Stability conditions on blowups
Nantao Zhang
TL;DR
The paper analyzes how perverse stability conditions interact with geometric (tilt) stability under blowups, using semiorthogonal decompositions to transfer stability from the ambient category to its components. It proves that a perverse stability condition on $D^b(Coh(X))$ induces a geometric stability condition on $D^b(Coh(Y))$, and shows that the generalized Bogomolov inequality for perverse tilted stability on $X$ follows from the corresponding geometric inequality on $Y$ in several key cases, providing evidence for Toda's conjecture in these settings. The authors treat both surface and threefold blow-ups, extending the framework of BLM+2023 to derive stability on the contracted base and discuss limitations involving higher-dimensional generalizations. Collectively, the results clarify how perverse and geometric stability notions interact with birational geometry and offer concrete instances where Toda's conjecture holds (e.g., Fano rank 1, abelian 3-folds, quintic 3-folds).
Abstract
We study the relation between perverse stability conditions and geometric stability conditions under blow up. We confirm a conjecture of Toda in some special cases and show that geometric stability conditions can be induced from perverse stability conditions from semiorthogonal decompositions associated to blowups.