Rational equivalence and Lagrangian tori on 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 $L\subset X$ to the skyscraper sheaf of a point $y\in Y$. 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.