Autoequivalences and stability conditions on a degenerate K3 surface
Hayato Arai
TL;DR
This work analyzes autoequivalences and stability conditions on the derived category of a singular surface X that sits as an open part of a type III Kulikov degeneration of K3s. By relating stability conditions on the compact component Z0 via a smoothing family, the authors describe a geometric chamber and the wall structure, showing the central stability component is simply connected and organized by half-spherical twists. They determine the subgroup of autoequivalences preserving this component as ${\mathbb{Z}} \times \Gamma_1(3) \times {\mathrm{Aut}}(X)$ and prove all autoequivalences are Fourier–Mukai; they also propose that the whole autoequivalence group should preserve the same stability component, aligning with Bridgeland’s expectations for K3s. The results illuminate how stability conditions behave on singular/open degenerations, contribute to homological mirror symmetry perspectives for degenerations, and provide a concrete computation of the autoequivalence group in this setting.
Abstract
We study autoequivalences and stability conditions on the derived category of coherent sheaves on a singular surface $X$ which arises as an open subvariety of a type III Kulikov degeneration of K3 surfaces. The surface $X$ consists of four irreducible components, one of which is $\mathbb{P}^2$, and the others are non-compact rational surfaces. Using a comparison with the total space of the degeneration, we show that the connected component $\mathrm{Stab}^\dagger(D^b_{\mathbb{P}^2}(X))$ of the space of stability conditions on the supported derived category $D^b_{\mathbb{P}^2}(X)$ containing geometric stability conditions is simply connected, and describe its wall-and-chamber structure via half-spherical twists. As consequences, we determine the subgroup of the autoequivalence group $\mathrm{Aut}(D^b(X))$ that preserves this component; it is isomorphic to $\mathbb{Z} \times Γ_1(3) \times \mathrm{Aut}(X)$, where $Γ_1(3) \subset \mathrm{SL}(2,\mathbb{Z})$ is the congruence subgroup of level~3.