Title: Non-geodesic variations of Hodge structure of maximum dimension,
Abstract: There are a number of examples of variations of Hodge structure of maximum
dimension. However, to our knowledge, those that are global on the level of the
period domain are totally geodesic subspaces that arise from an orbit of a
subgroup of the group of the period domain. That is, they are defined by Lie
theory rather than by algebraic geometry. In this note, we give an example of a
variation of maximum dimension which is nowhere tangent to a geodesic
variation. The period domain in question, which classifies weight two Hodge
structures with $h^{2,0} = 2$ and $h^{1,1} = 28$, is of dimension $57$. The
horizontal tangent bundle has codimension one, thus it is an example of a
holomorphic contact structure, with local integral manifolds of dimension 28.
The group of the period domain is $SO(4,28)$, and one can produce global
integral manifolds as orbits of the action of subgroups isomorphic to
$SU(2,14)$. Our example is given by the variation of Hodge structure on the
second cohomology of weighted projective hypersurfaces of degree $10$ in a
weighted projective three-space with weights $1, 1, 2, 5$