Rational equivalence and Lagrangian tori on K3 surfaces

TL;DR

This paper explores the relationship between Lagrangian tori in symplectic K3 surfaces and algebraic cycles on mirror K3 surfaces, revealing new instances of rational equivalence and non-generation by spheres.

Contribution

It demonstrates the existence of Lagrangian tori with vanishing Maslov class that are not generated by spheres in the Fukaya category, linking symplectic geometry with algebraic cycle theory.

Findings

Existence of special Lagrangian tori with vanishing Maslov class

Counterexamples to generation by Lagrangian spheres in Fukaya category

Connection between symplectic geometry and Chow groups of mirror K3 surfaces

Abstract

Fix a symplectic K3 surface X homologically mirror to an algebraic K3 surface Y by an equivalence taking a graded Lagrangian torus L in X to the skyscraper sheaf of a point y of Y. We show there are Lagrangian tori with vanishing Maslov class in X whose class in the Grothendieck group of the Fukaya category is not generated by Lagrangian spheres. This is mirror to a statement about the `Beauville--Voisin subring' in the Chow groups of Y, and fits into a conjectural relationship between Lagrangian cobordism and rational equivalence of algebraic cycles.