Decomposition of Lagrangian classes on K3 surfaces

TL;DR

This paper investigates the conditions under which Lagrangian homology classes on K3 surfaces can be decomposed into sums of special Lagrangian classes, providing criteria based on lattice theory and demonstrating density results.

Contribution

It introduces lattice-theoretic criteria for decomposability of Lagrangian classes on K3 surfaces and proves their density in the Kähler cone, with implications for special Lagrangian fibrations.

Findings

Decomposability of Lagrangian classes is dense in the Kähler cone.

Kähler classes admitting special Lagrangian fibrations are dense.

Infinitely many special Lagrangian 3-tori exist in any log Calabi-Yau 3-fold.

Abstract

We study the decomposability of a Lagrangian homology class on a K3 surface into a sum of classes represented by special Lagrangian submanifolds, and develop criteria for it in terms of lattice theory. As a result, we prove the decomposability on an arbitrary K3 surface with respect to the K\"ahler classes in dense subsets of the K\"ahler cone. Using the same technique, we show that the K\"ahler classes on a K3 surface which admit a special Lagrangian fibration form a dense subset also. This implies that there are infinitely many special Lagrangian 3-tori in any log Calabi-Yau 3-fold.