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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 1419.14034
The 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.

##### 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)
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##### References:
 [1] Atiyah, Michael, Riemann surfaces and spin structures, Ann. Sci. Éc. Norm. Supér., 4, 47-62, (1971) · Zbl 0212.56402 [2] Bainbridge, Matt; Chen, Dawei; Gendron, Quentin; Grushevsky, Samuel; Möller, Martin, A smooth compactification of strata of abelian differentials · Zbl 1403.14058 [3] Bainbridge, Matt; Chen, Dawei; Gendron, Quentin; Grushevsky, Samuel; Möller, Martin, Strata of $$k$$-differentials, (2016) [4] Bainbridge, Matt; Chen, Dawei; Gendron, Quentin; Grushevsky, Samuel; Möller, Martin, Compactification of strata of abelian differentials, Duke Math. J., 167, 12, 2347-2416, (2018) · Zbl 1403.14058 [5] Bauer, Max; Goujard, Élise, Geometry of periodic regions on flat surfaces and associated Siegel-Veech constants, Geom. Dedicata, 174, 203-233, (2015) · Zbl 1308.30052 [6] Boissy, Corentin, Connected components of the strata of the moduli space of meromorphic differentials, Comment. Math. Helv., 90, 2, 255-286, (2015) · Zbl 1323.30060 [7] Chen, Dawei, Degenerations of Abelian differentials, J. Differ. Geom., 107, 3, 395-453, (2017) · Zbl 1388.14080 [8] Chen, Dawei, Surveys on recent developments in algebraic geometry, 95, Teichmüller dynamics in the eyes of an algebraic geometer, 171-197, (2017), American Mathematical Society · Zbl 1393.14021 [9] Chen, Dawei; Chen, Qile, Spin and hyperelliptic structures of log twisted abelian differentials, (2016) · Zbl 1409.14048 [10] Cornalba, Maurizio, Proceedings of the First College on Riemann Surfaces held in Trieste, November 9-December 18, 1987, Moduli of curves and theta-characteristics, 560-589, (1989), World Scientific · Zbl 0800.14011 [11] Eskin, Alex; Masur, Howard, Asymptotic formulas on flat surfaces, Ergodic Theory Dyn. Syst., 21, 2, 443-478, (2001) · Zbl 1096.37501 [12] Eskin, Alex; Masur, Howard; Zorich, Anton, Moduli spaces of Abelian differentials: the principal boundary, counting problems, and the Siegel-Veech constants, Publ. Math., Inst. Hautes Étud. Sci., 97, 61-179, (2003) · Zbl 1037.32013 [13] Farkas, Gavril; Pandharipande, Rahul, The moduli space of twisted canonical divisors, with an appendix by Felix Janda, Rahul Pandharipande, Aaron Pixton, and Dimitri Zvonkine, J. Inst. Math. Jussieu, 17, 3, 615-672, (2018) · Zbl 06868654 [14] Gendron, Quentin, The Deligne-Mumford and the incidence variety compactifications of the strata of $$\Omega \mathcal{M}_{g}$$, Ann. Inst. Fourier, 68, 3, 1169-1240, (2018) · Zbl 1403.14059 [15] Goujard, Élise, Siegel-Veech constants for strata of moduli spaces of quadratic differentials, Geom. Funct. Anal., 25, 5, 1440-1492, (2015) · Zbl 1332.30064 [16] Guéré, Jérémy, A generalization of the double ramification cycle via log-geometry, (2016) · Zbl 1354.14081 [17] Harris, Joe; Mumford, David, On the Kodaira dimension of the moduli space of curves, Invent. Math., 67, 1, 23-88, (1982) · Zbl 0506.14016 [18] Johnson, Dennis, Spin structures and quadratic forms on surfaces, J. Lond. Math. Soc., 22, 365-373, (1980) · Zbl 0454.57011 [19] Kontsevich, Maxim; Zorich, Anton, Connected components of the moduli spaces of Abelian differentials with prescribed singularities, Invent. Math., 153, 3, 631-678, (2003) · Zbl 1087.32010 [20] Masur, Howard; Zorich, Anton, Multiple saddle connections on flat surfaces and the principal boundary of the moduli spaces of quadratic differentials, Geom. Funct. Anal., 18, 3, 919-987, (2008) · Zbl 1169.30017 [21] Mirzakhani, Maryam; Wright, Alex, The boundary of an affine invariant submanifold, Invent. Math., 209, 3, 927-984, (2017) · Zbl 1378.37069 [22] Mumford, David, Theta characteristics of an algebraic curve, Ann. Sci. Éc. Norm. Supér., 4, 181-192, (1971) · Zbl 0216.05904 [23] Veech, William A., Siegel measures, Ann. Math., 148, 3, 895-944, (1998) · Zbl 0922.22003 [24] Wright, Alex, Translation surfaces and their orbit closures: an introduction for a broad audience, EMS Surv. Math. Sci., 2, 1, 63-108, (2015) · Zbl 1372.37090 [25] Zorich, Anton, Frontiers in number theory, physics, and geometry I. On random matrices, zeta functions, and dynamical systems. Papers from the meeting, Les Houches, France, March 9-21, 2003, Flat surfaces, 437-583, (2006), Springer · Zbl 1129.32012
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