A partial compactification of the Bridgeland stability manifold
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Barbara Bolognese
Abstract
Bridgeland stability manifolds of Calabi–Yau categories are of noticeable interest both in mathematics and physics. By looking at some of the known examples, a pattern clearly emerges and gives a fairly precise description of how they look like. In particular, they all seem to have missing loci, which tend to correspond to degenerate stability conditions vanishing on spherical objects. Describing such missing strata is also interesting from a mirror-symmetric perspective, as they conjecturally parametrize interesting types of degenerations of complex structures. All the naive attempts at constructing modular partial compactifications show how elusive and subtle the problem in fact is: ideally, the missing strata would correspond to stability manifolds of quotient triangulated categories, but establishing such a correspondence on the geometric level and viewing stability conditions on quotients of the original triangulated category as suitable degenerations of stability conditions is not straightforward. In this paper, we will present a method to construct such partial compactifications if some additional hypotheses are satisfied, by realizing our space of interest as a suitable metric completion of the stability manifold.
Funding statement: This project was partially supported by the ERC Advanced Grant Stability conditions, Donaldson–Thomas invariants and cluster varieties, P. I. Tom Bridgeland.
Acknowledgement
The author wishes to heartily thank Tom Bridgeland: this paper is the result of extensive and frequent conversations with him, and has much benefited from his ideas, his knowledge and his suggestions. The author would also like to thank Domenico Fiorenza and the anonymous referee for pointing out a few mistakes in an early version of the manuscript: their comments have been extremely useful.
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Communicated by: R. Cavalieri
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Articles in the same Issue
- Frontmatter
- Exploring tropical differential equations
- On the bond polytope
- Universal convex covering problems under translations and discrete rotations
- Ehrhart theory of paving and panhandle matroids
- A partial compactification of the Bridgeland stability manifold
- The geometry of discrete L-algebras
- Projective self-dual polygons in higher dimensions
Articles in the same Issue
- Frontmatter
- Exploring tropical differential equations
- On the bond polytope
- Universal convex covering problems under translations and discrete rotations
- Ehrhart theory of paving and panhandle matroids
- A partial compactification of the Bridgeland stability manifold
- The geometry of discrete L-algebras
- Projective self-dual polygons in higher dimensions
