Strong McKay correspondence, string-theoretic Hodge numbers and mirror symmetry

We propose a new higher dimensional version of the McKay correspondence which enables us to understand the “Hodge numbers” assigned to singular Gorenstein varieties by physicists. Our results lead to the conjecture that string theory indicates the existence of some new cohomology theory Hst∗(X) for algebraic varieties with Gorenstein singularities. We give a formal mathematical definition of the Hodge numbers hstp,q(X) inspired from the consideration of strings on orbifolds and from this new (conjectural) version of the McKay correspondence. The numbers hstp,q(X) are expected to give the spectrum of orbifoldized Landau-Ginzburg models and mirror duality relations for higher dimensional Calabi-Yau varieties with Gorenstein toroidal or quotient singularities.