Mirror symmetry and rigid structures of generalized K3 surfaces

TL;DR

This paper develops a new formulation of mirror symmetry for generalized K3 surfaces using Mukai lattice polarizations, addressing issues in traditional models and solving the mirror symmetry problem for singular K3 surfaces.

Contribution

It introduces a novel approach to mirror symmetry for generalized K3 surfaces that overcomes limitations of previous formulations, especially for singular cases.

Findings

New formulation of mirror symmetry using Mukai lattice polarizations

Solution to mirror symmetry for singular K3 surfaces

Analysis of complex and Kähler rigid structures

Abstract

The present article is concerned with mirror symmetry for generalized K3 surfaces, with particular emphasis on complex and K\"ahler 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 K\"ahler rigid structures of generalized K3 surfaces.