We study closures of ${\rm GL}^+_2(\mathbb{R})$-orbits in the total space $\Omega M_g$ of the Hodge bundle over the moduli space of curves under the assumption that they are algebraic manifolds. We show that in the generic stratum, such manifolds are the whole stratum, the hyperelliptic locus, or parameterize curves whose Jacobian has additional endomorphisms. This follows from a cohomological description of the tangent bundle to $\Omega M_g$. For nongeneric strata, similar results can be shown by a case-by-case inspection. We also propose to study a notion of linear manifold that comprises Teichmüller curves, Hilbert modular surfaces, and the ball quotients of Deligne and Mostow [DM]. Moreover, we give an explanation for the difference between Hilbert modular surfaces and Hilbert modular threefolds with respect to this notion of linearity
References
E. Arbarello, M. Cornalba, P. A. Griffiths, and J. Harris, Geometry of Algebraic Curves, Vol. I, Grundlehren Math. Wiss. 267, Springer, Berlin, 1985.
M. F. Atiyah, Riemann surfaces and spin structures, Ann. Sci. École Norm. Sup. (4) 4 (1971), 47--62.
I. Bouw and M. MöLler, Teichmüller curves, triangle groups, and Lyapunov exponents, preprint,\arxivmath/0511738v2[math.AG]
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.
P. Deligne, Équations différentielles à points singuliers réguliers, Lecture Notes in Math. 163, Springer, Berlin, 1970.
—, ``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.
P. Deligne and G. D. Mostow, Monodromy of hypergeometric functions and nonlattice integral monodromy, Inst. Hautes Études Sci. Publ. Math. 63 (1986), 5--89.
B. Farb and H. Masur, Superrigidity and mapping class groups, Topology 37 (1998), 1169--1176.
J. Harris and I. Morrison, Moduli of Curves, Grad. Texts in Math. 187, Springer, New York, 1998.
J. Hubbard and H. Masur, Quadratic differentials and foliations, Acta Math. 142 (1979), 221--274.
P. Hubert, E. Lanneau, and M. MöLler, The Arnoux-Yoccoz Teichmüller disc, preprint,\arxivmath/0611655v1[math.GT]
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.
M. Kontsevich and A. Zorich, Connected components of the moduli spaces of Abelian differentials with prescribed singularities, Invent. Math. 153 (2003), 631--678.
H. Lange and C. Birkenhake, Complex Abelian Varieties, Grundlehren Math. Wiss. 302, Springer, Berlin, 1992.
P. Lochak, On arithmetic curves in the moduli space of curves, J. Inst. Math. Jussieu 4 (2005), 443--508.
H. Masur and S. Tabachnikov, ``Rational billiards and flat structures'' in Handbook of Dynamical Systems, Vol. 1A, North-Holland, Amsterdam, 2002, 1015--1089.
C. T. Mcmullen, Billiards and Teichmüller curves on Hilbert modular sufaces, J. Amer. Math. Soc. 16 (2003), 857--885.
—, Prym varieties and Teichmüller curves, Duke Math. J. 133 (2006), 569--590.
—, Dynamics of, $\SL_2(\RR)$ over moduli space in genus two, Ann. of Math. (2) 165 (2007), 397--456.
M. MöLler, Periodic points on Veech surfaces and the Mordell-Weil group over a Teichmüller curve, Invent. Math. 165 (2006), 633--649.
—, Variations of Hodge structures of a Teichmüller curve, J. Amer. Math. Soc. 19 (2006), 327--344.
D. Mumford, Hirzebruch's proportionality theorem in the noncompact case, Invent. Math. 42 (1977), 239--277.
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.
G. Shimura, On analytic families of polarized abelian varieties and automorphic functions, Ann. of Math. (2) 78 (1963), 149--192.
W. A. Veech, The Teichmüller geodesic flow, Ann. of Math. (2) 124 (1986), 441--530.
—, Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards, Invent. Math. 97 (1989), 533--583.
E. Viehweg and K. Zuo, A characterization of Shimura curves in the moduli stack of abelian varieties, J. Differential Geom. 66 (2004), 233--287.
—, Arakelov inequalities and the uniformization of certain rigid Shimura varieties, J. Differential Geom. 77 (2007), 291--352.
G. E. Welters, Polarized abelian varieties and the heat equations, Compositio Math. 49 (1983), 173--194.