# Linear manifolds in the moduli space of one-forms

Martin Möller

Number | 3 |
---|---|

Author | Martin Möller |

Year | 2007 |

We study closures of GL_{2}^{+} (R)-orbits on the total space ΩM_{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. Moreover,

we give an explanation for the
difference between Hilbert modular surfaces and Hilbert modular
threefolds with respect to this notion of linearity.