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 $\mathcal{C}$ of the bounded derived category of $X$, which consists of those complexes that derive pushforward to zero, and also on the affine subcategory $\mathcal{D}$, which consists of complexes supported on the exceptional locus. We show that a connected component of stability conditions on $\mathcal{C}$ is the universal cover of the complexified complement of the real hyperplane arrangement associated to $X$ via the Homological MMP, and more generally that a connected component of normalised stability conditions on $\mathcal{D}$ is a regular covering space of the infinite hyperplane arrangement constructed in Iyama-Wemyss [IW9]. 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.

Minor changes. Final version, to appear in Duke Math Journal