# A superficial working guide to deformations and moduli

^{1}

^{1}1I owe to David Buchsbaum the joke that an expert on algebraic surfaces is a ‘superficial’ mathematician.

Dedicated to David Mumford with admiration.

###### Contents

## Introduction

There are several ways to look at moduli theory, indeed the same name can at a first glance disguise completely different approaches to mathematical thinking; yet there is a substantial unity since, although often with different languages and purposes, the problems treated are substantially the same.

The most classical approach and motivation is to consider moduli theory as the fine part of classification theory: the big quest is not just to prove that certain moduli spaces exist, but to use the study of their structure in order to obtain geometrical informations about the varieties one wants to classify; and using each time the most convenient incarnation of ‘moduli’.

For instance, as a slogan, we might think of moduli theory and deformation theory as analogues of the global study of an algebraic variety versus a local study of its singularities, done using power series methods. On the other hand, the shape of an algebraic variety is easily recognized when it has singularities!

In this article most of our attention will be cast on the case of complex algebraic surfaces, which is already sufficiently intricate to defy many attempts of investigation. But we shall try, as much as possible, to treat the higher dimensional and more general cases as well. We shall also stick to the world of complex manifolds and complex projective varieties, which allows us to find so many beautiful connections to related fields of mathematics, such as topology, differential geometry and symplectic geometry.

David Mumford clarified the concept of biregular moduli through a functorial definition, which is extremely useful when we want a precise answer to questions concerning a certain class of algebraic varieties.

The underlying elementary concepts are the concept of normal forms, and of quotients of parameter spaces by a suitable equivalence relation, often given by the action of an appropriate group. To give an idea through an elementary geometric problem: how many are the projective equivalence classes of smooth plane curves of degree 4 admitting 4 distinct collinear hyperflexes?

A birational approach to moduli existed before, since, by the work of Cayley, Bertini, Chow and van der Waerden, varieties in a fixed projective space, having a fixed dimension and a fixed degree are parametrized by the so called Chow variety , over which the projective group acts. And, if is an irreducible component of , the transcendence degree of the field of invariant rational functions was classically called the number of polarized moduli for the class of varieties parametrized by . This topic: ‘embedded varieties’ is treated in the article by Joe Harris in this Handbook.

A typical example leading to the concept of stability was: take the fourfold symmetric product of , parametrizing 4-tuples of points in the plane. Then has dimension 8 and the field of invariants has transcendence degree 0. This is not a surprise, since 4 points in linear general position are a projective basis, hence they are projectively equivalent; but, if one takes 4 point to lie on a line, then there is a modulus, namely, the cross ratio. This example, plus the other basic example given by the theory of Jordan normal forms of square matrices (explained in [Mum-Suom] in detail) guide our understanding of the basic problem of Geometric Invariant Theory: in which sense may we consider the quotient of a variety by the action of an algebraic group. In my opinion geometric invariant theory, in spite of its beauty and its conceptual simplicity, but in view of its difficulty, is a foundational but not a fundamental tool in classification theory. Indeed one of the most difficult results, due to Gieseker, is the asymptotic stability of pluricanonical images of surfaces of general type; it has as an important corollary the existence of a moduli space for the canonical models of surfaces of general type, but the methods of proof do not shed light on the classification of such surfaces (indeed boundedness for the families of surfaces with given invariants had followed earlier by the results of Moishezon, Kodaira and Bombieri).

We use in our title the name ‘working’: this may mean many things, but in particular here our goal is to show how to use the methods of deformation theory in order to classify surfaces with given invariants.

The order in our exposition is more guided by historical development and by our education than by a stringent logical nesting.

The first guiding concepts are the concepts of Teichmüller space and moduli space associated to an oriented compact differentiable manifold of even dimension. These however are only defined as topological spaces, and one needs the Kodaira-Spencer-Kuranishi theory in order to try to give the structure of a complex space to them.

A first question which we investigate, and about which we give some new results (proposition 15 and theorem LABEL:kur=teich-surf), is: when is Teichmüller space locally homeomorphic to Kuranishi space?

This equality has been often taken for granted, of course under the assumption of the validity of the so called Wavrik condition (see theorem 5), which requires the dimension of the space of holomorphic vector fields to be locally constant under deformation .

An important role plays the example of Atiyah about surfaces acquiring a node: we interpret it here as showing that Teichmüller space is non separated (theorem LABEL:nonseparated). In section 4 we see that it also underlies some recent pathological behaviour of automorphisms of surfaces, recently discovered together with Ingrid Bauer: even if deformations of canonical and minimal models are essentially the same, up to finite base change, the same does not occur for deformations of automorphisms (theorems LABEL:main1 and LABEL:path). The connected components for deformation of automorphisms of canonical models are bigger than the connected components for deformation of automorphisms of minimal models , the latter yielding locally closed sets of the moduli spaces which are locally closed but not closed.

To describe these results we explain first the Gieseker coarse moduli space for canonical models of surfaces of general type, which has the same underlying reduced space as the coarse moduli stack for minimal models of surfaces of general type. We do not essentially talk about stacks (for which an elementary presentation can be found in [fantechi]), but we clarify how moduli spaces are obtained by glueing together Kuranishi spaces, and we show the fundamental difference for the étale equivalence relation in the two respective cases of canonical and minimal models: we exhibit examples showing that the relation is not finite (proper) in the case of minimal models (a fact which underlies the definition of Artin stacks given in [artinstacks]).

We cannot completely settle here the question whether Teichmüller space is locally homeomorphic to Kuranishi space for all surfaces of general type, as this question is related to a fundamental question about the non existence of complex automorphisms which are isotopic to the identity, but different from the identity (see however the already mentioned theorem LABEL:kur=teich-surf).

Chapter five is dedicated to the connected components of moduli spaces, and to the action of the absolute Galois group on the set of irreducible components of the moduli space, and surveys many recent results.

We end by discussing concrete issues showing how one can determine a connected component of the moduli space by resorting to topological or differential arguments; we overview several results, without proofs but citing the references, and finally we prove a new result, theorem LABEL:doublecover, obtained in collaboration with Ingrid Bauer.

There would have been many other interesting topics to treat, but these should probably better belong to a ‘part 2’ of the working guide.

## 1. Analytic moduli spaces and local moduli spaces: Teichmüller and Kuranishi space

### 1.1. Teichmüller space

Consider, throughout this subsection, an oriented real differentiable manifold of real dimension (without loss of generality we may a posteriori assume and all the rest to be or even , i.e., real-analytic).

At a later point it will be convenient to assume that is compact.

Ehresmann ([ACS]) defined an almost complex structure on as the structure of a complex vector bundle on the real tangent bundle : namely, the action of on is provided by an endomorphism

It is completely equivalent to give the decomposition of the complexified tangent bundle as the direct sum of the , respectively eigenbundles:

In view of the second condition, it suffices to give the subbundle , or, equivalently, a section of the associated Grassmannian bundle whose fibre at a point is the variety of -dimensional vector subspaces of the complex tangent space at , (note that the section must take values in the open set of subspaces such that and generate).

The space of almost complex structures, once (hence all associated bundles) is endowed with a Riemannian metric, has a countable number of seminorms (locally, the sup norm on a compact of all the derivatives of the endomorphism ), and is therefore a Fréchet space. One may for instance assume that is embedded in some .

Assuming that is compact, one can also consider the Sobolev k-norms (i.e., for derivatives up order k).

A closed subspace of consists of the set of complex structures: these are the almost complex structures for which there are at each point local holomorphic coordinates, i.e., functions whose differentials span the dual of for each point in a neighbourhood of .

In general, the splitting

yields a decomposition of exterior differentiation of functions as , and a function is said to be holomorphic if its differential is complex linear, i.e., .

This decomposition extends to higher degree differential forms.

The theorem of Newlander-Nirenberg ([NN]), first proven by Eckmann and Frölicher in the real analytic case ([E-F], see also [montecatini] for a simple proof) characterizes the complex structures through an explicit equation:

###### Theorem 1.

(Newlander-Nirenberg) An almost complex structure yields the structure of a complex manifold if and only if it is integrable, which means

Obviously the group of oriented diffeomorphisms of acts on the space of complex structures, hence one can define in few words some basic concepts.

###### Definition 2.

Let be the group of orientation preserving diffeomorphisms of , and let the space of complex structures on . Let be the connected component of the identity, the so called subgroup of diffeomorphisms which are isotopic to the identity.

Then Dehn ([dehn]) defined the mapping class group of as

while the Teichmüller space of , respectively the moduli space of complex structures on are defined as

These definitions are very clear, however they only show that these objects are topological spaces, and that

The simplest examples here are two: complex tori and compact complex curves.

The example of complex tori sheds light on the important question concerning the determination of the connected components of , which are called the deformation classes in the large of the complex structures on (cf. [cat02], [cat04]).

Complex tori are parametrized by an open set of the complex Grassmann Manifold , image of the open set of matrices

This parametrization is very explicit: if we consider a fixed lattice , to each matrix as above we associate the subspace

so that and

Finally, to we associate the torus , being the projection onto the first addendum.

Not only we obtain in this way a connected open set inducing all the small deformations (cf. [k-m71]), but indeed, as it was shown in [cat02] (cf. also [cat04]) is a connected component of Teichmüller space (as the letter suggests).

It was observed however by Kodaira and Spencer already in their first article ([k-s58], and volume II of Kodaira’s collected works) that for the mapping class group does not act properly discontinuously on . More precisely, they show that for every non empty open set there is a point such that the orbit intersects in an infinite set.

This shows that the quotient is not Hausdorff at each point, probably it is not even a non separated complex space.

Hence the moral is that for compact complex manifolds it is better to consider, rather than the Moduli space, the Teichmüller space.

Moreover, after some initial constructions by Blanchard and Calabi (cf. [bla53], [bla54], , [bla56], [cal58]) of non Kähler complex structures on manifolds diffeomorphic to a product , where is a compact complex curve and is a 2-dimensional complex torus, Sommese generalized their constructions, obtaining ([somm75]) that the space of complex structures on a six dimensional real torus is not connected.

These examples were then generalized in [cat02] [cat04] under the name of Blanchard-Calabi manifolds showing (corollary 7.8 of [cat04]) that also the space of complex structures on the product of a curve of genus with a four dimensional real torus is not connected, and that there is no upper bound for the dimension of Teichmüller space (even when is fixed).

The case of compact complex curves is instead the one which was originally considered by Teichmüller.

In this case, if the genus is at least , the Teichmüller space is a bounded domain, diffeomorphic to a ball, contained in the vector space of quadratic differentials on a fixed such curve .

In fact, for each other complex structure on the oriented 2-manifold underlying we obtain a complex curve , and there is a unique extremal quasi-conformal map , i.e., a map such that the Beltrami distortion has minimal norm (see for instance [hubbard] or [ar-cor]).

The fact that the Teichmüller space is homeomorphic to a ball (see [Tro] for a simple proof) is responsible for the fact that the moduli space of curves is close to be a classifying space for the mapping class group (see [mumshaf] and the articles by Edidin and Wahl in this Handbook).

### 1.2. Kuranishi space

Interpreting the Beltrami distortion as a closed - form with values in the dual of the cotangent bundle , we obtain a particular case of the Kodaira-Spencer-Kuranishi theory of local deformations.

In fact, by Dolbeault ’s theorem, such a closed form determines a cohomology class in , where is the sheaf of holomorphic sections of the holomorphic tangent bundle : these cohomology classes are interpreted, in the Kodaira-Spencer-Kuranishi theory, as infinitesimal deformations (or derivatives of a family of deformations) of a complex structure: let us review briefly how.

Local deformation theory addresses precisely the study of the small deformations of a complex manifold .

We shall use here unambiguously the double notation to refer to the splitting determined by the complex structure .

is a point in , and a neighbourhood in the space of almost complex structures corresponds to a distribution of subspaces which are globally defined as graphs of an endomorphism

called a small variation of complex structure, since one then defines

In terms of the new operator, the new one is simply obtained by considering

and the integrability condition is given by the Maurer-Cartan equation

where denotes the Schouten bracket, which is the composition of exterior product of forms followed by Lie bracket of vector fields, and which is graded commutative.

Observe for later use that the form is closed, if , since then

Recall also the theorem of Dolbeault: if is the sheaf of holomorphic sections of , then is isomorphic to the quotient space of the space of closed -forms with values in modulo the space of -exact -forms with values in .

Our is a map of degree 2 between two infinite dimensional spaces, the space of -forms with values in the bundle , and the space of -forms with values in .

Observe that, since our original complex structure corresponds to , the derivative of the above equation at is simply

hence the tangent space to the space of complex structures consists of the space of -closed forms of type and with values in the bundle . Moreover the derivative of surjects onto the space of -exact -forms with values in .

We are now going to show why we can restrict our consideration only to the class of such forms in the Dolbeault cohomology group

This is done by answering the question: how does the group of diffeomorphisms act on an almost complex structure ?

This is in general difficult to specify, but we can consider the infinitesimal action of a 1-parameter group of diffeomorphisms

corresponding to a differentiable vector field with values in ; from now on, we shall assume that is compact, hence the diffeomorphism is defined .

We refer to [kur3] and [huy], lemma 6.1.4 , page 260, for the following calculation of the Lie derivative:

###### Lemma 3.

Given a 1-parameter group of diffeomorphisms

, corresponds to the small variation .

The lemma says, roughly speaking, that locally at each point the orbit for the group of diffeomorphisms in contains a submanifold, having as tangent space the forms in the same Dolbeault cohomology class of , which has finite codimension inside another submanifold with tangent space the space of -closed forms . Hence the tangent space to the orbit space is the space of such Dolbeault cohomology classes.

Even if we ‘heuristically’ assume it looks like we are still left with another equation with values in an infinite dimensional space. However, the derivative surjects onto the space of exact forms, while the restriction of to the subspace of -closed forms ( takes values in the space of -closed forms: this is the moral reason why indeed one can reduce the above equation , associated to a map between infinite dimensional spaces, to an equation for a map , called the Kuranishi map.

This is done explicitly via a miracolous equation (see [k-m71], [Kodbook],[kur4] and [montecatini] for details) set up by Kuranishi in order to reduce the problem to a finite dimensional one (here Kuranishi, see [kur3], uses the Sobolev r- norm in order to be able to use the implicit function theorem for Banach spaces).

Here is how the Kuranishi equation is set up.

Let be a basis for the space of harmonic (0,1)-forms with values in , and set , so that establishes an isomorphism .

Then the Kuranishi slice (see [palais] for a general theory of slices) is obtained by associating to the unique power series solution of the following equation:

satisfying moreover higher order terms ( denotes here the Green operator).

The upshot is that for these forms the integrability equation simplifies drastically; the result is summarized in the following definition.

###### Definition 4.

The Kuranishi space is defined as the germ of complex subspace of defined by , where is the harmonic projector onto the space of harmonic forms of type and with values in .

Kuranishi space parametrizes exactly the set of small variations of complex structure which are integrable. Hence over we have a family of complex structures which deform the complex structure of .

It follows from the above arguments that the Kuranishi space surjects onto the germ of the Teichmüller space at the point corresponding to the given complex structure .

It fails badly to be a homeomorphism, and my favourite example for this is (see [catrav]) the one of the Segre ruled surfaces , obtained as the blow up at the origin of the projective cone over a rational normal curve of degree , and described by Hirzebruch biregularly as

Kuranishi space is here the vector space

parametrizing projectivizations , where the rank 2 bundle occurs as an extension

By Grothendieck’s theorem, however, is a direct sum of two line bundles, hence we get as a possible surface only a surface , for each . Indeed Teichmüller space, in a neighbourhood of the point corresponding to consists just of a finite number of points corresponding to each , and where is in the closure of if and only if .

The reason for this phenomenon is the following. Recall that the form can be infinitesimally changed by adding ; now, for , nothing is changed if . i.e., if is a holomorphic vector field. But the exponentials of these vector fields, which are holomorphic on , but not necessarily for , act transitively on each stratum of the stratification of given by isomorphism type (each stratum is thus the set of surfaces isomorphic to ).

In other words, the jumping of the dimension of for is responsible for the phenomenon.

Indeed Kuranishi, improving on a result of Wavrik ([Wav]) obtained in [kur3] the following result.

###### Theorem 5.

( Kuranishi’s third theorem) Assume that the dimension of for is a constant function in a neighbourhood of .

Then there is and a neighbourhood of the identity map in the group , with respect to the k-th Sobolev norm, and a neighbourhood of in such that, for each , and , cannot yield a holomorphic map between and .

Kuranishi’s theorem ([kur1],[kur2]) shows that Teichmüller space can be viewed as being locally dominated by a complex space of locally finite dimension (its dimension, as we already observed, may however be unbounded, cf. cor. 7.7 of [cat04]).

A first consequence is that Teichmüller space is locally connected by holomorphic arcs, hence the determination of the connected components of , respectively of , can be done using the original definition of deformation equivalence, given by Kodaira and Spencer in [k-s58].

###### Corollary 6.

Let , be two different complex structures on .

Define deformation equivalence as the equivalence relation generated by direct deformation equivalence, where , are said to be direct disk deformation equivalent if and only if there is a proper holomorphic submersion with connected fibres , where is a complex manifold, is the unit disk, and moreover there are two fibres of biholomorphic to , respectively .

Then two complex structures on yield points in the same connected component of if and only if they are in the same deformation equivalence class.

In the next subsections we shall illustrate the meaning of the condition that the vector spaces have locally constant dimension, in terms of deformation theory. Moreover, we shall give criteria implying that Kuranishi and Teichmüller space do locally coincide.

### 1.3. Deformation theory and how it is used

One can define deformations not only for complex manifolds, but also for complex spaces. The technical assumption of flatness replaces then the condition that be a submersion.

###### Definition 7.

1) A deformation of a compact complex space is a pair consisting of

1.1) a flat proper morphism between connected complex spaces (i.e., is a flat ring extension for each with )

1.2) an isomorphism of with a fibre of .

2.1) A small deformation is the germ of a deformation.

2.2) Given a deformation and a morphism with , the pull-back is the fibre product endowed with the projection onto the second factor (then ).

3.1) A small deformation is said to be versal or complete if every other small deformation is obtained from it via pull back; it is said to be semi-universal if the differential of at is uniquely determined, and universal if the morphism is uniquely determined.

4) Two compact complex manifolds are said to be direct deformation equivalent if there are a deformation of with irreducible and where all the fibres are smooth, and an isomorphism of with a fibre of .

Let’s however come back to the case of complex manifolds, observing that in a small deformation of a compact complex manifold one can shrink the base and assume that all the fibres are smooth.

We can now state the results of Kuranishi and Wavrik (([kur1], [kur2], [Wav]) in the language of deformation theory.

###### Theorem 8.

(Kuranishi). Let be a compact complex manifold: then

I) the Kuranishi family of is semiuniversal.

II) is unique up to (non canonical) isomorphism, and is a germ of analytic subspace of the vector space , inverse image of the origin under a local holomorphic map (called Kuranishi map and denoted by ) whose differential vanishes at the origin.

Moreover the quadratic term in the Taylor development of the Kuranishi map is given by the bilinear map , called Schouten bracket, which is the composition of cup product followed by Lie bracket of vector fields.

III) The Kuranishi family is a versal deformation of for .

IV) The Kuranishi family is universal if

V) (Wavrik) The Kuranishi family is universal if is reduced and is constant for in a suitable neighbourhood of .

In fact Wavrik in his article ([Wav]) gives a more general result than V); as pointed out by a referee, the same criterion has also been proven by Schlessinger (prop. 3.10 of [FAR]).

Wavrik says that the Kuranishi space is a local moduli space under the assumption that is locally constant. This terminology can however be confusing, as we shall show, since in no way the Kuranishi space is like the moduli space locally, even if one divides out by the action of the group of biholomorphisms of .

The first most concrete question is how one can calculate the Kuranishi space and the Kuranishi family. In this regard, the first resource is to try to use the implicit functions theorem.

For this purpose one needs to calculate the Kodaira Spencer map of a family of complex manifolds having a smooth base . This is defined as follows: consider the cotangent bundle sequence of the fibration

and the direct image sequence of the dual sequence of bundles,

Evaluation at the point yields a map of the tangent space to at into , which is the derivative of the variation of complex structure (see [k-m71] for a more concrete description, but beware that the definition given above is the most effective for calculations).

###### Corollary 9.

Let be a compact complex manifold and assume that we have a family with smooth base , such that , and such that the Kodaira Spencer map surjects onto .

Then the Kuranishi space is smooth and there is a submanifold which maps isomorphically to ; hence the Kuranishi family is the restriction of to .

The key point is that, by versality of the Kuranishi family, there is a morphism inducing as a pull back, and is the derivative of .

This approach clearly works only if is unobstructed, which simply means that is smooth. In general it is difficult to describe the Kuranishi map, and even calculating the quadratic term is nontrivial (see [quintics] for an interesting example).

In general, even if it is difficult to calculate the Kuranishi map, Kuranishi theory gives a lower bound for the ‘number of moduli’ of , since it shows that has dimension . In the case of curves , hence curves are unobstructed; in the case of a surface

The above is the Enriques’ inequality ([enr], observe that Max Noether postulated equality), proved by Kuranishi in all cases and also for non algebraic surfaces.

There have been recently two examples where resorting to the Kuranishi theorem in the obstructed case has been useful.

The first one appeared in a preprint by Clemens ([clemens1]), who then published the proof in [clemens]; it shows that if a manifold is Kählerian, then there are fewer obstructions than foreseen, since a small deformation of a Kähler manifold is again Kähler, hence the Hodge decomposition still holds for .

Another independent proof was given by Manetti in [Manobs].

###### Theorem 10.

(Clemens-Manetti) Let be a compact complex Kähler manifold.

Then there exists an analytic automorphism of with linear part equal to the identity, such that the Kuranishi map takes indeed values in the intersection of the subspaces

(the linear map is induced by cohomology cup product and tensor contraction).

Clemens’ proof uses directly the Kuranishi equation, and a similar method was used by Sönke Rollenske in [rol1], [rol2] in order to consider the deformation theory of complex manifolds yielding left invariant complex structures on nilmanifolds. Rollenske proved, among other results, the following

###### Theorem 11.

(Rollenske) Let be a compact complex manifold corresponding to a left invariant complex structure on a real nilmanifold. Assume that the following condition is verified:

(*) the inclusion of the complex of left invariant forms of pure antiholomorphic type in the Dolbeault complex

yields an isomorphism of cohomology groups.

Then every small deformation of the complex structure of consists of left invariant complex structures.

The main idea, in spite of the technical complications, is to look at Kuranishi’s equation, and to see that everything is then left invariant.

Rollenske went over in [rol3] and showed that for the complex structures on nilmanifolds which are complex parallelizable Kuranishi space is defined by explicit polynomial equations, and most of the time singular.

There have been several attempts to have a more direct approach to the understanding of the Kuranishi map, namely to do things more algebraically and giving up to consider the Kuranishi slice. This approach has been pursued for instance in [SS] and effectively applied by Manetti. For instance, as already mentioned, Manetti ([Manobs]) gave a nice elegant proof of the above theorem 10 using the notion of differential graded Lie algebras, abbreviated by the acronym DGLA ’s.

The typical example of such a DGLA is provided by the Dolbeault complex

further endowed with the operation of Schouten bracket (here: the composition of exterior product followed by Lie bracket of vector fields), which is graded commutative.

The main thrust is to look at solutions of the Maurer Cartan equation modulo gauge transformations, i.e., exponentials of sections in .

The deformation theory concepts generalize from the case of deformations of compact complex manifolds to the more general setting of DGLA’s , which seem to govern almost all of the deformation type problems (see for instance [man09]).

### 1.4. Kuranishi and Teichmüller

Returning to our setting where we considered the closed subspace of consisting of the set of complex structures on , it is clear that there is a universal tautological family of complex structures parametrized by , and with total space

on which the group naturally acts, in particular .

A rather simple observation is that acts freely on if and only if for each complex structure on the group of biholomorphisms contains no automorphism which is differentiably isotopic to the identity (other than the identity).

###### Definition 12.

A compact complex manifold is said to be rigidified if

A compact complex manifold is said to be cohomologically rigidified if is injective, and rationally cohomologically rigidified if is injective.

The condition of being rigidified is obviously stronger than the condition , which is necessary, else there is a positive dimensional Lie group of biholomorphic self maps, and is weaker than the condition of being cohomologically rigidified.

Compact curves of genus are rationally cohomologically rigidified since if is an automorphism acting trivially on cohomology, then in the product the intersection number of the diagonal with the graph equals the self intersection of the diagonal, which is the Euler number . But, if is not the identity, and are irreducible and distinct, and their intersection number is a non negative number, equal to the number of fixed points of , counted with multiplicity: a contradiction.

It is an interesting question whether compact complex manifolds of general type are rigidified. It is known that already for surfaces of general type there are examples which are not rationally cohomologically rigidified (see a partial classification done by Jin Xing Cai in [cai]), while examples which are not cohomologically rigidified might exist among surfaces isogenous to a product (potential candidates have been proposed by Wenfei Liu).

Jin Xing Cai pointed out to us that, for simply connected (compact) surfaces, by a result of Quinn ([quinn]), every automorphism acting trivially in rational cohomology is isotopic to the identity, and that he conjectures that simply connected surfaces of general type are rigidified (equivalently, rationally cohomologically rigidified).

###### Remark 13.

Assume that the complex manifold has , or satisfies Wavrik’s condition, but is not rigidified: then by Kuranishi’ s third theorem, there is an automorphism which lies outside of a fixed neighbourhood of the identity. acts therefore on the Kuranishi space, hence, in order that the natural map from Kuranishi space to Teichmüller space be injective, must act trivially on , which means that remains biholomorphic for all small deformations of .

At any case, the condition of being rigidified implies that the tautological family of complex structures descends to a universal family of complex structures on Teichmüller space:

on which the mapping class group acts.

Fix now a complex structure yielding a compact complex manifold , and compare with the Kuranishi family

Now, we already remarked that there is a locally surjective continuous map of to the germ of at the point corresponding to the complex structure yielding . For curves this map is a local homeomorphism, and this fact provides a complex structure on Teichmüller space.

###### Remark 14.

Indeed we observe that more generally, if

1) the Kuranishi family is universal at any point

2) is a local homeomorphism at every point, then

Teichmüller space has a natural structure of complex space.

Moreover

3) since is surjective, it is a local homeomorphism iff it is injective; in fact, since has the quotient topology and it is the quotient by a group action, and is a local slice for a subgroup of , the projection is open.

The simple idea used by Arbarello and Cornalba ([ar-cor]) to reprove the result for curves is to establish the universality of the Kuranishi family for continuous families of complex structures.

In fact, if any family is locally induced by the Kuranishi family, and we have rigidified manifolds only, then there is a continuous inverse to the map , and we have the desired local homeomorphism between Kuranishi space and Teichmüller space.

Since there are many cases (for instance, complex tori) where Kuranishi and Teichmüller space coincide, yet the manifolds are not rigidified, we give a simple criterion.

###### Proposition 15.

1) The continuous map is a local homeomorphism between Kuranishi space and Teichmüller space if there is an injective continuous map , where is Hausdorff, which factors through .

2) Assume that is a compact Kähler manifold and that the local period map is injective: then is a local homeomorphism.

3) In particular, this holds if is Kähler with trivial canonical divisor ^{2}^{2}2As observed by a referee, the same proof works when
is Kähler with torsion canonical divisor, since one can consider the local period map of the canonical cover of .

Proof. 1) : observe that, since is locally compact and is Hausdorff, it follows that is a homeomorphism with its image . Given the factorization , then the inverse of is the composition , hence is a homeomorphism.

2) : if is Kähler, then every small deformation of is still Kähler, as it is well known (see [k-m71]).

Therefore one has the Hodge decomposition

and the corresponding period map , where is the period domain classifying Hodge structures of type

As shown by Griffiths in [griff1], see also [griff2] and [vois], the period map is indeed holomorphic, in particular continuous, and is a separated complex manifold, hence 1) applies.

3) the previous criterion applies in several situations, for instance, when is a compact Kähler manifold with trivial canonical bundle.

In this case the Kuranishi space is smooth (this is the so called Bogomolov-Tian-Todorov theorem, compare [bogomolov], [tian], [todorov], and see also [ran] and [kawamata] for more general results) and the local period map for the period of holomorphic n-forms is an embedding, since the derivative of the period map, according to [griff1] is given by cup product

If we apply it for , we get that is injective, since by Serre duality and cup product with yields an isomorphism with which is by Serre duality exactly isomorphic to .

∎

As we shall see later, a similar criterion applies to show ‘Kuranishi= Teichmüller’ for most minimal models of surfaces of general type.

For more general complex manifolds, such that the Wavrik condition holds, then the Kuranishi family is universal at any point, so a program which has been in the air for a quite long time has been the one to glue together these Kuranishi families, by a sort of analytic continuation giving another variant of Teichmüller space.

We hope to be able to return on this point in the future.

## 2. The role of singularities

### 2.1. Deformation of singularities and singular spaces

The basic analytic result is the generalization due to Grauert of Kuranishi’s theorem ([grauert], see also [sernesi] for the algebraic analogue)

###### Theorem 16.

Grauert’s Kuranishi type theorem for complex spaces. Let be a compact complex space: then

I) there is a semiuniversal deformation of , i.e., a deformation such that every other small deformation is the pull-back of for an appropriate morphism whose differential at is uniquely determined.

II) is unique up to isomorphism, and is a germ of analytic subspace of the vector space of first order deformations.

is the inverse image of the origin under a local holomorphic map (called Kuranishi map and denoted by )

to the finite dimensional vector space (called obstruction space), and whose differential vanishes at the origin (the point corresponding to the point ).

If is reduced, or if the singularities of are local complete intersection singularities, then

If the singularities of are local complete intersection singularities, then .

Recall once more that this result reproves the theorem of Kuranishi ([kur1], [kur2]), which dealt with the case of compact complex manifolds, where , being the sheaf of holomorphic vector fields.

There is also the local variant, concerning isolated singularities, which was obtained by Grauert in [grauert1] extending the earlier result by Tyurina in the unobstructed case where ([tyurina1]).

###### Theorem 17.

Grauert’ s theorem for deformations of isolated singularities.. Let be the germ of an isolated singularity of a reduced complex space: then

I) there is a semiuniversal deformation of , i.e., a deformation such that every other small deformation is the pull-back of for an appropriate morphism whose differential at is uniquely determined.

II) is unique up to isomorphism, and is a germ of analytic subspace of the vector space inverse image of the origin under a local holomorphic map (called Kuranishi map and denoted by )

to the finite dimensional vector space (called obstruction space), and whose differential vanishes at the origin (the point corresponding to the point ).

The obstruction space equals if the singularity of is normal.

For the last assertion, see [sernesi], prop. 3.1.14, page 114.

The case of complete intersection singularities was shown quite generally to be unobstructed by Tyurina in the hypersurface case ([tyurina1]), and then by Kas-Schlessinger in [k-s].

This case lends itself to a very explicit description.

Let be the complete intersection , where

Then the ideal sheaf of is generated by and the conormal sheaf is locally free of rank on .

Dualizing the exact sequence

we obtain (as )

which represents as a quotient of , and as a finite dimensional vector space (whose dimension will be denoted as usual by , which is the so called Tyurina number).

Let , represent a basis of .

Consider now the complete intersection

where

Then