Towards mirror symmetry for varieties of general type

Abstract: Assuming the natural compactification S of a hypersurface in (C^*)^n is smooth (e.g. a general hypersurface of some fixed degree in projective space), it can exhibit any Kodaira dimension depending on the size and shape of the Newton polyhedron of S. In a joint work with Ludmil Katzarkov, we give a construction for the expected mirror symmetry partner of a complete intersection S in a toric variety. The mirror dual of S may be reducible and is equipped with a sheaf of vanishing cycles. We show that the Hodge numbers then fulfil the expected mirror symmetry relation for the case when S is a hypersurface. The leading example will be the mirror of a genus two curve. If time permits, we will explain relations to homological mirror symmetry and the Gross-Siebert construction.