Stability conditions on K3 surfaces via mass of spherical objects
Kohei Kikuta, Naoki Koseki, Genki Ouchi
TL;DR
The paper proves that on a K3 surface $X$, the stability condition data in the distinguished component $\mathrm{Stab}^*(X)/\mathbb{C}$ is determined by the masses of spherical objects, via the injectivity of the projectivized mass map $\mathbb{P} m: \mathrm{Stab}^*(X)/\mathbb{C} \to \mathbb{P}^{\mathcal{S}}_{\ge 0}$. It develops a program to approach compactifications of stability manifolds by correlating masses with central charges and phases, and constructs lax stability conditions associated to spherical bundles, enabling a boundary analysis and partial compactification picture. The work leverages the Mukai lattice framework, Huybrechts’ criteria, and the theory of lax stability to relate boundary points defined by hom-functionals to degenerations of stability data, including explicit degenerations $\sigma_{\alpha}$ and $\sigma_{\alpha_0}$. Overall, the results illuminate how spherical objects encode enough information to recover stability structures and lay groundwork for further compactification and autoequivalence studies, with potential extensions to rank-one Picard cases and elliptic Situations. The findings have significance for understanding stability manifolds, their boundaries, and how mass data interacts with the geometry of $D^b(X)$ on K3 surfaces.
Abstract
We prove that a stability condition on a K3 surface is determined by the masses of spherical objects up to a natural $\mathbb{C}$-action. This is motivated by the result of Huybrechts and the recent proposal of Bapat-Deopurkar-Licata on the construction of a compactification of a stability manifold. We also construct lax stability conditions in the sense of Broomhead-Pauksztello-Ploog-Woolf associated to spherical bundles.