Connected components of the moduli spaces of Abelian differentials with prescribed singularities. (English) Zbl 1087.32010

Kontsevich, Maxim; Zorich, Anton

Invent. Math. 153, No. 3, 631-678 (2003).

For \(g > 1\) define the space \(H_{g}\) as the moduli space of pairs \((C, \omega)\) where \(C\) is a smooth compact complex curve of genus \(g\) and \(\omega\) is a holomorphic 1-form on \(C\) which is not equal identically to zero. \(H_{g}\) is a complex algebraic orbifold of dimension \(4g - 3.\) It is fibered over the moduli space \(M_{g}\) of curves with the fiber over \([C]\in M_{g}\) equal to the punctured vector space \(\Gamma(C, \Omega^{1}_{C}) \backslash {0}\) (modulo the action of a finite group Aut\((C)).\) Let \(k_{1},\dots , k_{n}\) be a sequence of positive integers, with the sum \(\sum_{i}k_{i}\) equal to \(2g - 2.\) Denote by \(H(k_{1},\dots , k_{n})\) the subspace of \(H\) consisting of equivalence classes of pairs \((C, \omega)\) where \(\omega\) has exactly \(n\) zeros and their multiplicities are equal to \(k_{1},\dots , k_{n}.\) One has then \[ H_{g} = \sqcup_{n, (k_{1},\dots , k_{n})} H_{k_{1},\dots , k_{n}}. \]
Thus we have a stratification of the moduli space \(H_{g}.\)
The goal of this paper is to describe the set of connected components of all strata \(H_{k_{1},\dots , k_{n}}.\) This classification of extended Rauzy classes does not use only combinatorics but also tools of algebraic geometry, topology and of dynamical systems. This is important in the study of dynamics of interval exchange transformations and billiards in rational polygons, and in the study of geometry of translation surfaces.
Theorem 1. All connected components of any stratum of Abelian differentials on a curve of genus \(g \geq 4\) are described by the following list:
The stratum H(2g - 2) has three connected components: the hyperelliptic one, \(H^{\text{hyp}}(2g - 2),\) and two other components \(H^{\text{even}}(2g - 2)\) and \(H^{\text{odd}}(2g - 2)\) corresponding to even and odd spin structures.
The stratum \(H(2l, 2l), l \geq 2\) has three connected components: the hyperelliptic one, \(H^{\text{hyp}}(2l, 2l),\) and two other components \(H^{\text{even}}(2l, 2l)\) and \(H^{\text{odd}}(2l, 2l).\)
All the other strata of the form \(H(2l_{1},\dots ,2l_{n}),\) where all \(l_{i} \geq 1,\) have two connected components: \(H^{\text{even}}(2l_{1},\dots ,2l_{n})\) and \(H^{\text{odd}}(2l_{1},\dots ,2l_{n})\) corresponding to even and odd spin structures.
The strata \(H(2l - 1, 2l - 1), l \geq 2\) has two connected components; one of them: \(H^{\text{hyp}}(2l - 1, 2l - 1)\) is hyperelliptic; the other \(H^{\text{nonhyp}}(2l - 1, 2l - 1)\) is not.
All the other strata of Abelian differentials on the curves of genera \(g \geq 4\) are nonempty and connected.

Reviewer: V. V. Chueshev (Kemerovo)

MSC:

32G15	Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
37D40	Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
37D50	Hyperbolic systems with singularities (billiards, etc.) (MSC2010)
37F99	Dynamical systems over complex numbers

Keywords:

moduli space of pairs; smooth compact complex curve of genus; stratification of the moduli space; extended Rauzy classes