Mirror symmetry and rigid structures of generalized K3 surfaces

Abstract

The present article is concerned with mirror symmetry for generalized K3 surfaces, with particular emphasis on complex and Kähler rigid structures. Inspired by the works of Dolgachev, Aspinwall–Morrison and Huybrechts, we introduce a formulation of mirror symmetry for generalized K3 surfaces by using Mukai lattice polarizations. This approach solves issues in the conventional formulations of mirror symmetry for K3 surfaces. In particular, we provide a solution to the problem of mirror symmetry for singular K3 surfaces. Along the way, we investigate complex and Kähler rigid structures of generalized K3 surfaces.