# ANNALES DE L'INSTITUT FOURIER

Principal boundary of moduli spaces of abelian and quadratic differentials
Annales de l'Institut Fourier, to appear, 38 p.

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.

Le travail fondateur d’Eskin–Masur–Zorich a décrit la limite principale des espaces de modules des différentielles abéliennes qui paramètre les surfaces plates possédant une configuration générique de petites connexions de selles parallèles prescrite. Dans cet article, nous décrivons la limite principale pour chaque configuration en terme de différentielles entrelacées sur les courbes stables pointées de Deligne–Mumford. Nous décrivons également la limite principale des espaces de modules des différentielles quadratiques étudiée à l’origine par Masur–Zorich. Nos principaux outils sont la dégénérescence géométrique plate et le lissage développés par Bainbridge–Chen–Gendron–Grushevsky–Möller.

Revised : 2017-09-24
Accepted : 2018-02-02
Published online : 2019-03-08
Classification:  14H10,  14H15,  30F30,  32G15
Keywords: Abelian differential, principal boundary, moduli space of stable curves, spin and hyperelliptic structures
@unpublished{AIF_0__0_0_A3_0,
author = {Chen, Dawei and Chen, Qile},
title = {Principal boundary of moduli spaces of abelian and quadratic differentials},
note = {to appear in \emph{Annales de l'Institut Fourier}},
}

Chen, Dawei; Chen, Qile. Principal boundary of moduli spaces of abelian and quadratic differentials. Annales de l'Institut Fourier, to appear, 38 p.

[1] Atiyah, Michael Riemann surfaces and spin structures, Ann. Sci. Éc. Norm. Supér., Tome 4 (1971), pp. 47-62 | Zbl 0212.56402

[2] Bainbridge, Matt; Chen, Dawei; Gendron, Quentin; Grushevsky, Samuel; Möller, Martin A smooth compactification of strata of abelian differentials (in preparation)

[3] Bainbridge, Matt; Chen, Dawei; Gendron, Quentin; Grushevsky, Samuel; Möller, Martin Strata of $k$-differentials (2016) (to appear in Algebr. Geom., https://arxiv.org/abs/1610.09238)

[4] Bainbridge, Matt; Chen, Dawei; Gendron, Quentin; Grushevsky, Samuel; Möller, Martin Compactification of strata of abelian differentials, Duke Math. J., Tome 167 (2018) no. 12, pp. 2347-2416

[5] Bauer, Max; Goujard, Élise Geometry of periodic regions on flat surfaces and associated Siegel-Veech constants, Geom. Dedicata, Tome 174 (2015), pp. 203-233 | Zbl 1308.30052

[6] Boissy, Corentin Connected components of the strata of the moduli space of meromorphic differentials, Comment. Math. Helv., Tome 90 (2015) no. 2, pp. 255-286 | Zbl 1323.30060

[7] Chen, Dawei Degenerations of Abelian differentials, J. Differ. Geom., Tome 107 (2017) no. 3, pp. 395-453 | Zbl 1388.14080

[8] Chen, Dawei Teichmüller dynamics in the eyes of an algebraic geometer, Surveys on recent developments in algebraic geometry, American Mathematical Society (Proceedings of Symposia in Pure Mathematics) Tome 95 (2017), pp. 171-197 | Zbl 1393.14021

[9] Chen, Dawei; Chen, Qile Spin and hyperelliptic structures of log twisted abelian differentials (2016) (https://arxiv.org/abs/1610.05345 )

[10] Cornalba, Maurizio Moduli of curves and theta-characteristics, Proceedings of the First College on Riemann Surfaces held in Trieste, November 9–December 18, 1987, World Scientific (1989), pp. 560-589 | Zbl 0800.14011

[11] Eskin, Alex; Masur, Howard Asymptotic formulas on flat surfaces, Ergodic Theory Dyn. Syst., Tome 21 (2001) no. 2, pp. 443-478 | 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., Tome 97 (2003), pp. 61-179 | 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, Tome 17 (2018) no. 3, pp. 615-672 | Zbl 06868654

[14] Gendron, Quentin The Deligne-Mumford and the incidence variety compactifications of the strata of $\Omega {ℳ}_{g}$, Ann. Inst. Fourier, Tome 68 (2018) no. 3, pp. 1169-1240 | Article

[15] Goujard, Élise Siegel-Veech constants for strata of moduli spaces of quadratic differentials, Geom. Funct. Anal., Tome 25 (2015) no. 5, pp. 1440-1492 | Zbl 1332.30064

[16] Guéré, Jérémy A generalization of the double ramification cycle via log-geometry (2016) (https://arxiv.org/abs/1603.09213 )

[17] Harris, Joe; Mumford, David On the Kodaira dimension of the moduli space of curves, Invent. Math., Tome 67 (1982) no. 1, pp. 23-88 | Article | Zbl 0506.14016

[18] Johnson, Dennis Spin structures and quadratic forms on surfaces, J. Lond. Math. Soc., Tome 22 (1980), pp. 365-373 | Zbl 0454.57011

[19] Kontsevich, Maxim; Zorich, Anton Connected components of the moduli spaces of Abelian differentials with prescribed singularities, Invent. Math., Tome 153 (2003) no. 3, pp. 631-678 | Article | 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., Tome 18 (2008) no. 3, pp. 919-987 | Zbl 1169.30017

[21] Mirzakhani, Maryam; Wright, Alex The boundary of an affine invariant submanifold, Invent. Math., Tome 209 (2017) no. 3, pp. 927-984 | Zbl 1378.37069

[22] Mumford, David Theta characteristics of an algebraic curve, Ann. Sci. Éc. Norm. Supér., Tome 4 (1971), pp. 181-192 | Zbl 0216.05904

[23] Veech, William A. Siegel measures, Ann. Math., Tome 148 (1998) no. 3, pp. 895-944 | Zbl 0922.22003

[24] Wright, Alex Translation surfaces and their orbit closures: an introduction for a broad audience, EMS Surv. Math. Sci., Tome 2 (2015) no. 1, pp. 63-108 | Zbl 1372.37090

[25] Zorich, Anton Flat surfaces, 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, Springer (2006), pp. 437-583 | Zbl 1129.32012