The heart fan of an abelian category

Abstract

We apply convex geometry (cones, fans) to homological input (abelian categories, hearts of bounded t-structures) to construct a new invariant of an abelian category, its heart fan. This can be viewed as a `universal phase diagram' for Bridgeland stability conditions with the given heart. When the abelian category is the module category of a finite-dimensional algebra, the heart fan is complete and contains the g-fan as the subfan of full-dimensional cones. The heart fan is also closely related to the wall-and-chamber structure for King semistability.

v1 30 pages. v2 major revision: replaced section on scattering diagrams by universal phase diagrams; introduce and use cofans over dual face fans; 13 fan diagrams, 36 pages