## Linear manifolds in the moduli space of one-forms.(English)Zbl 1148.32007

Let $$M_g$$ denote the moduli space of curves of genus $$g$$. The total space of the Hodge bundle $$\Omega M_g$$ over $$M_g$$ is the space of pairs $$(X,\omega)$$ where $$X$$ is a Riemann surface of genus $$g$$ and $$\omega$$ a holomorphic one-form on $$X$$. The space $$\Omega M_g$$ admits a linear structure, induced by the operation of integration of one-forms over a basis of the first homology relative to the zeros of one-forms. From this linear structure, one deduces a natural action of $$\mathrm{GL}_2^+(\mathbb{R})$$ on $$M_g$$, and an important question that has been raised in the last few years is the classification of the closures of the orbits of this action. The question is partly motivated by a result by Marina Ratner on the classification of orbit closures in homogeneous manifolds.
The question studied in the paper under review has been answered by C. T. McMullen for the case $$g=2$$, in his paper [Ann. Math. (2) 165, 397–456 (2007; Zbl 1131.14027)]. The author analyzes the case $$g\geq 3$$. For that purpose, he splits the question of orbit closure into three subquestions: (1) to show that the orbit closures are complex manifolds; (2) to show that these complex manifolds are algebraic; (3) to classify these algebraic manifolds. The author shows the importance of the hyperelliptic loci in this classification problem. The main result is the following:
Theorem: Let $$B$$ be an algebraic manifold of the generic stratum $$S$$ of $$\Omega M_g$$ with linear structure. Then, there are three possibilites: 6mm
(1)
$$B$$ is a connected component of $$S$$;
(2)
$$g\geq 3$$ and $$B$$ is the preimage in $$S$$ of the hyperelliptic locus in $$M_g$$;
(3)
$$B$$ parametrizes curves with a Jacobian whose endomorphism ring is strictly larger than $$\mathbb{Z}$$.
The author also obtains statements for nongeneric strata, mostly by a case-by-case inspection.
The results obtained in this paper are based on a cohomological description of the tangent bundle to $$\Omega M_g$$.
The author also introduces a notion of linear manifold that includes Teichmüller curves, Hilbert modular surfaces, and the ball quotients of Deligne and Mostow. He proposes an explanation for the difference between the linearity of the eigenform bundle over a Hilbert modular surface and the nonlinearity of the Hilbert modular threefold parametrizing Abelian varieties of dimension three with real multiplication by $$\mathbb{Z}[\zeta_7+\zeta^{-1}_7]$$. This difference was pointed out by C. T. McMullen in his paper [J. Am. Math. Soc. 16, 857–885 (2003; Zbl 1030.32012)].

### MathOverflow Questions:

Tangent Space of the Hodge bundle on the moduli space of curves

### MSC:

 32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) 14D07 Variation of Hodge structures (algebro-geometric aspects) 32G20 Period matrices, variation of Hodge structure; degenerations

### Citations:

Zbl 1131.14027; Zbl 1030.32012
Full Text:

### References:

 [1] E. Arbarello, M. Cornalba, P. A. Griffiths, and J. Harris, Geometry of Algebraic Curves, Vol. I , Grundlehren Math. Wiss. 267 , Springer, Berlin, 1985. · Zbl 0559.14017 [2] M. F. Atiyah, Riemann surfaces and spin structures , Ann. Sci. École Norm. Sup. (4) 4 (1971), 47–62. · Zbl 0212.56402 [3] I. Bouw and M. MöLler, Teichmüller curves, triangle groups, and Lyapunov exponents , preprint,\arxivmath/0511738v2[math.AG] [4] J. De Jong and S.-W. Zhang, “Generic abelian varieties with real multiplication are not Jacobians” in Diophantine Geometry (Pisa, 2005) , CRM Series 4 , Ed. Norm., Pisa, 2007, 165–172. · Zbl 1138.14027 [5] P. Deligne, Équations différentielles à points singuliers réguliers , Lecture Notes in Math. 163 , Springer, Berlin, 1970. · Zbl 0244.14004 [6] -, “Un théorème de finitude pour la monodromie” in Discrete Groups in Geometry and Analysis (New Haven, 1984) , Progr. Math. 67 , Birkhäuser, Boston, 1987, 1–19. [7] P. Deligne and G. D. Mostow, Monodromy of hypergeometric functions and nonlattice integral monodromy , Inst. Hautes Études Sci. Publ. Math. 63 (1986), 5–89. · Zbl 0615.22008 [8] B. Farb and H. Masur, Superrigidity and mapping class groups , Topology 37 (1998), 1169–1176. · Zbl 0946.57018 [9] J. Harris and I. Morrison, Moduli of Curves , Grad. Texts in Math. 187 , Springer, New York, 1998. · Zbl 0913.14005 [10] J. Hubbard and H. Masur, Quadratic differentials and foliations , Acta Math. 142 (1979), 221–274. · Zbl 0415.30038 [11] P. Hubert, E. Lanneau, and M. MöLler, The Arnoux-Yoccoz Teichmüller disc , preprint,\arxivmath/0611655v1[math.GT] [12] N. M. Katz and T. Oda, On the differentiation of de Rham cohomology classes with respect to parameters , J. Math. Kyoto Univ. 8 (1986), 199–213. · Zbl 0165.54802 [13] 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 [14] H. Lange and C. Birkenhake, Complex Abelian Varieties , Grundlehren Math. Wiss. 302 , Springer, Berlin, 1992. · Zbl 0779.14012 [15] P. Lochak, On arithmetic curves in the moduli space of curves , J. Inst. Math. Jussieu 4 (2005), 443–508. · Zbl 1094.14018 [16] H. Masur and S. Tabachnikov, “Rational billiards and flat structures” in Handbook of Dynamical Systems, Vol. 1A , North-Holland, Amsterdam, 2002, 1015–1089. · Zbl 1057.37034 [17] C. T. Mcmullen, Billiards and Teichmüller curves on Hilbert modular sufaces , J. Amer. Math. Soc. 16 (2003), 857–885. · Zbl 1030.32012 [18] -, Prym varieties and Teichmüller curves , Duke Math. J. 133 (2006), 569–590. · Zbl 1099.14018 [19] -, Dynamics of, $$\SL_2(\RR)$$ over moduli space in genus two , Ann. of Math. (2) 165 (2007), 397–456. · Zbl 1131.14027 [20] M. MöLler, Periodic points on Veech surfaces and the Mordell-Weil group over a Teichmüller curve , Invent. Math. 165 (2006), 633–649. · Zbl 1111.14019 [21] -, Variations of Hodge structures of a Teichmüller curve , J. Amer. Math. Soc. 19 (2006), 327–344. · Zbl 1090.32004 [22] D. Mumford, Hirzebruch’s proportionality theorem in the noncompact case , Invent. Math. 42 (1977), 239–277. · Zbl 0365.14012 [23] F. Oort and J. Steenbrink, “The local Torelli problem for algebraic curves” in Journées de Géométrie Algébrique d’Angers, Juillet 1979/Algebraic Geometry, Angers 1979 (Angers, France, 1979) , Sijthoff and Noordhoff, Germantown, Md., 1980, 157–204. · Zbl 0444.14007 [24] G. Shimura, On analytic families of polarized abelian varieties and automorphic functions , Ann. of Math. (2) 78 (1963), 149–192. JSTOR: · Zbl 0142.05402 [25] W. A. Veech, The Teichmüller geodesic flow , Ann. of Math. (2) 124 (1986), 441–530. JSTOR: · Zbl 0658.32016 [26] -, Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards , Invent. Math. 97 (1989), 533–583. · Zbl 0676.32006 [27] E. Viehweg and K. Zuo, A characterization of Shimura curves in the moduli stack of abelian varieties , J. Differential Geom. 66 (2004), 233–287. · Zbl 1078.11043 [28] -, Arakelov inequalities and the uniformization of certain rigid Shimura varieties , J. Differential Geom. 77 (2007), 291–352. · Zbl 1133.14010 [29] G. E. Welters, Polarized abelian varieties and the heat equations , Compositio Math. 49 (1983), 173–194. · Zbl 0576.14042 [30] A. Zorich, “Flat surfaces” in Frontiers in Number Theory, Physics, and Geometry, I (Les Houches, France, 2003) , Springer, Berlin, 2006, 437–583. · Zbl 1129.32012
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