Hodge Theory in Combinatorics and Mirror Symmetry

Abstract

Hodge theory, in its broadest sense, encompasses the study of the decomposition of cohomology groups of complex manifolds, as well as related fields such as periods, motives, and algebraic cycles.

In this thesis, ideas from Hodge theory have been incorporated into two seemingly unrelated projects, namely mathematical mirror symmetry and combinatorics.

Papers I-II explore an instance of genus one mirror symmetry for the complete intersection of two cubics in five-dimensional projective space. The mirror family for this complete intersection is constructed, and it is demonstrated that the BCOV-invariant of the mirror family is related to the genus one Gromov-Witten invariants of the complete intersection of two cubic. This proves new cases of genus one mirror symmetry.

Paper III defines Hodge-theoretic structures on triangulations of a special type. It is shown that if a polytope admits a regular, unimodular triangulation with a particular additional property, its $\delta$-vector from Ehrhart theory is unimodal.