  We study closures of GL_2(R)-orbits on the total space 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 strata. For non-generic strata similar results can be shown by a
case-by-case inspection.
  We also propose to study a notion of 'linear manifold' that comprises
Teichmueller 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.
