Stability conditions, torsion theories and tilting

By J. Woolf. 2009

68 citations recorded by the source corpus.

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, that is, 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 1. 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.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/68c060eef7084547d271ff7c74d9c5e2955d7744

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

1. I. Smith
2. Dong Ping Yang
3. T. Bridgeland
4. S. Koenig
5. Sotaro Asai
6. M. Gorsky
7. Osamu Iyama
8. B. Keller

In Geometry and Topology

1. Dong Ping Yang
2. I. Smith
3. S. Koenig
4. T. Bridgeland
5. Yukinobu Toda
6. Osamu Iyama
7. Emanuele Macrì
8. B. Keller

In Mathematical Physics

1. I. Smith
2. T. Bridgeland
3. Dong Ping Yang
4. S. Koenig
5. Yukinobu Toda
6. Emanuele Macrì
7. Arend Bayer
8. D. Joyce

Citations in the corpus

• Silting objects, simple-minded collections, $t$-structures and co-$t$-structures for finite-dimensional algebras (2012, methodology): “These mutations provide an effective method to compute the space of Bridgeland’s stability conditions on C by gluing different charts, see [14, 48].”
• Quadratic differentials as stability conditions (2013, background): “Note however that it is not usually the case that StabA(D) is the union of the chambers Stab(B) for hearts B reachable from A (see [37] for a detailed discussion of this point).”