Mathematical Research Letters

Volume 28 (2021)

Number 6

Decomposition of Lagrangian classes on K3 surfaces

Pages: 1739 – 1763



Kuan-Wen Lai (Mathematisches Institut, Universität Bonn, Germany)

Yu-Shen Lin (Department of Mathematics and Statistics, Boston University, Boston, Massachusetts, U.S.A.)

Luca Schaffler (Dipartimento di Matematica e Fisica, Università degli Studi Roma Tre, Rome, Italy)


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