Stability conditions for polarised varieties

Dervan, R. (2023) Stability conditions for polarised varieties. Forum of Mathematics, Sigma, 11, e104. (doi: 10.1017/fms.2023.104)

Abstract

We introduce an analogue of Bridgeland’s stability conditions for polarised varieties. Much as Bridgeland stability is modelled on slope stability of coherent sheaves, our notion of Z-stability is modelled on the notion of K-stability of polarised varieties. We then introduce an analytic counterpart to stability, through the notion of a Z-critical Kähler metric, modelled on the constant scalar curvature Kähler condition. Our main result shows that a polarised variety which is analytically K-semistable and asymptotically Z-stable admits Z-critical Kähler metrics in the large volume regime. We also prove a local converse and explain how these results can be viewed in terms of local wall crossing. A special case of our framework gives a manifold analogue of the deformed Hermitian Yang–Mills equation.

• Item Type: Articles
• Additional Information: I was funded by a Royal Society University Research Fellowship (URF\R1\201041) for the duration of this work.
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Dervan, Dr Ruadhaí
• Authors: Dervan, R.
• College/School: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Journal Name: Forum of Mathematics, Sigma
• Publisher: Cambridge University Press
• ISSN: 2050-5094
• ISSN (Online): 2050-5094
• Published Online: 20 November 2023
• Copyright Holders: Copyright © 2023 The Author(s)
• First Published: First published in Forum of Mathematics, Sigma 11:e104
• Publisher Policy: Reproduced under a Creative Commons license

Project Code	Award No	Project Name	Principal Investigator	Funder's Name	Funder Ref	Lead Dept
318916		Moduli Spaces and Fibrations in Complex Geometry	Ian Strachan	The Royal Society (ROYSOC)	URF\R1\201041	M&S - Mathematics