Stability conditions, torsion theories and tilting

TL;DR

This paper explores the geometric structure of stability conditions on triangulated categories, showing how tilting at simple objects relates different stability regions and characterizing their intersections, with applications to the derived category of the projective line.

Contribution

It establishes a precise criterion for when the closures of stability condition subsets intersect, based on tilting relations, and describes the structure of the stability space in terms of simple tilts.

Findings

The intersection of stability condition subsets corresponds exactly to tilting relations.

The union of subsets obtained by finite simple tilts forms a connected component of the stability space.

Explicit computation of a stability space component for the derived category of the projective line.

Abstract

The space of stability conditions on a triangulated category is naturally partitioned into subsets $U(A)$ of stability conditions with a given heart $A$. If $A$ has finite length and $n$ simple objects then $U(A)$ has a simple geometry, depending only on $n$. Furthermore, Bridgeland has shown that if $B$ is obtained from $A$ by a simple tilt, i.e.\ by tilting at a torsion theory generated by one simple object, then the intersection of the closures of $U(A)$ and $U(B)$ has codimension one. Suppose that $A$, and any heart obtained from it by a finite sequence of (left or right) tilts at simple objects, has finite length and finitely many indecomposable objects. Then we show that the closures of $U(A)$ and $U(B)$ intersect if and only if $A$ and $B$ are related by a tilt, and that the dimension of the intersection can be determined from the torsion theory. In this situation the union of subsets $U(B)$, where $B$ is obtained from $A$ by a finite sequence of simple tilts, forms a component of the space of stability conditions. We illustrate this by computing (a component of) the space of stability conditions on the constructible derived category of the complex projective line stratified by a point and its complement.