On Lagrangian tori in K3 surfaces

TL;DR

This paper investigates the homological and Maslov index properties of Lagrangian tori in K3 surfaces, establishing new relationships between their Maslov indices and homology classes.

Contribution

It extends known results to Lagrangian tori with Maslov indices divisible by 4 and proves that non-trivial homology implies Maslov-zero in K3 surfaces.

Findings

Maslov-zero Lagrangian tori have non-trivial homology in K3 surfaces

Lagrangian tori with Maslov indices divisible by 4 are also considered

Homologically non-trivial Lagrangian tori are necessarily Maslov-zero

Abstract

Every Maslov-zero Lagrangian torus in a K3 surface has non-trivial homology class. This note aims to extend this result to Lagrangian tori with Maslov indices congruent to zero modulo 4. Conversely, we show that every homologically non-trivial Lagrangian torus is necessarily Maslov-zero.