Mirror Symmetry, Singularity Theory and Non-commutative Hodge Structures

We review a version of the mirror correspondence for smooth toric varieties with a numerically effective anticanonical bundle. We give a precise description of the so-called B-model, which involves the Gauß-Manin system of a family of Laurent polynomials. We show how to derive from these data a variation of non-commutative Hodge structures and describe general results on period maps and classifying spaces for these generalized Hodge structures. Finally, we explain a version of mirror symmetry as an isomorphism of Frobenius manifolds.