Duke Mathematical Journal

Compactification of strata of Abelian differentials

Matt Bainbridge, Dawei Chen, Quentin Gendron, Samuel Grushevsky, and Martin Möller

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We describe the closure of the strata of Abelian differentials with prescribed type of zeros and poles, in the projectivized Hodge bundle over the Deligne–Mumford moduli space of stable curves with marked points. We provide an explicit characterization of pointed stable differentials in the boundary of the closure, both a complex analytic proof and a flat geometric proof for smoothing the boundary differentials, and numerous examples. The main new ingredient in our description is a global residue condition arising from a full order on the dual graph of a stable curve.

Article information

Duke Math. J., Volume 167, Number 12 (2018), 2347-2416.

Received: 23 March 2017
Revised: 12 March 2018
First available in Project Euclid: 10 August 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14H10: Families, moduli (algebraic)
Secondary: 32G15: Moduli of Riemann surfaces, Teichmüller theory [See also 14H15, 30Fxx]

Abelian differentials flat surfaces moduli space of stable curves Teichmüller dynamics plumbing


Bainbridge, Matt; Chen, Dawei; Gendron, Quentin; Grushevsky, Samuel; Möller, Martin. Compactification of strata of Abelian differentials. Duke Math. J. 167 (2018), no. 12, 2347--2416. doi:10.1215/00127094-2018-0012. https://projecteuclid.org/euclid.dmj/1533866574

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  • [1] W. Abikoff, Degenerating families of Riemann surfaces, Ann. of Math. (2) 105 (1977), 29–44.
  • [2] M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra, Addison-Wesley, Reading, Mass., 1969.
  • [3] M. Bainbridge, Euler characteristics of Teichmüller curves in genus two, Geom. Topol. 11 (2007), 1887–2073.
  • [4] M. Bainbridge, D. Chen, Q. Gendron, S. Grushevsky, and M. Möller, Boundary structure of strata of abelian differentials, in preparation.
  • [5] C. Boissy, Connected components of the strata of the moduli space of meromorphic differentials, Comment. Math. Helv. 90 (2015), 255–286.
  • [6] C. Boissy, Moduli space of meromorphic differentials with marked horizontal separatrices, preprint, arXiv:1507.00555v4 [math.GT].
  • [7] D. Chen, Degenerations of Abelian differentials, J. Differential Geom. 107 (2017), 395–453.
  • [8] D. Chen and Q. Chen, Principal boundary of moduli spaces of abelian and quadratic differentials, to appear in Ann. Inst. Fourier (Grenoble), preprint, arXiv:1611.01591v1 [math.AG].
  • [9] D. Eisenbud and J. Harris, Limit linear series: Basic theory, Invent. Math. 85 (1986), 337–371.
  • [10] D. Eisenbud and J. Harris, Existence, decomposition, and limits of certain Weierstrass points, Invent. Math. 87 (1987), 495–515.
  • [11] A. Eskin, M. Kontsevich, and A. Zorich, Sum of Lyapunov exponents of the Hodge bundle with respect to the Teichmüller geodesic flow, Publ. Math. Inst. Hautes Études Sci. 120 (2007), 207–333.
  • [12] A. Eskin, H. Masur, and A. Zorich, Moduli spaces of abelian differentials: The principal boundary, counting problems, and the Siegel-Veech constants, Publ. Math. Inst. Hautes Études Sci. 97 (2003), 61–179.
  • [13] A. Eskin and M. Mirzakhani, Invariant and stationary measures for the $\operatorname{SL}(2,\mathbb{R})$ action on moduli space, Publ. Math. Inst. Hautes Études Sci. 127 (2018), 95–324.
  • [14] A. Eskin, M. Mirzakhani, and A. Mohammadi, Isolation, equidistribution, and orbit closures for the $\operatorname{SL}(2,\mathbb{R})$ action on moduli space, Ann. of Math. (2) 182 (2015), 673–721.
  • [15] A. Eskin, M. Mirzakhani, and K. Rafi, Counting closed geodesics in strata, to appear in Invent. Math., preprint, arXiv:1206.5574v1 [math.GT].
  • [16] E. Esteves and N. Medeiros, Limit canonical systems on curves with two components, Invent. Math. 149 (2002), 267–338.
  • [17] G. Farkas and R. Pandharipande, The moduli space of twisted canonical divisors, J. Inst. Math. Jussieu 17 (2018), 615–672.
  • [18] S. Filip, Splitting mixed Hodge structures over affine invariant manifolds, Ann. of Math. (2) 183 (2016), 681–713.
  • [19] O. Forster, Lectures on Riemann Surfaces, Grad. Texts in Math. 81, Springer, New York, 1991.
  • [20] Q. Gendron, The Deligne-Mumford and the incidence variety compactifications of the strata of $\Omega\mathcal{M}_{g}$, Ann. Inst. Fourier (Grenoble) 68 (2018), 1169–1240.
  • [21] S. Grushevsky, I. Krichever, and C. Norton, Real-normalized differentials: Limits on stable curves, preprint, arXiv:1703.07806v1 [math.AG].
  • [22] X. Hu, The locus of plane quartics with a hyperflex, Proc. Amer. Math. Soc. 145 (2017), 1399–1413.
  • [23] J. Hubbard and S. Koch, An analytic construction of the Deligne-Mumford compactification of the moduli space of curves, J. Differential Geom. 98 (2014), 261–313.
  • [24] M. Kontsevich, lecture, August 2008.
  • [25] M. Kontsevich and A. Zorich, Connected components of the moduli spaces of Abelian differentials with prescribed singularities, Invent. Math. 153 (2003), 631–678.
  • [26] S. Lang, Real and Functional Analysis, 3rd ed., Grad. Texts in Math. 142, Springer, New York, 1993.
  • [27] B. Lin and M. Ulirsch, “Towards a tropical Hodge bundle” in Combinatorial Algebraic Geometry, Fields Inst. Commun. 80, Fields Inst. Res. Math. Sci, Toronto, 2017, 353–368.
  • [28] M. Mirzakhani and A. Wright, The boundary of an affine invariant submanifold, Invent. Math. 209 (2017), 927–984.
  • [29] M. Möller, M. Ulirsch, and A. Werner, Realizability of tropical canonical divisors, preprint, arXiv:1710.06401v2 [math.AG].
  • [30] S. Mullane, On the effective cone of ${\overline{\mathcal{M}}}_{g,n}$, Adv. Math. 320 (2017), 500–519.
  • [31] K. Rafi, Thick-thin decomposition for quadratic differentials, Math. Res. Lett. 14 (2007), 333–341.
  • [32] A. Sauvaget, Cohomology classes of strata of differentials, preprint, arXiv:1701.07867v2 [math.AG].
  • [33] K. Strebel, Quadratic Differentials, Ergeb. Math. Grenzgeb. (3) 5, Springer, Berlin, 1984.
  • [34] E. F. Whittlesey, Analytic functions in Banach spaces, Proc. Amer. Math. Soc. 16 (1965), 1077–1083.
  • [35] S. A. Wolpert, Infinitesimal deformations of nodal stable curves, Adv. Math. 244 (2013), 413–440.