Special Lagrangians and Bridgeland stable objects beyond geometric stability conditions: the product case
Yu-Wei Fan
TL;DR
The paper constructs a family of non-geometric Bridgeland stability conditions on wrapped Fukaya categories via homological mirror symmetry and categorical Künneth, where the central charge is tied to a holomorphic volume form. It proves that every stable object under these conditions has a special Lagrangian representative in the mirror, yielding higher-dimensional instances of the stable-implies-special-Lagrangian phenomenon away from the large complex structure limit. The core method starts with product-type stability conditions on $ ext{D}^b((P^1)^n)$, showing stable objects must decompose as tensor products of stable factors, and then transports stability to the Fukaya category through HMS. This work extends product-type stability to more cases, including $ ext{D}^b(P^1 imesP^1)$ and products of elliptic curves, and analyzes embeddings of product stability spaces into the full stability spaces, highlighting both geometric and non-geometric aspects of stability in higher dimensions.
Abstract
We construct a family of non-geometric Bridgeland stability conditions on certain wrapped Fukaya categories, using homological mirror symmetry and categorical Künneth formulae. These stability conditions correspond to certain holomorphic volume forms, under which we prove that every stable object admits a special Lagrangian representative. This provides the first higher-dimensional examples of stability conditions away from the large complex structure limit for which ``stable implies special Lagrangian" is proved.