Fourier-Mukai transforms, mirror symmetry, and generalized K3 surfaces

TL;DR

This paper explores the relationships between generalized complex structures on K3 surfaces, demonstrating dualities and mirror symmetry through Fourier-Mukai transforms and deformation theory, with results connecting different generalized K3 structures.

Contribution

It introduces two families of generalized K3 surfaces parametrized by zeta, establishing their duality and mirror relationships, and links their deformation spaces via Fourier-Mukai equivalence.

Findings

Mukai duality at zeta=0 and infinity

Mirror symmetry for zeta not equal to 0 or infinity

Deformation spaces are isomorphic up to B-field correction

Abstract

We study generalized complex structures on K3 surfaces, in the sense of Hitchin. For each real parameter t between one and infinity we exhibit two families of generalized K3 surfaces, (M,cal{I}_{zeta}) and (M,cal{J}_{zeta}), parametrized by zeta in CP^1, which are Mukai dual for zeta=0 and infinity, amd mirror partners for zeta not equal to 0 and infinity. Moreover, the Fourier-Mukai equivalence D^b(M,cal{I}_0) -> D^b(M,cal{J}_0) induces an isomorphism phi_T between the spaces of first order deformations of (M,cal{I}_0) and (M,cal{J}_0) as generalized complex manifolds, and the deformations (M,cal{I}_{zeta}) and (M,cal{J}_{zeta}) agree under phi_T, up to a B-field correction which vanishes in the limit t -> infinity.