Coupled Contact Systems and Rigidity of Maximal Dimensional Variations of Hodge Structure

In this article we prove that locally Griffiths' horizontal distribution on the period domain is given by a generalized version of the familiar contact differential system. As a consequence of this description we obtain strong local rigidity properties of maximal dimensional variations of Hodge structure. For example, we prove that if the weight is odd then there is a unique germ of maximal dimensional variation of Hodge stucture though every point of the period domain. Similar results hold if the weight is even with the exception of one case.