The Teichmüller and Riemann Moduli Stacks
The aim of this paper is to study the structure of the Teichmüller and Riemann moduli spaces, viewed as stacks over the category of complex analytic spaces, for higher-dimensional manifolds. We show that both stacks are analytic in the sense that they admit a smooth analytic groupoid as atlas. We then show how to construct explicitly such an atlas as a sort of generalized holonomy groupoid for such a structure. This is achieved under the sole condition that the dimension of the automorphism group of each structure is bounded by a fixed integer. All this can be seen as an answer to Question 1.8 of .
1991 Mathematics Subject Classification:32G05, 58H05, 14D23
Let be a smooth oriented compact surface. The Teichmüller space is defined as the quotient space of the set of smooth integrable complex operators compatible with the orientation (o.c.)
by , the connected component of the identity in the oriented diffeomorphism group of .
The theory of Teichmüller spaces is a cornerstone in complex variables and Riemann surfaces. Originated by Riemann himself and followed by the fundamental works of Teichmüller, Ahlfors and Bers, it has moreover implications in many branches of mathematics as algebraic geometry, hyperbolic geometry, complex dynamics, discrete groups, …
Perhaps the most basic property of is that it has a natural structure of a complex manifold, making it a global moduli space of complex structures on .
Moreover, the mapping class group of acts on and the resulting quotient is a complex orbifold. This refined quotient coincides with the quotient of by the full group , the so-called Riemann moduli space .
Let now be a smooth oriented compact manifold of even dimension strictly greater than . The Teichmüller and Riemann moduli spaces can still be defined, but one now has to add the integrability condition in the definition
Although the literature about these higher dimensional Teichmüller and Riemann moduli spaces is much less developed than that about surfaces, it has grown significantly in the last years and these spaces play an increasing role in Complex Geometry. Catanese’s guide to deformations and moduli  as well as  gives some general local properties of and contains many results on the Teichmüller space of minimal surfaces of general type. And in the special case of hyperkähler manifolds, the Teichmüller space is used by Verbitsky in a prominent way in his proof of a global Torelli Theorem  and also to showing some important results on these manifolds .
However, the main difference with the case of surfaces is that and are just topological spaces and do not have any good geometric structure. Only for special classes such that the class of hyperkähler manifolds, an analytic structure is known on , but even in this case, it is not Hausdorff at all points. Perhaps the most dramatic example is given by being . Then , as a set, can be identified with , a point corresponding to the Hirzebruch surface (and each connected component of can be identified with , and encoding the same surface, see Examples 4.14 and 12.6). But, as a topological space, it is endowed with a non-Hausdorff topology. No two points are separated, as a consequence of the fact that can be deformed onto any with by an arbitrary small deformation. Equivalently, this comes from the fact that the dimension of the automorphism group of Hirzebruch surfaces jumps.
Indeed, in presence of this jumping phenomenon, and are not even locally Hausdorff hence not locally isomorphic to an analytic space (cf. Example 12.3). This explains why the classical approach developed in the fundamental works of Kodaira-Spencer and Kuranishi is based on the following principles.
in higher dimension the global point of view must be abandoned for the local point of view;
and the Teichmüller space replaced with the Kuranishi space which must be thought of as the best possible approximation in the analytic category for a local moduli space of complex structures.
Nevertheless, putting on and a global analytic structure in some sense is the only way to go beyond the classical local deformation theory. As we cannot expect a structure of analytic space, even a non-Hausdorff one, we have to view these quotient spaces as stacks. The aim of this paper is to develop this point of view. The question now becomes to showing that, as stacks, and are analytic. This can also be seen as an answer to Question 1.8 of . Since we work with arbitrary complex structures and not only with projective ones, we have to work with analytic stacks and not algebraic ones.
For surfaces of fixed genus , the classical setting coincide with the stack setting. Both stacks and are analytic and can be fully recovered from the Teichmüller and Riemann moduli spaces. In particular, the complex structure on , respectively the complex orbifold structure on are equivalent to the analytic structures on the corresponding stack. The case of genus is somewhat more complicated, because of the translations111To avoid this problem, it is customary to use marked complex tori, that is elliptic curves.. Here the stack structure contains strictly more information than the classical spaces and since it also encodes the translation group of each complex torus, but once again both stacks are analytic and their analytic structure comes from the complex structure on the corresponding spaces.
The main results of this paper show that, in any dimension, both the Teichmüller and the Riemann moduli stacks are analytic stacks. The only condition needed for this result to hold is that the dimension of the automorphism group of all structures of (or ) is bounded by a fixed integer222which is always the case for surfaces.. This is nevertheless a mild restriction since we may easily stratify into strata where this dimension is bounded. We emphasize that can be any compact manifold and that we consider all complex structures and not only projective or kähler ones333However, our results also apply to the set of kähler structures on modulo or ..
We postpone the precise statements of the main Theorems 2.9 and 2.10 as well as the strategy of proof to section 2 after defining precisely the involved notions. Let us just say that we will follow the same strategy that can be used for Riemann surfaces. Firstly, we define and as stacks of families of complex manifolds diffeomorphic to . This is the easy part. Secondly, we build an atlas with good analytic properties to show they are analytic. This is the difficult part which takes the rest of the paper.
We hope that this paper will serve as a source of motivation for studying global moduli problems in Complex Analytic Geometry and their interplay with analytic stacks. From the one hand, every abstract result on these stacks might apply to moduli problems and increase our knowledge of Complex Manifolds. From the other hand, examples of Teichmüller stacks are an unending source of examples of analytic stacks, showing all the complexity and richness of their structure, far from finite dimensional group actions and leaf spaces.
2. Definitions and statements of the main results
2.1. The Teichmüller and Riemann spaces
Let be a smooth (i.e. ) oriented compact connected manifold of even dimension. Let , respectively , be the space of smooth almost complex, respectively complex operators on which are compatible with the orientation. The definition of is given in (1.1). We assume that both sets are non-empty.
We topologize as a Fréchet manifold locally modelled onto the smooth sections of a vector bundle over (cf.  for the encoding of structures,  and  for the Fréchet topology). We denote by , respectively , a connected component of , respectively . Points of will be denoted generically by .
For a topological space, we denote by the set of connected components of . The previous topology being countable, is a countable set.
The diffeomorphism group acts on the right on by pullback of almost complex operators. It is a Fréchet Lie group  acting analytically444There is some subtle point here because the complex structure of depends on the choice of a complex structure on . We will just use the fact that, if we endow locally at identity with chart (3.1), then the map is analytic in a neighborhood of . onto . This action leaves invariant. It is given by
We focus on , the connected component of the identity in . We define the mapping class group
and we set
Both and are endowed with the quotient topology, making them topological spaces.
In the first version of this paper, we take for an unoriented smooth compact manifold and consider as the set of all integrable complex operators, regardless of orientation. Then is defined as in (2.3), and in (2.4), we have to replace the oriented diffeomorphism group by the full diffeomorphism group . This does not change substantially these two sets, and our results apply to this setting. In fact, the main drawback of forgetting the orientation is that the notion of Teichmüller space does not coincide to the classical one for surfaces. Especially, the unoriented Teichmüller space of a compact surface has two connected components, corresponding to the two possible orientations.
More generally, if admits a diffeomorphism reversing orientation, then the unoriented Teichmüller space has twice more connected components as the classical one. However, the two Riemann spaces coincide. Finally, if does not admit any orientation reversing diffeomorphism, then the unoriented Teichmüller and Riemann spaces are the disjoint union of the classical ones for both orientations. Notice that, in this last case, changing the orientation may completely change the Teichmüller and Riemann spaces. It is even possible that they become empty (think of and .
2.2. Stacks and groupoids.
Before getting into the definition of the Teichmüller and Riemann moduli stacks, let us define precisely the notions of stacks and groupoids we will use.
First, a warning. We insist on the fact that we work exclusively in the -analytic context, since we deal with arbitrary compact complex manifolds. This forces us to adapt and sometimes to transform the definitions of stacks coming from algebraic geometry. Also, since the literature on stacks over the category of analytic spaces is very scarce, we shall keep the required facts from stack theory to a minimum and give complete proofs even of some routine facts (for example in Proposition 2.5).
Moreover, our construction of atlas is inspired in the construction of the étale holonomy groupoid of a foliation. So for the groupoid point of view, we stick to the literature in foliation theory and Lie groupoids, especially . The conventions are somewhat different from those of algebraic geometry and we have to adapt ourselves to these differences. Especially, we will not make use of the notion of representability (see however Remark 2.4).
Let denote the category of -analytic spaces. We include analytic spaces that are everywhere non-reduced in . We consider it as a site for the euclidean topology: our families of coverings are just standard topological open coverings. We emphasize that we will not use other coverings as étale or analytic ones. At some points (for example in Section 12), we may restrict the base category to be that of complex manifolds, still with the euclidean coverings.
In this paper, a stack is a stack in groupoids over the site in the sense of [37, Def. 8.5.1]. In brief, is a stack if
comes equipped with a morphism whose fibers are groupoids.
is a category fibred in groupoids, i.e. pull-backs exist and are unique up to unique isomorphisms.
Isomorphisms form a sheaf, i.e. one can glue a compatible collection of isomorphisms defined over an open covering of an analytic space into a single isomorphism over .
Every descent data is effective, i.e. one can glue objects defined on an open covering of an analytic space into a single object over by means of a cocycle of morphisms.
The groupoids we consider are analytic, that is
the set of objects and the set of morphisms are complex analytic spaces
and all the structure maps are analytic morphisms.
In the setting of Lie groupoids and foliation theory, the space of morphisms is possibly non-Hausdorff, since such phenomena occur when constructing holonomy groupoids of finite-dimensional foliations. For example, the holonomy groupoid of the Reeb foliation of the sphere is non-Hausdorff. In classical foliation theory, this is linked to the existence of so-called vanishing cycles. Recall also that the Hausdorffness/Non Hausdorffness of the set of morphisms is preserved by Morita equivalence. We refer to [32, §5.2] for more details.
Even if our construction is inspired in that of holonomy groupoids of foliations, all the groupoids we construct will be proved to be Hausdorff. We note that in previous versions of this work, we authorized non-Hausdorff groupoids since at that time we did not succeed in proving our groupoids are Hausdorff.
Analytic groupoids are in particular topological so that it makes sense to localize them on an open covering of the set of objects . The geometric quotient associated to such a groupoid is the topological space obtained by taking the quotient of the set of objects by the equivalence relation defined by the set of morphisms. Connected components of the groupoid refer to connected components of the geometric quotient.
Such a groupoid is étale, respectively smooth, if both source and target maps are étale, respectively smooth, morphisms. Here smoothness refers to smoothness of morphisms in analytic/algebraic geometry, not to differentiability. We emphasize that a smooth analytic groupoid is not a complex Lie groupoid, since we allow singularities of both the set of objects and the set of morphisms, but it is the exact singular counterpart of a complex Lie groupoid, cf. [32, §5].
Given a stack, we are above all interested in knowing if it admits an analytic groupoid as atlas (or presentation) in the following sense: the stackification of this groupoid by torsors as explained in , Chapters 3 and 4, is isomorphic to the initial stack. In that case, the stack can be fully recovered from the atlas through this long process of stackification. We will detail this process in the proofs of Theorems 2.9 and 2.10. Among analytic presentations, the following ones are of special interest.
We call a stack étale analytic (respectively Artin analytic or simply analytic) if it admits a presentation by an étale (respectively smooth) analytic groupoid; Deligne-Mumford analytic if it is étale with finite stabilizers.
We take as definition of Morita equivalence that given in [32, §5.4], with the obvious adaptations to the groupoids we use (e.g. replace map with -analytic map, submersion with smooth morphism, …). It follows from carefully adapting  to the analytic context that two smooth atlases of the same stack are Morita equivalent.
Standard definitions of algebraic stacks (see for example [37, Def. 84.12.1]) does not involve directly the existence of an atlas but asks for representability of the diagonal and existence of a surjective, smooth morphism from a scheme or an algebraic space onto the stack. It is of course possible to adapt these definitions to the analytic context, but as mentioned in the warning we do not follow this way. However, both notions are not far each from the other. In the algebraic definitions, both conditions are used to ensure the existence of an algebraic atlas. From the surjective and smooth morphism, one constructs a symmetry groupoid which is an atlas for the stack. The set of objects of this groupoid is the scheme or algebraic space on which the smooth morphism is defined. Then, the condition on the diagonal ensures that the smooth morphism is itself representable and so that the set of morphisms of this symmetry groupoid has also a structure of scheme or algebraic space, depending on the precise definition that was taken. So in short, an algebraic stack admits a presentation by an algebraic groupoid. Besides, an analytic stack as defined above surely admits a surjective smooth morphism from an analytic space onto it: just take for the set of objects of the atlas and for the induced morphism. But we do not know if the diagonal is always representable.
2.3. The Teichmüller and Riemann moduli stacks as stacks of deformations.
Let be an open set of . Define the following category over .
Objects are -families
is a smooth and proper morphism with reduced fibers all diffeomorphic to .
Each fiber can be encoded as with .
In other words, a -family is nothing else than an analytic deformation of complex structures of such that the structure of each fiber is isomorphic to a point of . Of course, if denotes the image of through the action of , then and are equal. However, it is interesting to have this flexibility, for example we will often take for a connected component of , even if it is not saturated.
Morphisms are cartesian diagrams
between -families. Observe that the pull-back of a -family is a -family.
We now pass to the construction of , which is more delicate. Observe that any family can be seen locally over some sufficiently small open set as
for some smooth family of complex operators of . Over an intersection , two trivializations (2.7) are glued using a family of diffeomorphisms of whose differential commute with and . As a consequence, is diffeomorphically a bundle over with fiber and structural group . In particular, once such an identification with a bundle is fixed, it makes sense to speak of the structural group of , and to make a reduction of the structural group to some subgroup of . And it makes also sense to speak of -isomorphism of the family , that is isomorphism of such that, in each fiber, the induced diffeomorphism of is in .
We define as the category whose objects are -families with a marking
with a bundle with fiber and structural group reduced to and whose morphisms are cartesian diagrams (2.6) such that the canonical isomorphism between and induces a -isomorphism of the markings.
Alternatively, one may use -framings, that is -isotopy classes of maps
Here is any point of and isotopies are -maps from to such that is a point for all . In particular, we may replace by any other point using an isotopy. Set . Since is diffeomorphic to a -bundle, then, given an isotopy with , the diffeomorphism of belongs to . In other words, the framing induces in that case a coherent identification of the fibers with up to an element of .
This forms a category over and a subcategory of . In general, it contains stricly less objects, since some -families do not admit -markings.
Let be an open subset of . Then both and are stacks.
It is straightforward but we sketch it for sake of completeness. First, the natural morphism is obviously a category fibred in groupoids. The fiber over is the groupoid formed by -families over as objects and isomorphisms of families as morphisms. Then given two -families and and an open covering of , any collection of isomorphisms from the restriction of to onto the restriction of to such that and are equal on the intersections obviously glue to give an isomorphism of families between and . So isomorphisms form a sheaf. Finally, starting from a collection of -families and from a cocycle of isomorphisms of families between and , then
where is the equivalence relation given by the cocycle , is a -family over . Every descent data is effective. The proof for is similar. ∎
We may thus define
We call Riemann moduli stack the stack . The stack is the Riemann moduli stack for complex structures belonging to .
By abuse of notation, we denote simply by the Riemann moduli stack. No confusion should arise with (2.4). In the same way,
We call Teichmüller stack the stack . The stack is the Teichmüller stack for complex structures belonging to .
By abuse of notation, we denote simply by the Teichmüller stack. No confusion should arise with (2.3).
2.4. Statement of the main results
Let and set
To avoid cumbersome notations, we write for , and for , …
Let be the sheaf of germs of holomorphic vector fields on . For , we consider the function
The and are open sets of , see (4.5).
Let be an open set of (for example, is a connected component of ). For all , define as in (2.12). Then,
For all , the stack is Artin analytic.
Assume that the function is bounded on , resp. on . Then, the stack , resp. , is Artin analytic.
Let be an open set of (for example, is a connected component of ). For all , define as in (2.12). Then,
For all , the stack is Artin analytic.
Assume that the function is bounded on , resp. on . Then, the stack , resp. , is Artin analytic.
2.5. Strategy of proof and organization of the paper.
We want to construct smooth analytic atlases of and . Since we deal with arbitrary complex structures, one has to use as a starting point the classical deformation theory of Kodaira-Spencer and to build such an atlas from the local data encoded in the Kuranishi space. But, in the higher dimensional case, the jumping phenomenon causes many troubles. When it occurs, a positive-dimensional subspace of the Kuranishi space encode a single complex structure and we have to encode in our atlas that all these points are the same complex structure. And this is not the only problem. One should expect that knowing, at a complex structure , the Kuranishi space of and the identifications induced by its automorphisms, respectively by its automorphisms which are -isotopic to the identity is enough to get a local model of the Riemann moduli stack, respectively of the Teichmüller stack.
This is however not correct. A third element is missing. Some orbits of may a priori have a complicated geometry and accumulate onto . This induces additional identifications to be done in the Kuranishi space, and thus to encode in the atlas of the Riemann or Teichmüller stack, even in the absence of automorphisms.
The main problem behind this atlas construction is to understand how to glue the bunch of Kuranishi spaces, in other words how to keep track of all identifications to be done not only on a single Kuranishi space but also between different ones.
This is achieved here by describing the space of complex structures as a foliated space in a generalized sense. Then, we describe the stacky structure of the leaf space. A natural source of stacks is given by (leaf spaces of) foliations. Such stacks admit atlases given by an étale groupoid, the étale holonomy groupoid [32, §5.2]. In general, the action of onto does not define a foliation, nor a lamination. But we show that it defines a more complicated foliated structure. We then turn to the construction of an associated holonomy groupoid. It is however much more involved than the classical construction and it constitutes the bulk of the paper. Indeed, the transverse structure of this generalized foliation being stacks, the holonomy morphisms are stacks morphisms and do not fit easily into a nice groupoid. A lot of work is needed for that.
The paper is organized as follows. Recall that we defined the Teichmüller and the Riemann moduli stacks and stated precisely the main results in section 2. We collect some facts about the Kuranishi space in Section 3 and we explain how to turn it into a Kuranishi stack that encodes also the identifications induced by the automorphisms. We then give some general properties of in section 4, putting emphasis on connectedness properties, and introducing a graph, called the graph of -homotopy. The foliated structure of is introduced in section 5. The technical core of the paper is constituted by sections 8 and 9, where we perform the construction of the analogue for the holonomy groupoid. We call it the Teichmüller groupoid. To smoothe the difficulties of the construction, a sketch of it is given in section 6 and a very simple case is treated in section 7. All this culminates in the proof of Theorem 11.1, stating that the Teichmüller groupoid is an analytic smooth presentation of the Teichmüller stack. Analogous construction and statement for the Riemann moduli stack are done in sections 10 and 11. Complete examples are given in section 12.
3. The Kuranishi stack.
3.1. The Kuranishi space and Theorem.
Fix a riemannian metric on and let denote the exponential associated to this metric.
For any , a complex chart for at is given by the map
where is the -vector space of -vector fields of and a neighborhood of .
We denote by the group of automorphisms of . The connected component of the identity in is tangent to . We define
(Kuranishi, 1962). For any choice of a closed complex vector space such that
there exists a connected open neighborhood of in , a finite-dimensional analytic subspace of containing and an analytic isomorphism (onto its image)
The inverse map is given by
The composition of the maps
is the identity.
Indeed, Kuranishi always uses the -orthogonal complement of the space as . However, it is easy to see that everything works with any other closed complement, cf. .
Theorem 3.2 is proved using the inverse function Theorem. To do that, one extends to operators of Sobolev class (with big), so that becomes a Hilbert manifold. Then one may use the classical inverse function Theorem for Banach spaces to obtain the isomorphism (3.4). Finally, because is tangent to the kernel of a strongly elliptic differential operator, then it only consists of operators and the isomorphism (3.4) is still valid when restricting to operators, see ,  and  for more details.
Following , we call such a pair a Kuranishi domain based at . We make the following assumption
The image of is contained in a product with an open and connected neighborhood of in .
Moreover, we call the natural retraction map
and the other projection
Given , we denote by the Kuranishi space of . We use the same convention for as that stated for in Remark 2.8.
It is a classical fact that the germ of at is unique up to isomorphism. However, in this paper, we consider as an analytic subspace of , not as a germ. By abuse of terminology, we nevertheless speak of the Kuranishi space.
3.2. Automorphisms and the Kuranishi stack
The complex Lie group (respectively ) is the isotropy group for the action of at (respectively ). We focus on the connected component of the identity in this isotropy group. It acts on , and so locally on . This action induces a local action of each -parameter subgroup on . In other words, let now be an element of . There exists some maximal open set such that
is a well defined analytic map. Observe that fixes . We want to encode all these maps (3.9) in an analytic groupoid
Although it is the case in many examples, the groupoid (3.10) will not in general describe a local -action. This comes from the fact that there is no reason for to equal . In particular, there is no reason for the isotropy groups of the groupoid to be subgroups of . They are just submanifolds. Hence we will need some work to define it precisely.
We start with the following Lemma. We recall that is the neighborhood of in appearing in Hypothesis 3.5.
If is small enough, then there exist an open and connected neighborhood of the identity in and an open and connected neighborhood of the identity in such that
is an isomorphism.
Set . Then (3.11) extends as an isomorphism
Pass to vector fields and diffeomorphisms of Sobolev class for some big and extend the map. Since consists of holomorphic elements, this map is of class and a simple computation shows that its differential at is an isomorphism. Hence we may apply the local inverse Theorem and get the result for this Sobolev class. To finish with point (i), it is enough to remark that, since is holomorphic, is of class if and only is.
We say that is -admissible if belongs to and is a finite composition of diffeomorphisms of such that belongs to for between and and with the convention .
In particular, we have , so replacing with if necessary, we obtain a new -admissible couple such that belongs to . In the same way, replacing with , then with and so on, we may assume that every belongs to . In the sequel, we always assume that an -admissible couple has this property.
where denotes the set of diffeomorphisms from to a fiber of the Kuranishi family . Here by , we mean that we consider as a diffeomorphism from to the complex manifold . We also consider the two maps from to
There is a subtle point here. In order to define (3.10) as a smooth analytic groupoid, we will realize as an analytic subspace of . We emphasize that the complex structure on is not a product structure. Indeed, the -completion of can be endowed with a complex structure as an open set of the complex Banach manifold of -maps from to , but this complex structure depends on , that is depends on the choice of a complex structure on . If we cover it with (3.1) as complex chart at identity and with complex chart at , then the changes of charts depend on and thus on . We set , resp. for this Banach manifold and more generally , resp. . Now the completion (and thus the completion of as open subset of ) can be endowed with a structure of a complex Banach analytic space such that the natural projection onto is smooth with fiber over equal to , see . As a consequence, we will show in the proof of Lemma 3.12 that is locally modelled onto but is not realized in general as an open submanifold of it (cf. Remark 3.7). For example, if is an elliptic curve and is a neighborhood of in the upper half-plane , then is diffeomorphic to that is to but, as a complex manifold, is in fact the universal family over , that is the family whose fiber over is (cf. ).
Then we set
and . We have
The space is an analytic subspace of as an open subset of the set of in such that satisfies the analytic equations defining as an analytic subspace of (the completion of) and thus of (the completion of) . Also, is just the restriction to this analytic subspace of the projection of onto , hence is analytic. And is given by the action hence is also analytic.
Let now belong to . Let be a neighborhood of in such that belongs to for all points of . Consider the following composition of analytic maps
The first one is the restriction of the inverse of the chart to , hence satisfies ; the second one is given by Lemmas 3.8 and 3.11, hence ; and the third one is just the projection. The first two maps are obviously analytic isomorphisms onto their image. For the third one, its inverse is given by the formula
We note that the composition in (3.17) is independent of and that is locally modelled on the product of a neighborhood of a point in with some open neighborhood of the identity in .
Moreover, it shows that is a smooth morphism, since it is given by the projection map
Now, observe that and are isomorphic Banach manifolds. Moreover, composition in is not analytic (cf. Remark 3.9) but it is when restricted to finite-dimensional complex submanifolds/subspaces containing only structures. These two observations show that the multiplication is analytic. In the same way the inverse map of the groupoid is analytic. Finally, since the source map is smooth and the inverse map is analytic, this implies that the target map is also smooth. ∎
The Kuranishi stack associated to is the stackification of (3.10).
However, the geometric quotient of the Kuranishi stack has no reason to be homeomorphic to the topological space for the equivalence relation generated by for , because there may exist with and in but such that is not -admissible. Rephrasing this important remark, the Teichmüller stack is not locally isomorphic to the Kuranishi stack, cf. Remark 11.6.
In many cases, the groupoid (3.10) is a translation groupoid, although its structure is much more complicated in general. For that reason, in previous versions of this paper, we denote it abusively by .
We now want to link the structure of (3.10) with the foliated structure of described in . Recall that the leaf through a point is the maximal connected subset of all of whose points encode up to isotopy. We have
The space of connected components of the classes of in is homeomorphic to the leaf space of by its foliated structure.
Let be in the leaf through . Then there exists an isotopy such that
So is -admissible for all and belongs to the connected component containing of the equivalence class of . The converse is obvious. ∎
4. Connectedness properties of and the graph of -homotopy.
Observe that Kuranishi’s Theorem 3.2 implies that is locally -pathwise connected in . Therefore,
There are at most a countable number of connected components of in each .
Every connected component of is -pathwise connected.
The Teichmüller and Riemann moduli stacks have at most a countable number of connected components. Moreover,
The natural projection map from onto induces a bijection
The mapping class group acts on both and .
Passing to the quotient by the mapping class group , the bijection (4.1) descends as a bijection
Just use Proposition 4.1 and the fact that leaves the components of invariant. ∎
For further use, we let
denote the map given by the action of the mapping class group onto a fixed component .
For surfaces, the number of connected components of , that is the number of connected components of up to the action of the mapping class group, is finite as soon as it contains a projective manifold . However, it may be more than one, see .
In dimension , there are examples of manifolds with , henceforth having infinitely many connected components, as for , see , or the product of a K3 surface with , see .
In the above examples, we note that also has infinitely many connected components. Indeed each connected component of contains exactly one connected component of . This leads to the following problem:
Find a compact manifold with connected and having an infinite number of connected components.
Probably, for give such an example. In particular, it is proven in  that has a single connected component. And the structures of  should give the countably many connected components of . Since they have pairwise not biholomorphic universal covers, this should give the countably many connected components of and even of . But proving this is the case seems to be out of reach for the moment. Observe that the first step in showing this result would be to establish that any deformation in the large of a Hopf manifold is a Hopf manifold, which is still an open problem as far as we know.
The case of surfaces is somewhat different, see Remark 4.19.
Recall that Kodaira and Spencer defined in  the notion of -homotopy. Taking into account Kuranishi’s Theorem, it turns out that we may equivalently define it by saying that and are -homotopic if there exists a smooth path in joining them. That is if they belong to the same connected component . Similarly, we define
Let and be two points of the same . Then we say that they are -homotopic if there exists a smooth path in joining them such that the function is constant along it.
Recall also that, if denotes the Kuranishi space of some , then for any , the sets
are analytic subspaces of , cf. . Using Kuranishi’s Theorem, we immediately obtain that the sets
are analytic subspaces555To be more precise, one should pass to operators of class as in Remark 3.4 to have that and are Banach analytic spaces in the sense of . of . Observe that is the union of all -homotopy classes whose is greater than or equal to .
The analyticity of (4.4) comes indeed from the fact that the function is upper semi-continuous for the Zariski topology, see . But this also implies
There are at most a countable number of -homotopy classes in each .
Define a weighted and directed graph as follows. Each -homotopy class of corresponds to a vertex with weight equal to for . Two vertices and are related by an oriented edge if there exists a smooth path in such that
The structure belongs to .
For , the structure belongs to the class .
Observe that the edge is directed from the highest weight to the lowest weight.
The previous graph is called the graph of -homotopy of .
The graph of -homotopy has the following properties:
It has at most a countable number of connected components. Moreover, there is a correspondence between these connected components and the connected components of .
It has at most a countable number of vertices.
Each vertex is attached to at most a countable number of edges.
There is no directed loop.
Every directed path is finite.
The group acts on the graph of -homotopy. We detail in the following Proposition some trivial properties of this action.
The action of onto the graph of -homotopy
sends a connected component onto a connected component.
sends a vertex to a vertex of same weight.
respects the number and the orientation of the edges attached to a vertex.
Hence, the existence of diffeomorphisms acting non trivially on the graph implies strong symmetry properties of the graph. Indeed, if some sends a connected component of onto a different one, then these two connected components of must be completely isomorphic.
Hopf surfaces. Let . By classical results of Kodaira , , every complex surface diffeomorphic to is a (primary) Hopf surface. There is only one connected component of complex structures up to action of the mapping class group, since any Hopf surface is -homotopic to any other one, see . The mapping class group of is a non trivial group666 It was pointed out to me by A. Hatcher that no mapping class group of a closed -manifold seems to be known.. Indeed, observe that it contains at least the elements
for and a homotopically non trivial loop in , since both have non trivial action in homology. Even without knowing the mapping class group, we can characterize its action on . Following [41, p.24], we separate Hopf surfaces into five classes namely classes IV, III, IIa, IIb and IIc.
Let be a diffeomorphism of . Assume that leaves a connected component of invariant. Then is -isotopic to the identity.
Let represent a Hopf surface of type IIc, that is associated to a contracting diagonal matrix
Assume that belongs to the same connected component as . Then there exists a -homotopy of Hopf surfaces with endpoints and : just take the tautological family above a smooth path in joining to . By [30, Theorem 8.1], there exists an analytic space encoding the complex structures in a neighborhood of the path and obtained by gluing together a finite number of Kuranishi spaces of Hopf surfaces (up to taking the product with some vector space) such that the family maps onto a smooth path into . Using the description of the Kuranishi spaces of Hopf surfaces in [41, Theorem 2], it is easy to check that
is a manifold.
The points of encoding the type IIa Hopf surfaces belongs to a submanifold of complex codimension at least .
Hence, by transversality, we may replace the initial path defining the -homotopy with a new path and a thus a new -homotopy with same endpoints and such that all surfaces along this path are linear, that is not of type IIa. Such a family is locally and thus globally since the base is an interval isomorphic to the quotient of by the action generated by
induces a biholomorphism between and , which is smoothly isotopic to the identity. Composing with the inverse of this biholomorphism, this gives an automorphism of which corresponds to the same element of the mapping class group as .
Since every automorphism of every Hopf surface is isotopic to the identity (cf. [41, p.24] where all the automorphism groups are described), we are done. ∎
From Lemma 4.11, we deduce that decomposes into several identical connected components that are exchanged by action of the mapping class group. In particular,
The map (4.3) is a correspondence between the mapping class group of and the set of connected components of .
Let us focus on one of the connected components. It corresponds to a graph with an infinite number of vertices: one of weight (class IV), one of weight for each value of (class III of weight ) and one of weight (classes IIa, IIb and IIc together). There is an edge joining to and one joining to for each value of . There is no edge from to any vertex because it is not possible to deform a Hopf surface of class IV onto one of class III without crossing the -homotopy class of weight . In the same way, there is no edge between two different vertices of weight , because every -homotopy from a Hopf surface of type III with weight to a Hopf surface of type III with weight must pass through type II Hopf surfaces.
In Figure 1, we draw the graph in a synthetic way. The vertex encodes indeed the uncountable set of vertices of weight labelled by . The single edge from to remembers all the edges from vertices of label onto the vertex .
Using the five classes of Hopf surfaces, one obtains a graph of small deformations which is more precise and complicated than the graph of -homotopy, see , p.31. The graph of -homotopy must be considered as a very rough decomposition of .
Hirzebruch surfaces. Consider . It admits complex structures of even Hirzebruch surfaces . By , this exhausts the set of complex surfaces diffeomorphic to . Then there is only one connected component of complex structures up to action of the mapping class group. The mapping class group is not known (cf. footnote 6) but contains at least four elements generated by