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Compactification of strata of abelian differentials. (English) Zbl 1403.14058
Let $$g \geq 2$$ and $${\mathcal M}_{g}$$ be the moduli space of closed Riemann surfaces of genus $$g$$. The Hodge bundle $$\Omega{\mathcal M}_{g}$$ consists of classes of pairs $$(X,\omega)$$, where $$X$$ is a closed Riemann surface and $$\omega$$ is a holomorphic one-form on $$X$$ (if $$X$$ has trivial group of holomorphic automorphisms, then the fiber over a point $$[X] \in {\mathcal M}_{g}$$ can be identified with the $$g$$-dimensional complex vector space $$H^{1,0}(X)$$ of holomorphic one-forms on $$X$$). If $$\Omega{\mathcal M}_{g}^{*}$$ is the complement of the zero section, then the natural action of $${\mathbb C}^{*}={\mathbb C}-\{0\}$$ by multiplication on the second factor provides the projectivized Hodge structure $${\mathbb P}\Omega{\mathcal M}_{g}=\Omega{\mathcal M}_{g}^{*}/{\mathbb C}^{*}$$. For each tuple $$\mu=(m_{1},\ldots,m_{n})$$, where $$m_{i} \geq 1$$ are integers such that $$2g-2=m_{1}+\cdots+m_{n}$$, there is associated the stratum $${\mathbb P}\Omega{\mathcal M}_{g}(\mu)$$ consisiting of those classes for which the holomorphic one-form has $$n$$ zeroes, of respective orders $$m_{1},\ldots,m_{n}$$. If $$\overline{\mathcal M}_{g}$$ denotes the Deligne-Mumford compactification of $${\mathcal M}_{g}$$ (obtained by adding stable Riemann surfaces of genus $$g$$), then the Hodge bundle $$\Omega{\mathcal M}_{g}$$ extends to a bundle $$\Omega\overline{\mathcal M}_{g}$$ of $$\overline{\mathcal M}_{g}$$ (by considering stable differentials on stable Riemann surfaces). Again, one may consider the projectivized one $${\mathbb P}\Omega\overline{\mathcal M}_{g}$$. In this way, the closure of $${\mathbb P}\Omega{\mathcal M}_{g}(\mu)$$ in $${\mathbb P}\Omega\overline{\mathcal M}_{g}$$ provides a compactification of it, called the Hodge bundle compactification. Another compactification of $${\mathbb P}\Omega{\mathcal M}_{g}(\mu)$$ is obtained by lifting it to the moduli space $${\mathcal M}_{g,n}$$ (the $$n$$ marked points being the zeroes of the holomorphic forms) and by taking its closure in $$\overline{\mathcal M}_{g,n}$$, called the Deligne-Mumford compactification.
In the paper under review, the authors consider another compactification by looking $${\mathbb P}\Omega{\mathcal M}_{g}(\mu)$$ inside $${\mathbb P}\Omega\overline{\mathcal M}_{g,n}$$, called the incidence variety compactification. This compactification takes cares of the limit pointed stable holomorphic differentials and also takes cares of the limit of the corresponding zeroes. They also provides an explicit characterization of the points in the boundary of such a compactificaction (this is theorem 1.3). There are two natural projection from the incidence variety compactification to the other two compactifications above (this is explained in corollary 1.4). There are many explicative examples and the paper is very nicely written.

##### MSC:
 14H15 Families, moduli of curves (analytic) 14H10 Families, moduli of curves (algebraic) 32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
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##### References:
  W. Abikoff, Degenerating families of Riemann surfaces, Ann. of Math. (2) 105 (1977), 29–44. · Zbl 0347.32010  M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra, Addison-Wesley, Reading, Mass., 1969. · Zbl 0175.03601  M. Bainbridge, Euler characteristics of Teichmüller curves in genus two, Geom. Topol. 11 (2007), 1887–2073. · Zbl 1131.32007  M. Bainbridge, D. Chen, Q. Gendron, S. Grushevsky, and M. Möller, Boundary structure of strata of abelian differentials, in preparation.  C. Boissy, Connected components of the strata of the moduli space of meromorphic differentials, Comment. Math. Helv. 90 (2015), 255–286. · Zbl 1323.30060  C. Boissy, Moduli space of meromorphic differentials with marked horizontal separatrices, preprint, arXiv:1507.00555v4 [math.GT]. · Zbl 1323.30060  D. Chen, Degenerations of Abelian differentials, J. Differential Geom. 107 (2017), 395–453. · Zbl 1388.14080  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].  D. Eisenbud and J. Harris, Limit linear series: Basic theory, Invent. Math. 85 (1986), 337–371. · Zbl 0598.14003  D. Eisenbud and J. Harris, Existence, decomposition, and limits of certain Weierstrass points, Invent. Math. 87 (1987), 495–515. · Zbl 0606.14014  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. · Zbl 1305.32007  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. · Zbl 1037.32013  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. · Zbl 06914160  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. · Zbl 1357.37040  A. Eskin, M. Mirzakhani, and K. Rafi, Counting closed geodesics in strata, to appear in Invent. Math., preprint, arXiv:1206.5574v1 [math.GT].  E. Esteves and N. Medeiros, Limit canonical systems on curves with two components, Invent. Math. 149 (2002), 267–338. · Zbl 1046.14012  G. Farkas and R. Pandharipande, The moduli space of twisted canonical divisors, J. Inst. Math. Jussieu 17 (2018), 615–672. · Zbl 1455.14056  S. Filip, Splitting mixed Hodge structures over affine invariant manifolds, Ann. of Math. (2) 183 (2016), 681–713. · Zbl 1342.14015  O. Forster, Lectures on Riemann Surfaces, Grad. Texts in Math. 81, Springer, New York, 1991.  Q. Gendron, The Deligne-Mumford and the incidence variety compactifications of the strata of $$Ω\mathcal{M}_{g}$$, Ann. Inst. Fourier (Grenoble) 68 (2018), 1169–1240.  S. Grushevsky, I. Krichever, and C. Norton, Real-normalized differentials: Limits on stable curves, preprint, arXiv:1703.07806v1 [math.AG].  X. Hu, The locus of plane quartics with a hyperflex, Proc. Amer. Math. Soc. 145 (2017), 1399–1413. · Zbl 1364.14026  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. · Zbl 1318.32019  M. Kontsevich and A. Zorich, Connected components of the moduli spaces of Abelian differentials with prescribed singularities, Invent. Math. 153 (2003), 631–678. · Zbl 1087.32010  S. Lang, Real and Functional Analysis, 3rd ed., Grad. Texts in Math. 142, Springer, New York, 1993. · Zbl 0831.46001  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. · Zbl 1390.14197  M. Mirzakhani and A. Wright, The boundary of an affine invariant submanifold, Invent. Math. 209 (2017), 927–984. · Zbl 1378.37069  M. Möller, M. Ulirsch, and A. Werner, Realizability of tropical canonical divisors, preprint, arXiv:1710.06401v2 [math.AG].  S. Mullane, On the effective cone of $${\overline{\mathcal{M}}}_{g,n}$$, Adv. Math. 320 (2017), 500–519. · Zbl 1386.14108  K. Rafi, Thick-thin decomposition for quadratic differentials, Math. Res. Lett. 14 (2007), 333–341. · Zbl 1173.30031  A. Sauvaget, Cohomology classes of strata of differentials, preprint, arXiv:1701.07867v2 [math.AG].  K. Strebel, Quadratic Differentials, Ergeb. Math. Grenzgeb. (3) 5, Springer, Berlin, 1984.  E. F. Whittlesey, Analytic functions in Banach spaces, Proc. Amer. Math. Soc. 16 (1965), 1077–1083. · Zbl 0143.15302  S. A. Wolpert, Infinitesimal deformations of nodal stable curves, Adv. Math. 244 (2013), 413–440. · Zbl 1290.14019
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