the derived global sections of the sheaf $\mathcal{T}_X$, is to take a fine resolution? of it, and to take ordinary global sections of this (this is nothing but the general rule that a homotopy limit is the ordinary limit of a fibrant resolution). In particular, by taking as a fine resolution of $\mathcal{T}_X$ the Dolbeault resolution $(\mathcal{A}^{0,*}(T_X),\overline{\partial})$ we find the Newlander-Nierenberg description of $Def_X$.

## Gauge equivalent morphisms of dgla

### $L_\infty$-morphisms of dglas

Since we are now looking at dglas as to an (oo,1)-category, given two dglas $\mathfrak{g}$ and $\mathfrak{h}$, their hom-space $\mathbf{H}(\mathfrak{g},\mathfrak{h})$ is an $\infty$-groupoid. By the equivalence between dglas and (formal) $\infty$-groupoids stated at the end of the previous section, there must be a dgla $\underline{Hom}(\mathfrak {g},\mathfrak{h})$ such that $Def_\bullet(\underline{Hom}(\mathfrak{g},\mathfrak{h})$ is equivalent to $\mathbf{H}(\mathfrak{g},\mathfrak{h})$. And actually the dgla $\underline{Hom}(\mathfrak{g},\mathfrak{h})$ arises in a very natural way and admits a simple explicit description: it is the Chevalley-Eilenberg-type dgla given by the total dgla of the bigraded dgla

$\underline{Hom}^{p,q}(\mathfrak{g},\mathfrak{h})=\Hom^p(\wedge^q \mathfrak{g},\mathfrak{h}),$

endowed with the Lie bracket

$[\,,\,]_{\underline{Hom}}\colon \underline{Hom}^{p_1,q_1}(\mathfrak{g},\mathfrak{h})\otimes \underline{Hom}^{p_2,q_2}(\mathfrak{g},\mathfrak {h})\to \underline{Hom}^{p_1+p_2,q_1+q_2}(\mathfrak{g},\mathfrak{h})$

defined by

$\array{ [f,g]^{}_{\underline{Hom}}&(\gamma_1^{}\wedge\cdots\wedge\gamma_{q_1+q_2}^{})=\\ &=\sum_{\sigma\in{Sh}(q_1,q_2)}\pm[f( \gamma_{\sigma(1)}\wedge\cdots \wedge\gamma_{\sigma(q_1)}), g( \gamma_{\sigma(q_1+1)}\wedge\cdots \wedge\gamma_{\sigma(q_1+q_2)})]^{}_{\mathfrak{h}}, }$

and with the differentials

$d_{1,0}^{}\colon \underline{Hom}^{p,q}(\mathfrak{g},\mathfrak{h})\to \underline{Hom}^{p+1,q}(\mathfrak{g},\mathfrak{h})$

and

$d_{0,1}^{}\colon \underline{Hom}^{p,q}(\mathfrak{g},\mathfrak{h})\to \underline{Hom}^{p,q+1}(\mathfrak{g},\mathfrak{h})$

given by

$(d_{1,0}^{}{f})(\gamma_1^{}\wedge\cdots\wedge\gamma_q^{})=d_{\mathfrak {h}}(f(\gamma_1^{}\wedge\cdots\wedge\gamma_q^{}))+\sum_i \pm f(\gamma_1\wedge\cdots \wedge d_{\mathfrak{g}}\gamma_i\wedge \cdots\wedge\gamma_{q+1}^{})$

and

$(d_{0,1}^{}f)(\gamma_1\wedge\cdots\wedge\gamma_{q+1})= \sum_{i\lt j}\pm f([\gamma_i,\gamma_j]^{}_{\mathfrak{g}}\wedge \gamma_1\wedge\cdots \wedge\widehat{\gamma_i}\wedge\cdots\wedge\widehat{\gamma_j}\wedge \cdots\wedge\gamma_{q+1}^{}).$

These operations are best seen pictorially:

$svg graphics should go here$

Maurer-Cartan elements in $\underline{Hom}(\mathfrak{g},\mathfrak{h})$ are $L_\infty$-morphisms between $\mathfrak{g}$ and $\mathfrak{h}$. Explicitely, such an $L_\infty$-morphism is a collection of degree $1-n$ maps

$F_n:\bigotimes^n \mathfrak{g}\to \mathfrak{h}$

such that

$\array{ d_{\mathfrak{h}}F_n^1(\gamma_1^{}\wedge\cdots\wedge\gamma_n^{})&+\frac{1}{2}\!\!\!\sum_{\stackrel{\sigma\in{Sh}(q_1,q_2)}{q_1+q_2=n}}\!\!\!\!\!\!\pm[F_{q_1}^1( \gamma_{\sigma(1)}\wedge\cdots \wedge\gamma_{\sigma(q_1)}), F_{q_2}^1( \gamma_{\sigma(q_1+1)}\wedge\cdots \wedge\gamma_{\sigma(q_1+q_2)})]^{}_{\mathfrak{h}}\\ &=\sum_i \pm F_n^1(\gamma_1\wedge\cdots \wedge d_{\mathfrak{g}}\gamma_i\wedge \cdots\wedge\gamma_{n}^{})\\ &\qquad\qquad+ \sum_{i\lt j}\pm F_{n-1}^1([\gamma_i,\gamma_j]^{}_{\mathfrak{g}}\wedge \gamma_1\wedge\cdots \wedge\widehat{\gamma_i}\wedge\cdots\wedge\widehat{\gamma_j}\wedge \cdots\wedge\gamma_{q+1}^{}). }$

Note in particular that a dgla morphism $\varphi:\mathfrak{g}\to\mathfrak{h}$ is, in a natural way, an $L_\infty$-morphism between $\mathfrak{g}$ and $\mathfrak{h}$, of a very special kind: all but the first one of its Taylor coefficients vanish.

Passing to $\pi_{0}$‘s, the equivalence $\mathbf{H}(\mathfrak {g},\mathfrak {h})\simeq Def_\bullet(\underline{Hom}(\mathfrak{g},\mathfrak {h}))$ implies the following:

Proposition Let $f,g:\mathfrak{g} \to \mathfrak{h}$ be two $L_\infty$-morphisms of dglas. Then $f$ and $g$ are gauge equivalent in $MC(\underline{\Hom}(\mathfrak {g},\mathfrak{h}))$ if and only if $f$ and $g$ represent the same morphism in the homotopy category of dglas.

### Cartan homotopies

Let now $\mathfrak{g}$ and $\mathfrak{h}$ be dglas and $\boldsymbol{i}: \mathfrak{g}\to \mathfrak{h}[-1]$ be a morphism of graded vector spaces. Then $\boldsymbol{i}$, and so also $-\boldsymbol{i}$, is an element of $\underline{Hom}^{-1,1}(\mathfrak{g},\mathfrak{h})$, and so a degree zero element in the dgla $\underline{Hom}(\mathfrak{g},\mathfrak{h})$. The gauge transformation $e^{-\boldsymbol{i}}$ will map the $0$ dgla morphism to an $L_\infty$-morphism $e^{-\boldsymbol{i}}*0$ between $\mathfrak{g}$ and $\mathfrak{h}$.

This $L_\infty$-morphism will in general fail to be a dgla morphism since its nonlinear components will be nontrivial. This is conveniently seen as follows: let $\boldsymbol{l}=d_{1,0}\boldsymbol{i}$; that is,

$\boldsymbol{l}_a= d_{\mathfrak{h}} \boldsymbol{i}_a + \boldsymbol{i}_{d_{\mathfrak{g}a}}$

for any $a\in\mathfrak{g}$. Then the $(0,1)$-component of

$\array{ e^{-\boldsymbol{i}}*0 &= \sum_{n=0}^{+\infty} \frac{{[-\boldsymbol{i},-]}^n}{(n+1)!}(d_{\underline{Hom}}\boldsymbol{i})= \sum_{n=0}^{+\infty} \frac{{[-\boldsymbol{i},-]}^n}{(n+1)!} (\boldsymbol{l}+\boldsymbol{i}_{[\,,\,]_{\mathfrak{g}}}) }$

is just $\boldsymbol{l}$; the $(-1,2)$-component is

$\boldsymbol{i}_{[\,,\,]_{\mathfrak{g}}}-\frac{1}{2}[\boldsymbol{i},\boldsymbol{l}]_{\underline{Hom}}$

and, for $n\geq 3$ the $(1-n,n)$-component has two contributions, one of the form

$[\boldsymbol{i},[\boldsymbol{i},\cdots,[\boldsymbol{i}, \boldsymbol{l}]_{\underline{Hom}}\cdots]_{\underline{Hom}}]_{\underline{Hom}}$

and the other of the form

$[\boldsymbol{i},[\boldsymbol{i},\cdots,[\boldsymbol{i}, \boldsymbol{i}_{[\,,\,]_{\mathfrak{g}}}]_{\underline{Hom}}\cdots]_{\underline{Hom}}]_{\underline{Hom}}.$

From this we see that all the nonlinear components of $e^{-\boldsymbol{i}}*0$ vanish as soon as one imposes the two simple conditions

$\boldsymbol{i}_{[a,b]_{\mathfrak{g}}}= \frac{1}{2}\bigl([\boldsymbol{i}_a, \boldsymbol{l}_b]_{\mathfrak{h}}\pm [\boldsymbol{i}_b, \boldsymbol{l}_a]_{\mathfrak{h}}\bigr)\qquad and \qquad [\boldsymbol{i}_a, [\boldsymbol{i}_b, \boldsymbol{l}_c]_{\mathfrak{h}}]_{\mathfrak{h}}=0,\qquad \forall a, b,c \in \mathfrak{g}.$

These two condition may appear quite unnatural, but are actually implied by the following two, which are stronger, simpler, and familiar from differential geometry:

$\boldsymbol{i}_{[a,b]_{\mathfrak{g}}}= [\boldsymbol{i}_a, \boldsymbol{l}_b]_{\mathfrak{h}};\qquad [\boldsymbol{i}_a, \boldsymbol{i}_b]_{\mathfrak{h}}=0.$

Definition A linear map $\boldsymbol{i}\colon \mathfrak{g}\to \mathfrak{h}[-1]$ satisfying the two conditions above will be called a Cartan homotopy.

The name Cartan homotopy has an evident geometric origin: if $\mathcal{T}_X$ is the tangent sheaf of a smooth manifold $X$ and $\Omega^*_{X}$ is the sheaf of complexes of differential forms, then the contraction of differential forms with vector fields is a Cartan homotopy

$\boldsymbol{i}\colon \mathcal{T}_X\to \mathcal {E} n d^*(\Omega^*_X)[-1].$

In this case, $\boldsymbol{l}_a$ is the Lie derivative along the vector field $a$, and the conditions $\boldsymbol{i}_{[a,b]}= [\boldsymbol{i}_a, \boldsymbol{l}_b]$ and $[\boldsymbol{i}_a, \boldsymbol{i}_b]=0$, together with the defining equation $\boldsymbol{l}_a=[d_{\Omega^*_X},\boldsymbol{i}_a]$ and with the equations $\boldsymbol{l}_{[a,b]}=[\boldsymbol{l}_a,\boldsymbol{l}_b]$ and $[d_{\Omega^*_X},\boldsymbol{l}_a]=0$ expressing the fact that

$\boldsymbol{l}\colon \mathcal{T}_X\to \mathcal{E} n d^*(\Omega^*_X)$

is a dgla morphism, are nothing but the well-known Cartan identities involving contractions and Lie derivatives.

The above discussion can be summarized as follows.

Proposition Let $\mathfrak{g}$ and $\mathfrak{h}$ be two dglas. If $\boldsymbol{i}\colon \mathfrak{g}\to\mathfrak{h}[-1]$ is a Cartan homotopy, then $\boldsymbol{l}=d_{1,0}\boldsymbol{i}\colon \mathfrak {g}\to \mathfrak{h}$ is a dgla morphism gauge equivalent to the zero morphism via the gauge action of $e^{\boldsymbol{i}}$. In other wors, the pair $(\boldsymbol{l},e^{\boldsymbol{i}})$ defines a morphism form $\mathfrak{g}$ to the homotopy fiber of $\mathfrak{h}\stackrel{Id}{\to}\mathfrak{h}$, what is known as the inner derivations dgla $inn(\mathfrak{h})$.

### Homotopy fibers

By the above considerations we immediately see that, if the morphism $\boldsymbol{l}:\mathfrak{g}\to \mathfrak{h}$ factors through a dgla morphism $f:\mathfrak{n}\to \mathfrak{h}$, then we have a homotopy commutative diagram

$\array{ \mathfrak{g}&\to&\mathfrak{n}\\ \downarrow&\searrow^{\boldsymbol{l}}&\downarrow^f\\ \{0\}&\to&\mathfrak{h} }$

and so, by the universal property of the homotopy pullback, a dgla morphism

$\mathfrak{g}\to \mathfrak{n}\times_{\mathfrak{h}}^{f}\{0\}$

from $\mathfrak{g}$ to the homotopy fiber of $f:\mathfrak{n}\to \mathfrak{h}$.

Taking $Def_\bullet$‘s we therefore obtain a morphism of $\infty$-groupoids:

$Def_\bullet(\mathfrak{g})\to Def_\bullet(\mathfrak{n})\times_{Def_\bullet(\mathfrak{h})}^{f}*$

which, taking $\pi_0$‘s gives a natural morphism of sets

$\mathcal{P}\colon Def(\mathfrak{g})\to\pi_{0}(Def_\bullet(\mathfrak{n})\times_{Def_\bullet(\mathfrak{h})}^{f}*)$

from the classical deformation fuctor associated with the dgla $\mathfrak{g}$ to the set of connected components of the homotopy fiber of $Def_\bullet(\mathfrak{n})\to Def_\bullet(\mathfrak{h})$.

To investigate the geometry of $\pi_{0}(Def_\bullet(\mathfrak{n})\times_{Def_\bullet(\mathfrak{h})}^{f}*)$ it is sufficiet to recall how it fits into the homotopy exact sequence

$\pi_1Def_\bullet(\mathfrak{n}) \stackrel{f_*}{\to} \pi_1Def_\bullet(\mathfrak{h}) \to \pi_{0}(Def_\bullet(\mathfrak{n})\times_{Def_\bullet(\mathfrak{h})}^{f}*)\to *,$

inducing a canonical isomorphism

$\pi_{0}(Def_\bullet(\mathfrak{n})\times_{Def_\bullet(\mathfrak{h})}^{f}*)\cong \frac{ \pi_1 Def(\mathfrak{h})}{f_*\pi_1Def\bullet(\mathfrak{n})}.$

The group $\pi_1(Def_\bullet(\mathfrak{h})$ is the group of automorphisms of the distinguished object $0$ in the groupoid $\pi_{\leq 1}\Def_\bullet(\mathfrak{h}))$; as we have remarked, this groupoid is not equivalent to the Deligne groupoid of $\mathfrak{h}$, since the irrelevant stabilizer of a Maurer-Cartan element $x$ may be nontrivial. However, the group $\pi_1(Def_\bullet(\mathfrak{h})$ only sees the connected component of the distinguished element $0$, and on this connected component the irrelevant stabilizers are trivial as soon as the differential of the dgla $\mathfrak{h}$ vanishes on $\mathfrak{h}^{-1}$. This immediately follows from noticing that

$Stab_{irr}(0)= \{d h \mid h \in \mathfrak{g}^{-1}\}$

an by the fact that irrelevant stabilizers of gauge equivalent Maurer-Cartan elements are conjugate subgroups of $\exp(\mathfrak{h}^0)$. In particular, if $\mathfrak{h}$ is a graded Lie algebra (which we can consider as a dgla with trivial differential), then $\pi_1(\Def(\mathfrak{h});0)\simeq \exp(\mathfrak{h}^0)$. Then, if $j:\mathfrak {n}\hookrightarrow\mathfrak{h}$ is a subdgla of $\mathfrak{h}$, we also have $\pi_1\Def(\mathfrak{n})\cong \exp(\mathfrak{n}^0)$, and the map $Def(\mathfrak{g})\to\pi_{0}(Def_\bullet(\mathfrak{n})\times_{Def_\bullet(\mathfrak{h})}^{j}*)$ is just the natural map

$e^{\boldsymbol{i}}\colon \Def(\mathfrak{g})\to \exp(\mathfrak {h}^0)/\exp(\mathfrak{n}^0)$

which sends a Maurer-Cartan element $\xi\in \mathfrak{g}^1$ to $e^{\boldsymbol{i}_\xi}\mod \exp(\mathfrak {n}^0)$.

A particularly interesting case of this situation is when the morphism $f:\mathfrak{n}\to\mathfrak{h}$ is formal, i.e., when it is homotopy equivalent to an inclusion in cohomology $H^*(\mathfrak {n})\hookrightarrow H^*(\mathfrak{h})$. Indeed, in this case the morphism $f:Def_\bullet(\mathfrak{n})\to Def_\bullet(\mathfrak{h})$ will be homotopy equivalent to the morphism $j:Def_\bullet(H^*(\mathfrak{n}))\to Def_\bullet(H^*(\mathfrak{h}))$ and there will be an induced isomorphism between $\pi_1(Def(\mathfrak{ h})/f_*\pi_1\Def(\mathfrak{n})$ and the homogeneous space $exp(H^0(\mathfrak{h}))/exp(H^0(\mathfrak{n}))$. We can summarize the results described in this section as follows:

Proposition Let $\boldsymbol{i}\colon \mathfrak{g}\to\mathfrak{h}[-1]$ be a Cartan homotopy, let $\boldsymbol{l}\colon \mathfrak{g}\to\mathfrak{h}$ be the associated dgla morphism, and let $f:\mathfrak{n}\hookrightarrow\mathfrak{h}$ be a dgla morphism factorizing $\boldsymbol{l}$. Then, if the morphism $f$ is formal, we have a morphism

$\mathcal{P}\colon Def(\mathfrak{g}) \to \exp(H^0(\mathfrak {h}))/\exp(H^0(\mathfrak{n}))$

induced by the dgla map $\mathfrak{g} \to \mathfrak{n}\times_{\mathfrak{h}}^{f}\{0\}$.

It should be remarked that the morphism $\mathcal{P}$ is not canonical: it depends on the choice of a quasi isomorphism beween $f:\mathfrak {n}\to\mathfrak{h}$ and $j:H^*(\mathfrak{n})\hookrightarrow H^*(\mathfrak{n}))$.

The differential of $\mathcal{P}$ is easily computed: it is the linear map

$H^1(\mathfrak{g}) \stackrel{H^1((\boldsymbol{l}, e^{\boldsymbol{i}}))}{\longrightarrow} H^1(\mathfrak{n}\times_{\mathfrak{h}}^f{0}).$

Since the model category structure on dglas is the same as on differential complexes, we can compute the $H^1$ on the right hand side by computing the homotopy fiber of $f:\mathfrak{n}\to \mathfrak{h}$ in the category of complexes. In particular, if $j:\mathfrak{n}\hookrightarrow\mathfrak{h}$ is an inclusion, then the natural quasi-isomorphism of complexes

$\mathfrak{n}\times_{\mathfrak{h}}^j\{0\}\simeq (\mathfrak{h}/\mathfrak{n})[-1]$

tells us that the differential of $\mathcal{P}$ is just the map

$H^1({\boldsymbol{i}})\colon H^1(\mathfrak{g}) \to H^0(\mathfrak {h}/\mathfrak{n})$

induced by the morphism of complexes ${\boldsymbol{i}}\colon \mathfrak{g} \to (\mathfrak{h}/\mathfrak{n})[-1]$.

Also note that if the inclusion $j:\mathfrak{n}\hookrightarrow\mathfrak{h}$ is formal, then we have a (non canonical) isomorphism
$H^0(\mathfrak{h})/H^0(\mathfrak{n})$ and $H^0(\mathfrak{h}/\mathfrak{n})$, in agreement with the descripion of $\mathcal{P}$ given in the above proposition.

## The period map

Let again $X$ be a smooth projective manifold, and let $\mathcal{T}_X$ and $\Omega_X^*$ be the holomorphic tangent sheaf and the sheaf of holomorphic differential forms on $X$, respectively. The sheaf of complexes $(\Omega_X^*,d_{\Omega_X^*})$ is naturally filtered by setting $F^p\Omega_X^*=\oplus_{i\geq p}\Omega^i_X$. Finally, let $\mathcal{E} n d^*(\Omega_X^*)$ be the endomorphism sheaf of $\Omega_X^*$ and $\mathcal{E} n d^{\geq 0}(\Omega_X^*)$ be the subsheaf consisting of nonnegative degree elements. Note that $\mathcal{E} n d^{\geq 0}(\Omega_X^*)$ is a subdgla of $\mathcal{E} n d^{*}(\Omega_X^*)$, and can be seen as the subdgla of endomorphisms preserving the filtration on $\Omega_X^*$.

### Local description

Recall that the prototypical example of Cartan homotopy was the contraction of differential forms with vector fields ${\boldsymbol{i}}: \mathcal{T}_X \to \mathcal{E} n d^*(\Omega_X^*)[-1]$; the corresponding dgla morphism is $a\mapsto \boldsymbol{l}_a$, where $\boldsymbol{l}_a$ the Lie derivative along $a$. Explicitly, $\boldsymbol{l}_a= d_{\Omega_X^*}\circ {\boldsymbol{i}}_a+{\boldsymbol{i}}_{a}\circ d_{\Omega_X^*}$, and so $\boldsymbol{l}_a$ preserves the filtration and $\boldsymbol{l}:\mathcal{T}_X\to\mathcal{E} n d^*(\Omega_X^*)$ factors through the inclusion $j:\mathcal{E} n d^{\geq 0}(\Omega_X^*)\hookrightarrow \mathcal{E} n d^*(\Omega_X^*)$.

Therefore, we have a natural morphism of oo-groupoids

$Def_\bullet(\mathcal{T}_X) \stackrel{(\boldsymbol{l}, e^{\boldsymbol{i}})}{\longrightarrow} Def_\bullet(\mathcal{E} n d^{\geq 0}(\Omega_X^*))\times_{Def_\bullet(\mathcal{E} n d^{*}(\Omega_X^*))}*.$

The homotopy fiber on the right should be thought as a homotopy flag manifold. Let us briefly explain this: the functor $Def_\bullet(\mathcal{E} n d^{*}(\Omega_X^*))$ describes the infinitesimal deformations of the differential complex $\Omega_X^*$, whereas the functor $Def_\bullet(\mathcal{E} n d^{\geq0}(\Omega_X^*))$ describes the deformations of the filtered complex $(\Omega_X^*,F^\bullet\Omega_X^*)$, i.e., of the pair consisting of the complex $\Omega_X^*$ and of the filtration $F^\bullet\Omega_X^*$. Therefore, the homotopy fiber describes a deformation of the pair (complex, filtration) together with a trivialization of the deformation of the complex.

Summing up, the contraction of differential forms with vector fields induces a map of presheaves of oo-groupoids (on the category of open subsets of $X$)

$Def(\mathcal{T}_X) \to hoFlag(\Omega_X^*;F^\bullet\Omega_X^*),$

which we will call the local periods map of $X$.

### Recovering the classical picture

To recover from this the classical periods map, we just need to take global sections (clearly, since we are working in homotopy categories, these will be derived global sections). Globalizing the Cartan homotopy ${\boldsymbol{i}}: \mathcal{T}_X \to \mathcal{E} n d^*(\Omega_X^*)[-1]$, we get a Cartan homotopy

${\boldsymbol{i}}:\mathbf{R}\Gamma\mathcal{T}_X\to \mathcal{E} n d^*(\mathbf{R}\Gamma\Omega_X^*)[-1].$

The image of the corresponding dgla morphism $\boldsymbol{l}$ (the derived globalization of the Lie derivative) preserves the filtration $F^\bullet\mathbf{R}\Gamma\Omega_X^*$ induced by $F^\bullet\Omega_X^*$, so we have a natural map of $\infty$-groupoids

$Def_\bullet(\mathbf{R}\Gamma\T_X)\to {hoFlag}(\mathbf{R}\Gamma\Omega_X^*;F^\bullet\mathbf{R}\Gamma\Omega_X^*)$

and, at the $\pi_0$ level, a map of sets

$\mathcal{P}\colon Def(\mathbf{R}\Gamma\mathcal{T}_X)\to \pi_{0} hoFlag(\mathbf{R}\Gamma\Omega_X^*;F^\bullet\mathbf{R}\Gamma\Omega_X^*)$

On the left hand side we see $Def_X$, the Set-valued functor of (classical) infinitesimal deformations of $X$. On the right hand side we are considering the homotopy fiber of the inclusion $End^*(\mathbf{R}\Gamma\Omega_X^*;F^\bullet\mathbf{R}\Gamma\Omega_X^*)\hookrightarrow End^*(\mathbf{R}\Gamma\Omega_X^*)$ of the subdgla consisting of endomorhisms preserving the filtration. And we know from Hodge theory that this inclusion is formal, and that

$H^0(End^*(\mathbf{R}\Gamma\Omega_X^*; F^\bullet\mathbf{R}\Gamma\Omega_X^*))=\End^0(H^*(X;\mathbb{C}); F^\bullet H^*(X;\mathbb{C})),$

where $F^\bullet H^*(X;\mathbb{C})$ is the Hodge filtration on the cohomology of $X$. Moreover,

$H^0(End^*(\mathbf{R}\Gamma\Omega_X^*))=\End^0(H^*(X;\mathbb{C}))$

and so, by results described in the previous section,

$\pi_{0}{hoFlag}(\mathbf{R}\Gamma\Omega_X^*;F^\bullet\mathbf{R}\Gamma\Omega_X^*)\cong \frac{exp(End^0(H^*(X;\mathbb{C})))}{exp(End^0(H^*(X;\mathbb{C}); F^\bullet H^*(X;\mathbb{C})))}$

Thus we recover the classical periods map of $X$

$\mathcal{P}\colon\Def_X\to{Flag}(H^*(X;\mathbb{C}); F^\bullet H^*(X;\mathbb {C})).$

### The differential of the period map

Also, the general argument above tells us that the differential of $\mathcal {P}$ is the map induced in cohomology by the contraction of differential forms with vector fields,

$H^1(\boldsymbol{i})\colon H^1(X,\mathcal{T}_X)\to \int_p\Hom^0\left(F^p H^*(X;\mathbb{C});\frac{H^*(X;\mathbb{C})}{F^p H^*(X;\mathbb{C})}\right),$

a result originally proved by Griffiths [griffiths]. In the above formula, $\int_p$ denotes the end of the diagram

$\Hom^0\left(F^p H^*;\frac{H^*}{F^p H^*}\right) \rightarrow \Hom^0\left(F^p H^*;\frac{H^*}{F^{p+1} H^*}\right) \leftarrow \Hom^0\left(F^{p+1}H^*;\frac{H^*}{F^{p+1} H^*}\right)$

## References

Revised on March 11, 2011 at 12:18:21 by Urs Schreiber