Principal boundary of moduli spaces of abelian and quadratic differentials.
(Limite principale des espaces de modules des différentielles abéliennes et quadratiques.)

*(English. French summary)*Zbl 1419.14034The paper under review studies the flat structures of Riemann surfaces induced from abelian differentials, where the zeros of differentials correspond to the saddle points of flat surfaces.

The loci of abelian differentials for a given type of zeros form a natural stratification of the moduli space of abelian differentials. For a given configuration of saddle connections for a stratum of flat surfaces, the number of collections of saddle connections with bounded lengths has quadratic asymptotic growth, whose leading coefficient is called the Siegel-Veech constant. In [Publ. Math., Inst. Hautes Étud. Sci., 97, 61–179 (2003; Zbl. 1037.32013)], A. Eskin et al., gave a complete description of all possible configurations of parallel saddle connections on a generic flat surface, and also a method to compute the Siegel-Veech constant. A key step for this calculation is to describe the so-called principal boundary.

The goal of this paper is to give for each configuration a complete description of the principal boundary in terms of twisted differentials over pointed stable curves. This is obtained by splitting the study into principal boundary of type I and of type II according to the work by Eskin et al. [loc. cit.]. This study is performed in Sections 2 and 3 respectively, and it leads to Theorems 2.1 and 3.4, which jointly describe the principal boundary.

If a stratum is disconnected, then the extra connected components are due to spin and hyperelliptic structures. While Eskin et al. described in [loc. cit.] how to distinguish these structures by an analytic approach, Subsections 4.6 and 4.7 of the present paper give an algebraic proof for that distinction. Finally, in Section 5 a description of the principal boundary is given in terms of twisted quadratic differentials. A number of examples taken from related works are used throughout the paper in order to show how the method developed by the authors works explicitly. In this sense Subsection 5.3 with an example in genus 13 is specially noteworthy.

The loci of abelian differentials for a given type of zeros form a natural stratification of the moduli space of abelian differentials. For a given configuration of saddle connections for a stratum of flat surfaces, the number of collections of saddle connections with bounded lengths has quadratic asymptotic growth, whose leading coefficient is called the Siegel-Veech constant. In [Publ. Math., Inst. Hautes Étud. Sci., 97, 61–179 (2003; Zbl. 1037.32013)], A. Eskin et al., gave a complete description of all possible configurations of parallel saddle connections on a generic flat surface, and also a method to compute the Siegel-Veech constant. A key step for this calculation is to describe the so-called principal boundary.

The goal of this paper is to give for each configuration a complete description of the principal boundary in terms of twisted differentials over pointed stable curves. This is obtained by splitting the study into principal boundary of type I and of type II according to the work by Eskin et al. [loc. cit.]. This study is performed in Sections 2 and 3 respectively, and it leads to Theorems 2.1 and 3.4, which jointly describe the principal boundary.

If a stratum is disconnected, then the extra connected components are due to spin and hyperelliptic structures. While Eskin et al. described in [loc. cit.] how to distinguish these structures by an analytic approach, Subsections 4.6 and 4.7 of the present paper give an algebraic proof for that distinction. Finally, in Section 5 a description of the principal boundary is given in terms of twisted quadratic differentials. A number of examples taken from related works are used throughout the paper in order to show how the method developed by the authors works explicitly. In this sense Subsection 5.3 with an example in genus 13 is specially noteworthy.

Reviewer: José Javier Etayo (Madrid)

##### MSC:

14H10 | Families, moduli of curves (algebraic) |

14H15 | Families, moduli of curves (analytic) |

30F30 | Differentials on Riemann surfaces |

32G15 | Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) |

##### Keywords:

moduli space of stable curves; abelian differential; principal boundary; spin and hyperelliptic structures##### References:

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