Mathematical Research Letters
Volume 28 (2021)
Decomposition of Lagrangian classes on K3 surfaces
Pages: 1739 – 1763
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 Kähler classes in dense subsets of the Kähler cone. Using the same technique, we show that the Kähler 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.
The second author is supported by the Simons Collaboration grant #635846 and the NSF grant DMS #2204109.
Received 31 March 2020
Accepted 29 July 2020
Published 29 August 2022