Rigidity of Lagrangian embeddings into symplectic tori and K3 surfaces

TL;DR

This paper proves that Maslov-zero Lagrangian tori in symplectic tori and K3 surfaces with Kahler-type forms must have non-zero, primitive homology classes, extending previous results with new dynamical and arithmetic methods.

Contribution

It extends prior results by showing homology class constraints for Lagrangian tori in K3 surfaces and tori using dynamical systems and Ratner's theorem.

Findings

Homology class of Maslov-zero Lagrangian tori is non-zero and primitive.

Uses dynamical properties of diffeomorphism group actions.

Employs Ratner's orbit closure theorem in symplectic geometry.

Abstract

A Kahler-type form is a symplectic form compatible with an integrable complex structure. Let M be either a torus or a K3-surface equipped with a Kahler-type form. We show that the homology class of any Maslov-zero Lagrangian torus in M has to be non-zero and primitive. This extends previous results of Abouzaid-Smith (for tori) and Sheridan-Smith (for K3-surfaces) who proved it for particular Kahler-type forms on M. In the K3 case our proof uses dynamical properties of the action of the diffeomorphism group of M on the space of the Kahler-type forms. These properties are obtained using Shah's arithmetic version of Ratner's orbit closure theorem.