Canonical currents and heights for K3 surfaces

TL;DR

This paper develops canonical currents and height functions on the boundary of the ample cone of K3 surfaces, ensuring automorphism invariance and continuity, with applications to elliptically fibered surfaces.

Contribution

It introduces a new framework for canonical currents and heights on K3 surfaces' boundary, extending to an enlarged boundary and providing preferred representatives.

Findings

Constructed canonical positive currents for K3 surfaces.

Defined height functions compatible with automorphisms.

Extended the theory to elliptically fibered surfaces.

Abstract

We construct canonical positive currents and heights on the boundary of the ample cone of a K3 surface. These are equivariant for the automorphism group and fit together into a continuous family, defined over an enlarged boundary of the ample cone. Along the way, we construct preferred representatives for certain height functions and currents on elliptically fibered surfaces.