Degenerations and Fibrations of K3 Surfaces: Lattice Polarisations and Mirror Symmetry

Abstract: Tyurin degenerations of K3 surfaces are degenerations whose central fibre consists of a pair of rational surfaces glued along a smooth elliptic curve. We study the lattice theory of such Tyurin degenerations, establishing a notion of lattice polarisation that is compatible with existing definitions for the general fibre and the rational surfaces comprising the central fibre. We separately consider elliptically fibred K3 surfaces, where the base of the fibration admits a splitting into a pair of discs with specified monodromy around the boundary. In this setting we establish a notion of lattice polarisation for the induced elliptic fibrations over discs, which is compatible with the existing definition for K3 surfaces. Finally, we discuss the mirror symmetric correspondence between these two settings.