Rational equivalence and Lagrangian tori on K3 surfaces
Symplectic Geometry
2020-11-03 v2 Algebraic Geometry
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.
Keywords
Cite
@article{arxiv.1809.03892,
title = {Rational equivalence and Lagrangian tori on K3 surfaces},
author = {Nick Sheridan and Ivan Smith},
journal= {arXiv preprint arXiv:1809.03892},
year = {2020}
}
Comments
30 pages. Version 2: minor clarifications. This is the final version, to appear in Commentarii