# Non-geodesic variations of Hodge structure of maximum dimension

## TL;DR

This paper presents a novel example of a maximum dimension variation of Hodge structure that is not geodesic, arising from weighted projective hypersurfaces, challenging the typical Lie-theoretic geometric perspective.

## Contribution

It provides the first known example of a maximum dimension Hodge variation that is nowhere tangent to geodesic variations, constructed via weighted projective hypersurfaces.

## Key findings

- The example is of maximum dimension 28 in a 57-dimensional period domain.
- The variation is not geodesic, contrasting with previous examples.
- It is realized through the second cohomology of specific weighted hypersurfaces.

## 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 $1, 1, 2, 5$