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.
Here the basis is a (possibly singular) complex manifold, and is a proper holomorphic function. A morphism of deformations is a morphism of pullback diagrams.
We denote by the set of isomorphism classes of deformations of parametrized by the pointed base
If is a deformation of , and is a morphism of pointed complex, then the pullback diagram
exhibits as a deformation of over . Therefore is a functor
We are actually interested in the local behaviour of deformations in a neighborhood of the distinguished point . So is actually a functor
We pass to infinitesimal deformations by going from germs of pointed complex manifolds to formal pointed manifolds, i.e., to spectra of local -Artin algebras. Since is a contravariant functor, we finally have the functor of infinitesimal deformations of
where is the category of local Artin algebras with residue field . For an object in we will denote by its maxial ideal.
Newlander-Nierenberg approach
The Newlander-Nierenberg approach to consists in noticing that from a differential point of view, the family is trivial (Ehresmannβs theorem), so we can think of it as a fixedsmooth 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 of holomorphic functions inside the sheaf of smooth complex valued functions, what we are interested in is in deforming this subsheaf.
This is achieved as follows: for any in the vector space of sections of the sheaf of -forms with coefficients in the holomorphic tangent sheaf? of , consider the holomorphic Lie derivative
where
is the contraction of differential forms with the vector field (with coefficients in ) .
If , then is a degree derivation of and so we can use it to perturb . Since we want this perturbation to be infinitesimal, we actually fix an Artin algebra and pick in . Then
is our perturbed Dolbeault operator. By construction it is a derivation of , but it is in general not a differential. We have to impose the condition , i.e.,
where the bracket is the graded commutator in the graded associative algebra . 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
Also the graded vector space of differential forms of type with coefficients in has a natural dgla structure, and the holomorphic Lie derivative
is an morphism of dglas. Therefore the above condition on becomes
where is the differential in the dgla . Moreover, since in injective, this is equivalent to
i.e., we can naturally associate a deformation of the complex structure of to a Maurer-Cartan element in the dgla . To obtain the set we still have to quotient out isomorphic copies of the same complex structure. Two Maurer-Cartan elements and give isomorphic complex structures if there exist a diffeomorphism of of the form with in such that the following diagram commutes
In other words
Kodaira-Spencer approach
The Newlander-Nierenberg approach to deformations of the sheaf we described above was based on the remark that is the kernel of a differenial operator globally defined on . The Kodaira-Spencer approach is instead a local approach. Fix an open cover of , so that for any in . Then the global sheaf is obatined by glueing toghether the local sheaves via the trivial glueing maps
The Kodaira-Spencerβs idea is then that to deform we have to deform the local sheaves and the glueing maps between the restrictions and . So, one takes an open cover of such that the βpiecesβ have no nontrivial deformations (this happens, e.g., if is isomorphic as a ringed space to a polidisk in with its sheaf of holomorphic functions), then the datum of a deformation of is entirely encoded in the datum of a deformation of the glueing maps. Since we are working infinitesimally, an isomorphism
will be of the form , with an holomorphic vector field on . In order to define a global sheaf, the glueings have to satisfy the cocycle condition
Two cocycles and describe isomorphic deformations if and only if there are isomorphisms such that the diagram
commutes. And again, since we are working on an infinitesimal base, will be of the form with an holomorphic vector field on . Therefore we see that deformations of are given by cohomology classes in nonabelian cohomology
where is the holomorphic tangent sheaf of . This can be seen as the set of connetced components of the hom-space , by looking both at the manifold and to as simplicial presheaves over a suitable site of local models for complex geometry. Here is the presheaf represented by , and is the presheaf
Finally, is the presheaf given by the delooping of
From dglas to -groupoids
Given an dgla , we can functorially associate with any Artin algebra a nilpotent dgla by . With this in mind we will look at all dglas we will meet in what follows as functors .
One thinks element in this set as flat -connections: indeed
The subspace of is a Lie algebra; the group
acts on as a group of gauge transformations on the set of -connections (by conjugation), and this action preserves the subset of flat connections. Hence we have a gauge action of on :
We denote by the decategorification of , 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
Gauge vs. homotopy equivalent Maurer-Cartan elements
Consider now two gauge equivalent Maurer-Cartan elements and for a dgla . By definition, this means that there exists an element in sucht that . We can make this way of going from to a βcontinuous pathβ simply by going from the dgla to the dgla , where is the commutative differential graded algebra of differential forms of the geometric realization of the 1-simplex. Indeed, is a degree zero element in , and is an element in , and so is a Maurer-Cartan element in , with and .
Definition Two Maurer-Cartan elements ad are homotopy equivalent if there exists a maurer-Cartan element in such that and .
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
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 is of the form , for some in and some in . So if is a homotopy equivalence between and , then is a gauge equivalence between them, i.e.
The -groupoid of a dgla
With any nilpotent dgla is therefore naturally associated the simplicial set
where is the simplicial differential graded commutative associative algebra of polynomial (or piecewise smooth, smooth with sitting instnts, etc.) differential forms on the standard -simplexes, for . The crucial property of the simplicial set 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 .
If the dgla is concentrated in nonnegative degrees, then parallel transport induces a natural morphism of simplicial sets
Indeed, a Maurer-Cartan element in is of the form with a -valued 0-form on , and a -valued 1-form on . The 2-form component of the Maurer-Cartan equation for is , where is the differential on differential forms on the simplex. So is a flat -connection on the simplex, and parallel transport of from the vertex 0 of the simplex to the other vertices will define a map from to the -simplices of the (nerve of) the Deligne groupoid. Here we are using the fact that the connection is flat and the -simplex is simply connected.
It is a remarkable result by Hinich that the morphism is an acyclic fibration. In particular and are (weakly) homotopy equivalent. An immediate corollary of this is that , but note that by the
mentioned above, we have
for any dgla , i.e., even for dglas non concentrated in nonegative degrees. On the other hand, when is not concentrated in nonegative degrees, we do not have, in general, a morphism induced by parallel transport. Let us see why. If a degree component is present, then a Maurer-Cartan element in is of the form with a -valued 0-form on , a -valued 1-form on , and a -valued 2-form on . The 2-form component of the Maurer-Cartan equation for is , and we see that is not flat anymore, and that the obstruction to flatness is given by the term . 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 is defined as
(note that is actually a subgroup of the stabilizer ). So we see that the groupoid fails to be equivalent to precisely because of nontrivial irrelevant stabilizers. And so we also see that we do have a morphism of simlicial sets as soon as irrelevant stabilizers are trivial, and in this case we have
A particular case of this is the case in which is concentrated in nonnegative degrees we mentioned above.
A very good reason for working with -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) βniceβ topological spaces.
A sketchy proof of the above equivalence can be found in (lurie-moduli); see also (Pridham).
We will refer to a functor as to a formal -groupoid. In what follows, we will just say β-groupoidβ to mean βformal -groupoidβ.
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 . Consider the holomorphic tangent sheaf of ; it is a sheaf of Lie algebras. Then for any open subset of we can consider the oo-groupoid . This defines a presheaf of oo-groupoids on the site of open subsets of : the oo-presheaf of local deformations of . 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 of . Since is an equivalence of -categories, we have
On the left-hand side, since is a sheaf of Lie algebras, and so is concentrated in nonnegative degree as a dgla, we have natural homotopy equivalences , and so
and we find the Kodaira-Spencer description of the deformations of . On the right-hand side, a way of computing the homotopy limit holim , i.e., to compute the derived global sections of the sheaf , 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 the Dolbeault resolution we find the Newlander-Nierenberg description of .
Gauge equivalent morphisms of dgla
-morphisms of dglas
Since we are now looking at dglas as to an (oo,1)-category, given two dglas and , their hom-space is an -groupoid. By the equivalence between dglas and (formal) -groupoids stated at the end of the previous section, there must be a dgla such that is equivalent to . And actually the dgla 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
endowed with the Lie bracket
defined by
and with the differentials
and
given by
and
These operations are best seen pictorially:
Maurer-Cartan elements in are -morphisms between and . Explicitely, such an -morphism is a collection of degree maps
such that
Note in particular that a dgla morphism is, in a natural way, an -morphism between and , of a very special kind: all but the first one of its Taylor coefficients vanish.
Passing to βs, the equivalence implies the following:
Proposition Let be two -morphisms of dglas. Then and are gauge equivalent in if and only if and represent the same morphism in the homotopy category of dglas.
Cartan homotopies
Let now and be dglas and be a morphism of graded vector spaces. Then , and so also , is an element of , and so a degree zero element in the dgla . The gauge transformation will map the dgla morphism to an -morphism between and .
This -morphism will in general fail to be a dgla morphism since its nonlinear components will be nontrivial. This is conveniently seen as follows: let ; that is,
for any . Then the -component of
is just ; the -component is
and, for the -component has two contributions, one of the form
and the other of the form
From this we see that all the nonlinear components of vanish as soon as one imposes the two simple conditions
These two condition may appear quite unnatural, but are actually implied by the following two, which are stronger, simpler, and familiar from differential geometry:
Definition A linear map satisfying the two conditions above will be called a Cartan homotopy.
The name Cartan homotopy has an evident geometric origin: if is the tangent sheaf of a smooth manifold and is the sheaf of complexes of differential forms, then the contraction of differential forms with vector fields is a Cartan homotopy
In this case, is the Lie derivative along the vector field , and the conditions and , together with the defining equation and with the equations and expressing the fact that
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 and be two dglas. If is a Cartan homotopy, then is a dgla morphism gauge equivalent to the zero morphism via the gauge action of . In other wors, the pair defines a morphism form to the homotopy fiber of , what is known as the inner derivations dgla .
Homotopy fibers
By the above considerations we immediately see that, if the morphism factors through a dgla morphism , then we have a homotopy commutative diagram
and so, by the universal property of the homotopy pullback, a dgla morphism
Taking βs we therefore obtain a morphism of -groupoids:
which, taking βs gives a natural morphism of sets
from the classical deformation fuctor associated with the dgla to the set of connected components of the homotopy fiber of .
To investigate the geometry of it is sufficiet to recall how it fits into the homotopy exact sequence
inducing a canonical isomorphism
The group is the group of automorphisms of the distinguished object in the groupoid ; as we have remarked, this groupoid is not equivalent to the Deligne groupoid of , since the irrelevant stabilizer of a Maurer-Cartan element may be nontrivial. However, the group only sees the connected component of the distinguished element , and on this connected component the irrelevant stabilizers are trivial as soon as the differential of the dgla vanishes on . This immediately follows from noticing that
an by the fact that irrelevant stabilizers of gauge equivalent Maurer-Cartan elements are conjugate subgroups of . In particular, if is a graded Lie algebra (which we can consider as a dgla with trivial differential), then . Then, if is a subdgla of , we also have , and the map is just the natural map
which sends a Maurer-Cartan element to .
A particularly interesting case of this situation is when the morphism is formal, i.e., when it is homotopy equivalent to an inclusion in cohomology . Indeed, in this case the morphism will be homotopy equivalent to the morphism and there will be an induced isomorphism between and the homogeneous space . We can summarize the results described in this section as follows:
Proposition Let be a Cartan homotopy, let be the associated dgla morphism, and let be a dgla morphism factorizing . Then, if the morphism is formal, we have a morphism
induced by the dgla map .
It should be remarked that the morphism is not canonical: it depends on the choice of a quasi isomorphism beween and .
The differential of is easily computed: it is the linear map
Since the model category structure on dglas is the same as on differential complexes, we can compute the on the right hand side by computing the homotopy fiber of in the category of complexes. In particular, if is an inclusion, then the natural quasi-isomorphism of complexes
tells us that the differential of is just the map
induced by the morphism of complexes .
Also note that if the inclusion is formal, then we have a (non canonical) isomorphism and , in agreement with the descripion of given in the above proposition.
The period map
Let again be a smooth projective manifold, and let and be the holomorphic tangent sheaf and the sheaf of holomorphic differential forms on , respectively. The sheaf of complexes is naturally filtered by setting . Finally, let be the endomorphism sheaf of and be the subsheaf consisting of nonnegative degree elements. Note that is a subdgla of , and can be seen as the subdgla of endomorphisms preserving the filtration on .
Local description
Recall that the prototypical example of Cartan homotopy was the contraction of differential forms with vector fields ; the corresponding dgla morphism is , where the Lie derivative along . Explicitly, , and so preserves the filtration and factors through the inclusion .
Therefore, we have a natural morphism of oo-groupoids
The homotopy fiber on the right should be thought as a homotopy flag manifold. Let us briefly explain this: the functor describes the infinitesimal deformations of the differential complex , whereas the functor describes the deformations of the filtered complex , i.e., of the pair consisting of the complex and of the filtration . 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 )
which we will call the local periods map of .
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 , we get a Cartan homotopy
The image of the corresponding dgla morphism (the derived globalization of the Lie derivative) preserves the filtration induced by , so we have a natural map of -groupoids
and, at the level, a map of sets
On the left hand side we see , the Set-valued functor of (classical) infinitesimal deformations of . On the right hand side we are considering the homotopy fiber of the inclusion of the subdgla consisting of endomorhisms preserving the filtration. And we know from Hodge theory that this inclusion is formal, and that
where is the Hodge filtration on the cohomology of . Moreover,
and so, by results described in the previous section,
Thus we recover the classical periods map of
The differential of the period map
Also, the general argument above tells us that the differential of is the map induced in cohomology by the contraction of differential forms with vector fields,
a result originally proved by Griffiths [griffiths]. In the above formula, denotes the end of the diagram