Posted by: matheuscmss | November 2, 2010

## Lyapunov spectrum of Kontsevich-Zorich cocycle on the Hodge bundle of square-tiled cyclic covers II

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 ${\mathcal{L}_g}$ the set of Abelian differentials on a Riemann surface of genus ${g\geq 1}$, or more precisely, the set of pairs ${(\textrm{Riemann surface structure on } M, \omega)}$ where ${M}$ is a compact topological surface of genus ${g}$ and ${\omega}$ is a ${1}$-form which is holomorphic with respect to the underlying Riemann surface structure. In this notation, the Teichmüller space of Abelian differentials is the quotient ${\mathcal{TH}_g:=\mathcal{L}_g/\textrm{Diff}^+_0(M)}$ and the moduli space of Abelian differentials is the quotient ${\mathcal{H}_g:=\mathcal{L}_g/\Gamma_g}$. Here ${\textrm{Diff}^+_0(M)}$ and ${\Gamma_g:=\textrm{Diff}^+(M)/\textrm{Diff}^+_0(M)}$ (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 ${\mathcal{L}_g}$, ${\mathcal{TH}_g}$ and ${\mathcal{H}_g}$.

1.1. Stratification

Given a non-trivial Abelian differential ${\omega}$ on a Riemann surface of genus ${g\geq 1}$, we can form a list ${\kappa=(k_1,\dots,k_{\sigma})}$ by collecting the multiplicities of the (finite set of) zeroes of ${\omega}$. Observe that any such list ${\kappa=(k_1,\dots,k_{\sigma})}$ verifies the constraint ${\sum\limits_{l=1}^{\sigma} k_l=2g-2}$ in view of Poincaré-Hopf theorem (or alternatively Gauss-Bonnet theorem). Given an unordered list ${\kappa=(k_1,\dots,k_{\sigma})}$ with ${\sum\limits_{l=1}^{\sigma}k_l=2g-2}$, the set ${\mathcal{L}(\kappa)}$ of Abelian differentials whose list of multiplicities of its zeroes coincide with ${\kappa}$ is called a stratum of ${\mathcal{L}_g}$. Since the actions of ${\textrm{Diff}^+_0(M)}$ and ${\Gamma_g}$ respect the multiplicities of zeroes, the quotients ${\mathcal{TH}(\kappa):=\mathcal{L}(\kappa)/\textrm{Diff}^+_0(M)}$ and ${\mathcal{H}(\kappa):=\mathcal{L}(\kappa)/\Gamma_g}$ are well-defined. By obvious reasons, ${\mathcal{TH}(\kappa)}$ and ${\mathcal{H}(\kappa)}$ are called stratum of ${\mathcal{TH}_g}$ and ${\mathcal{H}_g}$ (resp.). Notice that, by definition,

$\displaystyle \mathcal{L}_g=\bigsqcup\limits_{\kappa=(k_1,\dots,k_{\sigma}), k_1+\dots+k_{\sigma}=2g-2}\mathcal{L}(\kappa),$

$\displaystyle \mathcal{TH}_g=\bigsqcup\limits_{\kappa=(k_1,\dots,k_{\sigma}), k_1+\dots+k_{\sigma}=2g-2}\mathcal{TH}(\kappa),$

and

$\displaystyle \mathcal{H}_g=\bigsqcup\limits_{\kappa=(k_1,\dots,k_{\sigma}), k_1+\dots+k_{\sigma}=2g-2}\mathcal{H}(\kappa).$

In other words, the sets ${\mathcal{L}_g}$, ${\mathcal{TH}_g}$ and ${\mathcal{H}_g}$ are naturally “decomposed” into the subsets (strata) ${\mathcal{L}(\kappa)}$, ${\mathcal{TH}(\kappa)}$ and ${\mathcal{H}(\kappa)}$. 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 ${X}$ is a decomposition ${X=\bigcup\limits_{i\in I} X_i}$ where ${I}$ is a finite set of indices and the strata ${X_i}$ are disjoint manifolds/orbifolds of distinct dimensions. Thus, one can’t call stratification our decomposition of ${\mathcal{L}_g}$, ${\mathcal{TH}_g}$ and ${\mathcal{H}_g}$ until we put nice manifold/orbifold structures (of different dimension) on the corresponding strata. The introduction of nice complex-analytic manifold/orbifold structures on ${\mathcal{TH}(\kappa)}$ and ${\mathcal{H}(\kappa)}$ are the topic of our next subsection.

Remark 1 The curious reader might be asking at this point whether the strata ${\mathcal{H}(\kappa)}$ are non-empty. In fact, this is a natural question because, while the condition ${\sum\limits_{s=1}^{\sigma}k_s=2g-2}$ 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 ${\mathcal{H}(\kappa)}$ of Abelian differentials are non-empty whenever ${\sum\limits_{s=1}^{\sigma}k_s=2g-2}$. For comparison, we note that exact analog result for strata of non-orientable quadratic differentials is false. Indeed, if we denote by ${\mathcal{Q}(d_1,\dots,d_m,-1^p)}$ the stratum of non-orientable quadratic differentials with ${m}$ zeroes of orders ${d_1,\dots,d_m\geq 1}$ and ${p}$ simple poles, the Poincaré-Hopf theorem says that a necessary condition is ${\sum\limits_{l=1}^m d_l -p = 4g-4}$, and a theorem of H. Masur and J. Smillie says that this condition is almost sufficient: except for the empty strata ${\mathcal{Q}(\emptyset), \mathcal{Q}(1,-1), \mathcal{Q}(1,3), \mathcal{Q}(4)}$ 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 ${\mathcal{TH}(\kappa)}$ be a stratum, say ${\kappa=(k_1,\dots,k_{\sigma})}$, ${\sum\limits_{l=1}^{\sigma} k_l=2g-2}$. Given an Abelian differential ${\omega\in\mathcal{TH}(\kappa)}$, we denote by ${\Sigma(\omega)}$ the set of zeroes of ${\omega}$. It is possible to prove that, for every ${\omega_0\in\mathcal{TH}(\kappa)}$, there is an open set ${\omega_0\in U_0\subset\mathcal{TH}(\kappa)}$ such that, after identifying, for all ${\omega\in U_0}$, the cohomology ${H^1(M,\Sigma(\omega),\mathbb{C})}$ with the fixed complex vector space ${H^1(M,\Sigma(\omega_0),\mathbb{C})}$ via the Gauss-Manin connection (i.e., through identification of the integer lattices ${H^1(M,\Sigma(\omega),\mathbb{Z}\oplus i\mathbb{Z})}$ and ${H^1(M,\Sigma(\omega_0),\mathbb{Z}\oplus i\mathbb{Z})}$), the period map ${\Theta: U_0\rightarrow H^1(M,\Sigma(\omega_0),\mathbb{C})}$ given by

$\displaystyle \Theta(\omega):=\left(\gamma\rightarrow\int_{\gamma}\omega\right)\in\textrm{Hom}(H_1(M,\Sigma(\omega),\mathbb{Z}),\mathbb{C})\simeq H^1(M,\Sigma(\omega),\mathbb{C})$

$\displaystyle \simeq H^1(M,\Sigma(\omega_0),\mathbb{C})$

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 ${\mathcal{TH}(\kappa)}$, it can be applied to understand the connected components of ${\mathcal{H}(\kappa)}$, make volume estimates for ${\mu_{\kappa}}$, study the dynamics of Teichmüller flow on ${\mathcal{H}(\kappa)}$ through combinatorial methods (Markov partitions), etc. In other words, Veech’s zippered rectangles is a powerful tool to study global geometry and dynamics on ${\mathcal{H}(\kappa)}$. However, since the long-term goal of our future discussions is not the study of the entire strata ${\mathcal{H}(\kappa)}$, 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 ${H^1(M,\Sigma(\omega_0),\mathbb{C})}$ is a complex vector space isomorphic to ${\mathbb{C}^{2g+\sigma-1}}$. An explicit way to see this fact is by taking ${\{\alpha_j,\beta_j\}_{j=1}^{g}}$ a basis of the absolute homology group ${H_1(M,\mathbb{Z})}$, and relative cycles ${\gamma_1, \dots, \gamma_{\sigma-1}}$ joining an arbitrarily chosen point ${p_0\in\Sigma(\omega_0)}$ to the other points ${p_1,\dots,p_{\sigma-1}\in\Sigma(\omega)}$. Then, the map

$\displaystyle \omega\in H^1(M,\Sigma(\omega_0),\mathbb{C})\mapsto \left(\int_{\alpha_1}\omega,\int_{\beta_1}\omega,\dots,\int_{\alpha_g}\omega,\int_{\beta_g}\omega, \int_{\gamma_1}\omega,\dots,\int_{\gamma_l}\omega\right)$

$\displaystyle \in\mathbb{C}^{2g+\sigma-1}$

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 ${\mathbb{C}^{2g+\sigma-1}}$. In particular, we can also pullback Lebesgue measure on ${\mathbb{C}^{2g+\sigma-1}}$ to get a natural measure ${\lambda_{\kappa}}$ on ${\mathcal{TH}(\kappa)}$: indeed, ${\lambda_{\kappa}}$ is well-defined modulo normalization because the changes of coordinates are affine maps; to remedy the normalization problem, we ask that the integral lattice ${H^1(M,\Sigma(\omega_0),\mathbb{Z}\oplus i\mathbb{Z})}$ has covolume 1, i.e., the volume of the torus ${H^1(M,\Sigma(\omega_0),\mathbb{C})/H^1(M,\Sigma(\omega_0),\mathbb{Z}\oplus i\mathbb{Z})}$ is 1.

In resume, by using the period maps as local coordinates, ${\mathcal{TH}(\kappa)}$ has a structure of affine complex manifold of complex dimension ${2g+\sigma-1}$ and a natural (Lebesgue) measure ${\lambda_{\kappa}}$. Also, it is not hard to check that the action of the modular group ${\Gamma_g:=\textrm{Diff}^+(M)/\textrm{Diff}_0^+(M)}$ is compatible with the affine complex manifold structure on ${\mathcal{TH}(\kappa)}$, so that, by passing to the quotient, one gets that the stratum ${\mathcal{H}(\kappa)}$ of the moduli space of Abelian differentials has the structure of a affine complex orbifold and a natural Lebesgue measure ${\mu_{\kappa}}$.

Remark 3 Note that, as we emphasized above, after passing to the quotient, ${\mathcal{H}(\kappa)}$ is an affine complex orbifold at best. In fact, we can’t expect ${\mathcal{H}(\kappa)}$ 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, ${\mathcal{H}(\emptyset)}$ 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 ${\mathcal{TH}(\kappa)}$ (and not of its cousin ${\mathcal{H}(\kappa)}$).

The following example shows a concrete way to geometrically interpret the role of period maps as local coordinates of the strata ${\mathcal{TH}(\kappa)}$.

Example 1 In the picture above (extracted from Zorich’s survey, Figure 12), we have a polygon ${P}$ whose corresponding sides ${v_j}$, ${j=1,2,3,4}$, are identified by translation. After performing these identification, we obtain a Riemann surface ${M}$ equipped with an Abelian differential ${\omega_0}$ (obtained as the quotient of ${dz}$ on the polygon ${P\subset\mathbb{C}}$ under the side identifications). It is not to hard to infer from the picture that the vertices of polygon ${P}$ are all identified to the same point ${p\in M}$. Furthermore, since the total angle around ${p}$ is ${6\pi}$ and ${p}$ is the unique zero of ${\omega_0}$ (of order 2), so that ${M}$ is a Riemann surface of genus 2 and ${\omega_0\in\mathcal{TH}(2)}$. Also, it can be checked that the four closed loops ${\alpha_1,\alpha_2,\alpha_3,\alpha_4}$ of ${M}$ obtained by projecting to ${M}$ the four sides ${v_1,v_2,v_3,v_4}$ from ${P}$ are a basis of the absolute homology group ${H_1(M,\mathbb{Z})}$. It follows that, in this case, the period map in a small neighborhood ${U_0}$ of ${\omega_0}$ is$\displaystyle \omega\in U_0\subset\mathcal{TH}(2)\mapsto \left(\int_{\alpha_1}\omega,\int_{\alpha_2}\omega,\int_{\alpha_3}\omega,\int_{\alpha_4}\omega\right)\in V_0\subset\mathbb{C}^4,$

where ${V_0}$ is a small neighborhood of ${(v_1,v_2,v_3,v_4)}$. Consequently, we see that any Abelian differential ${\omega\in\mathcal{H}(2)}$ sufficiently close to ${\omega_0}$ can be obtained geometrically by small arbitrary perturbations of the sides ${v_1,v_2,v_3,v_4}$ of our initial polygon ${P}$.

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 ${q\in\mathcal{Q}(\kappa)}$ on a genus ${g}$ Riemann surface ${M}$, there is a canonical (orienting) double-cover ${\pi_{\kappa}:\widehat{M}\rightarrow M}$ such that ${\pi_{\kappa}^*(q)=\widehat{\omega}^2}$, where ${\widehat{\omega}}$ is an Abelian differential on ${\widehat{M}}$. Here, ${\widehat{M}}$ is a connected Riemann surface because ${q}$ is non-orientable (otherwise it would be two disjoint copies of ${M}$). Also, by writing ${\kappa=(o_1,\dots,o_{\tau}, e_1,\dots, e_{\nu})}$ where ${o_j}$ are odd integers and ${e_j}$ are even integers, one can check that ${\pi_{\kappa}}$ is ramified over singularities of odd orders ${o_j}$ while ${\pi_{\kappa}}$ is regular (i.e., it has two pre-images) over singularities of even orders ${e_j}$. It follows that

$\displaystyle \widehat{\omega}\in\mathcal{H}\left(o_1+1,\dots,o_{\tau}+1,\frac{e_1}{2},\frac{e_1}{2},\dots,\frac{e_{\nu}}{2},\frac{e_{\nu}}{2}\right).$

In particular, Riemann-Hurwitz formula implies that the genus ${\widehat{g}}$ of ${\widehat{M}}$ is ${\widehat{g} = 2g-1+\tau/2}$. Denote by ${\sigma:\widehat{M}\rightarrow\widehat{M}}$ the non-trivial involution such that ${\pi_\kappa\circ\sigma=\pi_{\kappa}}$ (i.e., ${\sigma}$ interchanges the sheets of the double-covering ${\pi_{\kappa}}$). Observe that ${\sigma}$ induces an involution ${\sigma_*}$ acting on the cohomology group ${H^1(\widehat{M},\widehat{\Sigma_\kappa},\mathbb{C})}$, where ${\widehat{\Sigma_\kappa}:=\pi_{\kappa}^{-1}(\Sigma_\kappa)-\pi_\kappa^{-1}(\{p_1,\dots,p_{\eta}\})}$, ${\Sigma_{\kappa}}$ is the set of singularities of ${q}$ and ${p_1,\dots,p_{\eta}}$ are the (simple) poles of ${q}$. Since ${\sigma_*}$ is an involution, we can decompose

$\displaystyle H^1(\widehat{M},\widehat{\Sigma_\kappa},\mathbb{C})=H_+^1(\widehat{M},\widehat{\Sigma_\kappa},\mathbb{C})\oplus H_-^1(\widehat{M},\widehat{\Sigma_\kappa},\mathbb{C})$

where ${H_+^1(\widehat{M},\widehat{\Sigma_\kappa},\mathbb{C})}$ is the subspace of ${\sigma_*}$-invariant cycles and ${H_-^1(\widehat{M},\widehat{\Sigma_\kappa},\mathbb{C})}$ is the subspace of ${\sigma_*}$-anti-invariant cycles. Observe that the Abelian differential ${\widehat{\omega}}$ is ${\sigma_*}$anti-invariant: indeed, since ${\widehat{\omega}^2}$ is ${\pi_\kappa(q)}$, we have that ${\sigma_*(\widehat{\omega})=\pm\widehat{\omega}}$; if ${\widehat{\omega}}$ were ${\sigma_*}$-invariant, it would follow that ${\pi_{\kappa}^*(\widehat{\omega})^2=q}$, i.e., ${q}$ 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 ${q}$ can be identified with a small neighborhood of the ${\sigma_*}$-anti-invariant Abelian differential ${\widehat{\omega}}$ inside the anti-invariant subspace ${H_-^1(\widehat{M},\widehat{\Sigma_\kappa},\mathbb{C})}$. Again, the changes of coordinates are locally affine, so that ${\mathcal{Q}(\kappa)}$ 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 ${\mathcal{H}(\kappa)}$ are always connected, i.e., once we fix the list ${\kappa}$ of orders of zeroes, one might conjecture that there are no further obstructions to deform an arbitrary Abelian differential ${\omega_0\in\mathcal{H}(\kappa)}$ into another arbitrary Abelian differential ${\omega_1\in\mathcal{H}(\kappa)}$. However, W. Veech discovered that the stratum ${\mathcal{H}(4)}$ has two connected components. In fact, W. Veech distinguished the connected components of ${\mathcal{H}(4)}$ 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 ${d}$ symbols with ${d=2g+s-1}$, where ${g}$ is the genus and ${s}$ 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 ${\mathcal{H}(4)}$ 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 ${\mathcal{H}(6)}$ 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 ${g}$, one should perform several combinatorial operations with pairs of permutations on an alphabet of ${d\geq 2g}$ 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 ${g\geq 4}$.

• the minimal stratum ${\mathcal{H}(2g-2)}$ has ${3}$ connected components;
• ${\mathcal{H}(2l,2l)}$, ${l\geq 2}$, has ${3}$ connected components;
• any ${\mathcal{H}(2l_1,\dots,2l_n)\neq \mathcal{H}(2l,2l)}$, ${l_i\geq 1}$, has ${2}$ connected components;
• ${\mathcal{H}(2l-1,2l-1)}$, ${l\geq 2}$, has ${2}$ connected components;
• all other strata of Abelian differentials of genus ${g}$ are connected.

The classification of the connected components of strata of Abelian differentials of genus ${g=2}$ and ${3}$ are slightly different:

Theorem 2 (M. Kontsevich and A. Zorich) In genus ${2}$, the strata ${\mathcal{H}(2)}$ and ${\mathcal{H}(1,1)}$ are connected. In genus ${3}$, both of the strata ${\mathcal{H}(4)}$ and ${\mathcal{H}(2,2)}$ have two connected components, while the other strata in genus ${3}$ 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 ${g\geq 3}$, each of the four strata ${\mathcal{Q}(9,-1)}$, ${\mathcal{Q}(6,3,-1)}$, ${\mathcal{Q}(3,3,3,-1)}$, ${\mathcal{Q}(12)}$ have exactly two connected components, each of the following strata

${\mathcal{Q}(2j-1,2j-1,2k-1,2k-1)}$ (${j,k\geq 0}$, ${j+k=g}$),

${\mathcal{Q}(2j-1,2j-1,4k+2)}$ (${j,k\geq 0}$, ${j+k=g-1}$),

${\mathcal{Q}(4j+2,4k+2)}$ (${j,k\geq 0}$, ${j+k=g-2}$)

also have exactly two connected components, and all other strata are connected. For genus ${0\leq g\leq 2}$, we have that any strata in genus 0 and 1 are connected, and, in genus 2, ${\mathcal{Q}(6,-1^2)}$, ${\mathcal{Q}(3,3,-1^2)}$ 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.