Mirror symmetry for varieties of general type

I will discuss joint work with Ludmil Katzarkov and Helge Ruddat. Given a hypersurface X in a toric variety of positive Kodaira dimension, (with a certain number of hypotheses) we construct an object which we believe can be viewed as the mirror of X. In particular, it exhibits the usual interchange of Hodge numbers expected in mirror symmetry. This may seem puzzling at first. For example, a curve of genus g would be expected to have a mirror such that h^{0,0}=g, which is not possible for a variety. However, our mirror is a singular scheme Y along with a perverse sheaf F, whose cohomology carries a mixed Hodge structure. It then makes sense to compute Hodge numbers for F, and we find the traditional exchange of Hodge numbers.