Stability conditions for 3-fold flops

Abstract

Let $f:X\to \mathrm{Spec}R$ be a 3-fold flopping contraction, where X has at worst Gorenstein terminal singularities and R is complete local. We describe the space of Bridgeland stability conditions on the null subcategory 𝒞 of $D^b(\mathrm{coh}X)$, which consists of those complexes that derive pushforward to zero, and also on the affine subcategory 𝒟, which consists of complexes supported on the exceptional locus. We show that a connected component ${\mathrm{Stab}}^{\circ }𝒞$ of $\mathrm{Stab}𝒞$ is the universal cover of the complexified complement of the real hyperplane arrangement associated to X via the homological MMP, and more generally that ${\mathrm{Stab}}_n^{\circ }𝒟$ is a regular covering space of the infinite hyperplane arrangement constructed in an earlier preprint by Iyama and the second named author. Neither arrangement is Coxeter in general. As a consequence, we give the first description of the stringy Kähler moduli space (SKMS) for all smooth irreducible 3-fold flops. The answer is surprising: we prove that the SKMS is always a sphere, minus either 3, 4, 6, 8, 12, or 14 points, depending on the length of the curve.