Linear manifolds in the moduli space of one-forms

We study closures of GL₂⁺ (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.