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

Non-geodesic variations of Hodge structure of maximum dimension