Hi! After a month of silence, I’m back to the surface with the second post of the series “Lyapunov spectrum of Kontsevich-Zorich cocycle on the Hodge bundle of square-tiled cyclic covers”. For those of you who are curious to know why I was so quiet lately, the main reason is the significant amount of activity in Paris around moduli spaces of Abelian differentials. For instance, Don Zagier started (October 11, 2010) his course at Collège de France about Teichmüller curves and Hilbert modular surfaces (where he reports on a work in progress joint with Martin Möller), François Labourie started (October 21, 2010) a “groupe de travail” at Orsay about the recent preprint of A. Eskin, M. Kontsevich and A. Zorich about the computation of values of individual Lyapunov exponents of square-tiled cyclic covers (a topic we’ll cover here, of course 🙂 ), and Jean-Christophe Yoccoz will start his course at Collège de France about square-tiled surfaces soon (November 17, 2010). Evidently, the material of these activities affect the way I’m organizing the current series of posts, so that’s why I decided to slow down my blogging frequency.
In any case, after our initial post, we’ll continue our discussions by introducing nice structures (stratification by complex spaces) on the Teichmüller and moduli spaces of Abelian differentials and we’ll close today’s post by stating (without proofs) Kontsevich-Zorich’s classification of the connected components of (strata of) moduli spaces of Abelian differentials.
1. Some structures on the set of Abelian differentials
We denote by the set of Abelian differentials on a Riemann surface of genus , or more precisely, the set of pairs where is a compact topological surface of genus and is a -form which is holomorphic with respect to the underlying Riemann surface structure. In this notation, the Teichmüller space of Abelian differentials is the quotient and the moduli space of Abelian differentials is the quotient . Here and (the set of diffeomorphisms isotopic to the identity and the mapping class group resp.) act on the set of Riemann surface structure in the usual manner, while they act on Abelian differentials by pull-back.
Below we’ll put some structures on , and .
Given a non-trivial Abelian differential on a Riemann surface of genus , we can form a list by collecting the multiplicities of the (finite set of) zeroes of . Observe that any such list verifies the constraint in view of Poincaré-Hopf theorem (or alternatively Gauss-Bonnet theorem). Given an unordered list with , the set of Abelian differentials whose list of multiplicities of its zeroes coincide with is called a stratum of . Since the actions of and respect the multiplicities of zeroes, the quotients and are well-defined. By obvious reasons, and are called stratum of and (resp.). Notice that, by definition,
In other words, the sets , and are naturally “decomposed” into the subsets (strata) , and . However, at this stage, we can’t promote this decomposition into disjoint subsets to a stratification because, in the literature, a stratification of a set is a decomposition where is a finite set of indices and the strata are disjoint manifolds/orbifolds of distinct dimensions. Thus, one can’t call stratification our decomposition of , and until we put nice manifold/orbifold structures (of different dimension) on the corresponding strata. The introduction of nice complex-analytic manifold/orbifold structures on and are the topic of our next subsection.
Remark 1 The curious reader might be asking at this point whether the strata are non-empty. In fact, this is a natural question because, while the condition is a necessary condition (in view of Poincaré-Hopf theorem say), it is not completely obvious that this condition is also sufficient. In any case, it is possible to show that the strata of Abelian differentials are non-empty whenever . For comparison, we note that exact analog result for strata of non-orientable quadratic differentials is false. Indeed, if we denote by the stratum of non-orientable quadratic differentials with zeroes of orders and simple poles, the Poincaré-Hopf theorem says that a necessary condition is , and a theorem of H. Masur and J. Smillie says that this condition is almost sufficient: except for the empty strata in genera 1 and 2, any strata verifying the previous condition is non-empty. See Masur and Smillie paper for more details.
1.2. Period map and local coordinates
Let be a stratum, say , . Given an Abelian differential , we denote by the set of zeroes of . It is possible to prove that, for every , there is an open set such that, after identifying, for all , the cohomology with the fixed complex vector space via the Gauss-Manin connection (i.e., through identification of the integer lattices and ), the period map given by
is a local homeomorphism. See e.g. J.-C. Yoccoz’s survey for a proof of this fact based on an important construction called W. Veech’s zippered rectangles.
Remark 2 Concerning the possibility of using Veech’s zippered rectangles to show that the period maps are local homeomorphisms, we vaguely mentioned this particular strategy because the zippered rectangles is a fundamental tool: indeed, besides its usage to put nice structures on , it can be applied to understand the connected components of , make volume estimates for , study the dynamics of Teichmüller flow on through combinatorial methods (Markov partitions), etc. In other words, Veech’s zippered rectangles is a powerful tool to study global geometry and dynamics on . However, since the long-term goal of our future discussions is not the study of the entire strata , but only the geometry (and dynamics) of strictly smaller loci in these strata, Veech’s zippered rectangles are less effective in this context, so that they will not be part of our topics. Nevertheless, we strongly recommend the reader to consult Yoccoz’s survey for more details on Veech’s zippered rectangles and applications.
Recall that is a complex vector space isomorphic to . An explicit way to see this fact is by taking a basis of the absolute homology group , and relative cycles joining an arbitrarily chosen point to the other points . Then, the map
gives the desired isomorphism. Moreover, by composing the period maps with such isomorphisms, we see that all changes of coordinates are given by affine maps of . In particular, we can also pullback Lebesgue measure on to get a natural measure on : indeed, is well-defined modulo normalization because the changes of coordinates are affine maps; to remedy the normalization problem, we ask that the integral lattice has covolume 1, i.e., the volume of the torus is 1.
In resume, by using the period maps as local coordinates, has a structure of affine complex manifold of complex dimension and a natural (Lebesgue) measure . Also, it is not hard to check that the action of the modular group is compatible with the affine complex manifold structure on , so that, by passing to the quotient, one gets that the stratum of the moduli space of Abelian differentials has the structure of a affine complex orbifold and a natural Lebesgue measure .
Remark 3 Note that, as we emphasized above, after passing to the quotient, is an affine complex orbifold at best. In fact, we can’t expect to be a manifold since, as we saw in Subsection 1.4 of the previous post, even in the most simple example of genus 1, is an orbifold but not a manifold. In particular, the fact that the period maps are local homeomorphisms are intimately related to the fact that we were talking specifically about the Teichmüller space of Abelian differentials (and not of its cousin ).
The following example shows a concrete way to geometrically interpret the role of period maps as local coordinates of the strata .
Example 1 In the picture above (extracted from Zorich’s survey, Figure 12), we have a polygon whose corresponding sides , , are identified by translation. After performing these identification, we obtain a Riemann surface equipped with an Abelian differential (obtained as the quotient of on the polygon under the side identifications). It is not to hard to infer from the picture that the vertices of polygon are all identified to the same point . Furthermore, since the total angle around is and is the unique zero of (of order 2), so that is a Riemann surface of genus 2 and . Also, it can be checked that the four closed loops of obtained by projecting to the four sides from are a basis of the absolute homology group . It follows that, in this case, the period map in a small neighborhood of is
where is a small neighborhood of . Consequently, we see that any Abelian differential sufficiently close to can be obtained geometrically by small arbitrary perturbations of the sides of our initial polygon .
Remark 4 The introduction of nice affine complex structures in the case of non-orientable quadratic differentials is slightly different from the case of Abelian differentials. Given a non-orientable quadratic differential on a genus Riemann surface , there is a canonical (orienting) double-cover such that , where is an Abelian differential on . Here, is a connected Riemann surface because is non-orientable (otherwise it would be two disjoint copies of ). Also, by writing where are odd integers and are even integers, one can check that is ramified over singularities of odd orders while is regular (i.e., it has two pre-images) over singularities of even orders . It follows that
In particular, Riemann-Hurwitz formula implies that the genus of is . Denote by the non-trivial involution such that (i.e., interchanges the sheets of the double-covering ). Observe that induces an involution acting on the cohomology group , where , is the set of singularities of and are the (simple) poles of . Since is an involution, we can decompose
where is the subspace of -invariant cycles and is the subspace of -anti-invariant cycles. Observe that the Abelian differential is –anti-invariant: indeed, since is , we have that ; if were -invariant, it would follow that , i.e., would be the square of an Abelian differential, and hence an orientable quadratic differential (a contradiction). From this, it is not hard to believe that one can prove that a small neighborhood of can be identified with a small neighborhood of the -anti-invariant Abelian differential inside the anti-invariant subspace . Again, the changes of coordinates are locally affine, so that also comes equipped with a affine complex structure and a natural Lebesgue measure.
At this stage, the terminology stratification used in the previous Subsection is completely justified by now and we pass to a brief discussion of the topology of our strata.
1.3. Connectedness of strata
At first sight, it might be tempting to say that the strata are always connected, i.e., once we fix the list of orders of zeroes, one might conjecture that there are no further obstructions to deform an arbitrary Abelian differential into another arbitrary Abelian differential . However, W. Veech discovered that the stratum has two connected components. In fact, W. Veech distinguished the connected components of by means of a combinatorial invariants called extended Rauzy classes, a slightly modified version of Rauzy classes, a combinatorial invariant (composed of pair of permutations with symbols with , where is the genus and is the number of zeroes) introduced by G. Rauzy in his study of interval exchange transformations. Roughly speaking, W. Veech showed that there is a bijective correspondence between connected components of strata and extended Rauzy classes, and, using this fact, he concluded that has two connected components because he checked (by hand) that there are precisely two extended Rauzy classes associated to this stratum. By a similar strategy, P. Arnoux proved that the stratum has three connected components. In principle, one could think of pursing the strategy of describing extended Rauzy classes to determine the connected components of strata, but this is a hard combinatorial problem: for instance, when trying to compute extended Rauzy classes associated to connected components of strata of Abelian differentials of genus , one should perform several combinatorial operations with pairs of permutations on an alphabet of letters. Nevertheless, M. Kontsevich and A. Zorich managed to classify completely the connected components of strata of Abelian differentials with the aid of some invariants from algebraic and geometrical nature: technically speaking, there are exactly three types of connected components of strata — hyperelliptic, even spin and odd spin. The outcome of Kontsevich-Zorich classification are the following results:
Theorem 1 (M. Kontsevich and A. Zorich) Fix .
- the minimal stratum has connected components;
- , , has connected components;
- any , , has connected components;
- , , has connected components;
- all other strata of Abelian differentials of genus are connected.
The classification of the connected components of strata of Abelian differentials of genus and are slightly different:
Theorem 2 (M. Kontsevich and A. Zorich) In genus , the strata and are connected. In genus , both of the strata and have two connected components, while the other strata in genus are connected.
For more details, we strongly recommend the original article by M. Kontsevich and A. Zorich.
Remark 5The classification of connected components of strata of non-orientable quadratic differentials was performed by E. Lanneau. For genus , each of the four strata , , , have exactly two connected components, each of the following strata
also have exactly two connected components, and all other strata are connected. For genus , we have that any strata in genus 0 and 1 are connected, and, in genus 2, , have exactly two connected components each, while the other strata are connected.
This ends our post. Next time, we will begin talking about dynamics on the (connected components of strata of) moduli spaces of Abelian differentials.