Bridgeland/Weak Stability Conditions under Spherical Twist Associated to A Torsion Sheaf
Tristan C. Collins, Jason Lo, Yun Shi, Shing-Tung Yau
TL;DR
This work investigates how the spherical twist $ST_{ ext{O}_{C}(-1)}$ interacts with Bridgeland stability on a K3 surface and with generalized weak stability conditions. It constructs a Bridgeland stability condition tied to a non-nef divisor, conjecturally located in the geometric component, and analyzes destabilizing objects and line-bundle stability within nef- and weak-stability contexts. A key technical achievement is the explicit central-charge transformation under the twist, plus a precise correspondence between the hearts $ rak{B}_{ u,-2}$ and $ rak{A}_{ u,-2}$, enabling transfer of stability data across the autoequivalence. The paper further shows that the image of standard Bridgeland stability under the twist can be realized by a sequence of tilting leading to a non-nef stability condition, and it provides criteria for line-bundle stability in these settings, with implications for moduli interpretations and geometric components of stability manifolds.
Abstract
In this paper, we study the action of an autoequivalence, the spherical twist associated to a torsion sheaf, on the standard Bridgeland stability conditions and a generalized weak stability condition on the derived category of a K3 surface. As a special case, we construct a Bridgeland stability condition associated to a non-nef divisor, which conjecturally lies in the geometric component but outside the geometric chamber. We also discuss the destabilizing objects and stability of certain line bundles at the weak stability condition associated to a nef divisor.