The heart fan of an abelian category

TL;DR

This paper introduces the heart fan, a new geometric invariant of abelian categories derived from convex geometry, which generalizes stability conditions and relates to wall structures in representation theory.

Contribution

It constructs the heart fan as a universal phase diagram for Bridgeland stability, connecting convex geometry with homological algebra and stability conditions.

Findings

Heart fan is a complete invariant for module categories of finite-dimensional algebras.

Contains the g-fan as a subfan of full-dimensional cones.

Closely related to wall-and-chamber structures for King semistability.

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.