A finite dimensional approach to Donaldson’s J‑flow

Dervan, R. and Keller, J. (2019) A finite dimensional approach to Donaldson’s J‑flow. Communications in Analysis and Geometry, 27(5), pp. 1025-1085. (doi: 10.4310/CAG.2019.v27.n5.a2)

Abstract

Consider a projective manifold with two distinct polarisations L1 and L2. From this data, Donaldson has defined a natural flow on the space of Kähler metrics in c1(L1), called the J flow. The existence of a critical point of this flow is closely related to the existence of a constant scalar curvature Kähler metric in c1(L1) for certain polarisations L2. Associated to a quantum parameter k≫0, we define a flow over Bergman type metrics, which we call the J balancing flow. We show that in the quantum limit k→+∞, the rescaled J balancing flow converges towards the J flow. As corollaries, we obtain new proofs of uniqueness of critical points of the J flow and also that these critical points achieve the absolute minimum of an associated energy functional. We show that the existence of a critical point of the J flow implies the existence of J balanced metrics for k≫0. Defining a notion of Chow stability for linear systems, we show that this in turn implies the linear system |L2| is asymptotically Chow stable. Asymptotic Chow stability of |L2| implies an analogue of K semistability for the J flow introduced by Lejmi–Székelyhidi, which we call J-semistability. We prove also that J stability holds automatically in a certain numerical cone around L2, and that if L2 is the canonical class of the manifold that J semistability implies K stability. Eventually, this leads to new K stable polarisations of surfaces of general type.

• Item Type: Articles
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Dervan, Dr Ruadhaí
• Authors: Dervan, R., and Keller, J.
• College/School: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Journal Name: Communications in Analysis and Geometry
• Publisher: International Press
• ISSN: 1019-8385
• ISSN (Online): 1944-9992