Symplectic topology of K3 surfaces via mirror symmetry

TL;DR

This paper explores the symplectic topology of specific K3 surfaces using mirror symmetry, revealing an infinitely generated symplectic Torelli group and new constraints on Lagrangian tori.

Contribution

It demonstrates the infinite generation of the symplectic Torelli group and applies homological mirror symmetry to derive new topological constraints.

Findings

Symplectic Torelli group may be infinitely generated.

New constraints on Lagrangian tori in K3 surfaces.

Utilizes homological mirror symmetry and autoequivalence groups.

Abstract

We study the symplectic topology of certain K3 surfaces (including the "mirror quartic" and "mirror double plane"), equipped with certain K\"ahler forms. In particular, we prove that the symplectic Torelli group may be infinitely generated, and derive new constraints on Lagrangian tori. The key input, via homological mirror symmetry, is a result of Bayer and Bridgeland on the autoequivalence group of the derived category of an algebraic K3 surface of Picard rank one.