Masur\u2013Veech volumes and intersection theory on moduli spaces of Abelian differentials

We study relations between Rauzy classes coming from an interval exchange map and the corresponding connected components of strata of the moduli space of Abelian differentials. This gives a criterion to decide whether two permutations are in the same Rauzy class or not, without actually computing them. We prove a similar result for Rauzy classes corresponding to quadratic differentials.

For the moduli spaces of Abelian differentials, the Euler characteristic is one of the most intrinsic topological invariants. We give a formula for the Euler characteristic that relies on intersection theory on the smooth compactification by multi-scale differentials. It is a consequence of a formula for the full Chern polynomial of the cotangent bundle of the compactification.The main new technical tools are an Euler sequence for the cotangent bundle of the moduli space of multi-scale differentials and computational tools in the Chow ring, such as a description of normal bundles to boundary divisors.

The seminal work of Eskin–Masur–Zorich described the principal boundary of moduli spaces of abelian differentials that parameterizes flat surfaces with a prescribed generic configuration of short parallel saddle connections. In this paper we describe the principal boundary for each configuration in terms of twisted differentials over Deligne–Mumford pointed stable curves. We also describe similarly the principal boundary of moduli spaces of quadratic differentials originally studied by Masur–Zorich. Our main technique is the flat geometric degeneration and smoothing developed by Bainbridge–Chen–Gendron–Grushevsky–Möller.

A Correction to this paper has been published: https://doi.org/10.1007/s00222-020-00969-4

The strata of the moduli spaces of Abelian differentials are non-homogenous spaces carrying natural bi-algebraic structures. Partly inspired by the case of homogenous spaces carrying bi-algebraic structures (such as torii, Abelian varieties and Shimura varieties), Klingler and Lerer recently showed that any bi-algebraic curve in a stratum of the moduli space of Abelian differentials is linear provided that the so-called condition (☆) is fulfilled.In this note, we construct a non-linear bi-algebraic curve, resp. surface, of Abelian differentials of genus 7, resp. 10.

We present an approach for counting the Teichmuller volumes of the moduli spaces of Abelian differentials on a Riemann surface of genus g. We show that the volumes can be counted by means of counting the “integer points” in the corresponding moduli space. The “integer points” are represented by square tiled surfaces — the flat surfaces tiled by unit squares. Such tilings have several conical singularities with 8, 12,... adjacent unit squares. Counting the leading term in the asymptotics of the number of tilings having at most N unit squares, we get the volumes of the corresponding strata of the moduli spaces.

The Teichmuller flow gt on the moduli space of Abelian differentials with zeros of given orders on a Riemann surface of a given genus is considered. This flow is known to preserve a finite absolutely continuous measure and is ergodic on every connected component ℋ of the moduli space. The main result of the paper is that µ/µ(ℋ) is the unique measure with maximal entropy for the restriction of gt to ℋ. The proof is based on the symbolic representation of gt.

Large deviation rates are obtained for suspension flows over symbolic dynamical systems with a countable alphabet. We use a method employed previously by the first author [Large deviations bound for semiflows over a non-uniformly expanding base.Bull. Braz. Math. Soc. (N.S.)38(3) (2007), 335–376], which follows that of Young [Some large deviation results for dynamical systems.Trans. Amer. Math. Soc.318(2) (1990), 525–543]. As a corollary of the main results, we obtain a large deviation bound for the Teichmüller flow on the moduli space of abelian differentials, extending earlier work of Athreya [Quantitative recurrence and large deviations for Teichmuller geodesic flow.Geom. Dedicata119(2006), 121–140].

The main result of the paper is the statement that the 'smooth' measure of Masur and Veech is the unique measure of maximal entropy for the Teichmüller flow on the moduli space of Abelian differentials. The proof is based on the symbolic representation of the flow in Veech's space of zippered rectangles.Bibliography: 29 titles.

The main goal of this work is to construct and study a reasonable compactification of the strata of the moduli space of Abelian differentials. This allows us to compute the Kodaira dimension of some strata of the moduli space of Abelian differentials. The main ingredients to study the compactifications of the strata are a version of the plumbing cylinder construction for differential forms and an extension of the parity of the connected components of the strata to the differentials on curves of compact type. We study in detail the compactifications of the hyperelliptic minimal strata and of the odd minimal stratum in genus three.

We study the translation surfaces corresponding to meromorphic differentials on compact Riemann surfaces. Such geometric structures naturally appear when studying compactifications of the strata of the moduli space of Abelian differentials. We compute the number of connected components of the strata of the moduli space of meromorphic differentials. We show that in genus greater than or equal to two, one has up to three components with a similar description as the one of Kontsevich–Zorich for the moduli space of Abelian differentials. In genus one, one can obtain an arbitrarily large number of connected components that are distinguished by a simple topological invariant.

The aim of this paper is to prove a stretched-exponential bound for the decay of correlations for the Rauzy-Veech-Zorich induction map on the space of interval exchange transformations. A corollary is the Central Limit Theorem for the Teichmüller flow on the moduli space of abelian differentials with prescribed singularities.

A holomorphic 1-form on a compact Riemann surface S naturally defines a flat metric on S with cone-type singularities. We present the following surprising phenomenon: having found a geodesic segment (saddle connection) joining a pair of conical points one can find with a nonzero probability another saddle connection on S having the same direction and the same length as the initial one. A similar phenomenon is valid for the families of parallel closed geodesics.We give a complete description of all possible configurations of parallel saddle connections (and of families of parallel closed geodesics) which might be found on a generic flat surface S. We count the number of saddle connections of length less than L on a generic flat surface S; we also count the number of admissible configurations of pairs (triples,...) of saddle connections; we count the analogous numbers of configurations of families of closed geodesics. By the previous result of [EMa] these numbers have quadratic asymptotics c·(πL2). Here we explicitly compute the constant c for a configuration of every type. The constant c is found from a Siegel–Veech formula.To perform this computation we elaborate the detailed description of the principal part of the boundary of the moduli space of holomorphic 1-forms and we find the numerical value of the normalized volume of the tubular neighborhood of the boundary. We use this for evaluation of integrals over the moduli space.

We provide formulas for the Chern classes of linear submanifolds of the moduli spaces of Abelian differentials and hence for their Euler characteristic. This includes as special case the moduli spaces of k -differentials, for which we set up the full intersection theory package and implement it in the SageMath package . As an application, we give an algebraic proof of the theorems of Deligne–Mostow and Thurston that suitable compactifications of moduli spaces of k -differentials on the 5 -punctured projective line with weights satisfying the INT-condition are quotients of the complex two-ball.

The boundary of the multi-scale differential compactification of strata of abelian differentials admits an explicit combinatorial description. However, even for low-dimensional strata, the complexity of the boundary requires use of a computer. We give a description of the algorithms implemented in the SageMath package diffstrata to enumerate the boundary components and perform intersection theory on this space. In particular, the package can compute the Euler characteristic of strata using the methods developed by the same authors.