Topology of Algebraic Varieties

For an odd integer \(n > 3\) the data of generic n-dimensional subspace of the space of skew bilinear forms on an n-dimensional vector space define two different Calabi-Yau varieties of dimension \(n-4\). Specifically, one is a complete intersection of n hyperplanes in the Grassmannian \(G(2,n)\) and the other is a complete intersection of \(n(n-3)/2\) hyperplanes in the Pfaffian variety of degenerate skew forms. In \(n=7\) case, these have been investigated by Rodland and were (heuristically) found to have the same mirror family. As a result, these Calabi-Yau varieties share many common features. For example, it has been verified that they are derived equivalent even though they are not birational to each other. In the ongoing project, joint with Anatoly Libgober, we are trying to verify that elliptic genera of these Calabi-Yau varieties coincide. This becomes a subtle problem for large n, since the varieties on the Pfaffian side are singular and elliptic genus is defined in terms of log resolution of singularities. In the talk I will focus mostly on the background and related contructions to try to explain my interest in Pfaffian-Grassmannian double mirror phenomenon.