Mirror symmetry for cohomology with values in vector bundles?
Abstract:
Mirror symmetry implies a (slightly broken) symmetry of the set of cohomology groups Hp,q(X) for Calabi-Yau manifolds X under the interchange of p,q. A similar, and possible related symmetry exists for a large class of vanishing theorems for cohomology groups of positive vector bundles, with the Kodaira-Akizuki-Nakano theorem as simplest example.