A Thomas-Yau-Joyce result for Lagrangian spheres in K3 surfaces
Abstract
We consider objects in the Fukaya category of a class of K3 surfaces defined by certain Lagrangian spheres. Assuming that homological mirror symmetry for K3 surfaces holds with sufficiently strong (expected) properties, we prove that, in this case, stability with respect to a suitable Bridgeland stability condition implies the existence of an isomorphic special Lagrangian sphere, as predicted by the general Thomas-Yau-Joyce conjectures. In particular this holds unconditionally for suitable quartic surfaces in ℙ3 or sextics in ℙ(3,1,1,1) and their mirrors. A variant holds on the Calabi-Yau threefolds obtained by taking the product of our K3 surfaces with an elliptic curve. These seem to be the first results of this type on compact manifolds.
Related
- A semiampleness result for hyperkähler fourfolds and smoothings of Lagrangian subvarieties
- High-Dimensional Families of Minimal Surfaces of Arbitrary Genus in Round Spheres
- Photon spheres, elliptic surfaces, and Seiberg-Witten geometry
- Hamiltonian stationary Lagrangian surfaces of constant Gaussian curvature in complex space forms
- Hamiltonian stationary Lagrangian surfaces in complex space forms with constant-length gradient of the mean curvature
BibTeX
@misc{stoppa2026thomasyaujoyce,
title = {{A Thomas-Yau-Joyce result for Lagrangian spheres in K3 surfaces}},
author = {Jacopo Stoppa},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.39554},
eprint = {2609.39554},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.39554},
}
RIS
TY - UNPB
TI - A Thomas-Yau-Joyce result for Lagrangian spheres in K3 surfaces
AU - Jacopo Stoppa
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2609.39554
UR - https://arxiv.org/abs/2609.39554
AB - We consider objects in the Fukaya category of a class of K3 surfaces defined by certain Lagrangian spheres. Assuming that homological mirror symmetry for K3 surfaces holds with sufficiently strong (expected) properties, we prove that, in this case, stability with respect to a suitable Bridgeland stability condition implies the existence of an isomorphic special Lagrangian sphere, as predicted by the general Thomas-Yau-Joyce conjectures. In particular this holds unconditionally for suitable quartic surfaces in ℙ^3 or sextics in ℙ(3,1,1,1) and their mirrors. A variant holds on the Calabi-Yau threefolds obtained by taking the product of our K3 surfaces with an elliptic curve. These seem to be the first results of this type on compact manifolds.
ER -