A finite dimensional approach to Donaldson's J-flow

TL;DR

This paper introduces a finite-dimensional approach to the Donaldson's J-flow, establishing new links between stability notions and canonical metrics on complex manifolds, with implications for K-stability of surfaces.

Contribution

It develops a quantisation of the J-flow, proves new results on critical points and stability, and connects J-stability with K-stability for complex surfaces.

Findings

Existence of J-balanced metrics implies critical points of the J-flow.

J-semistability relates to K-semistability when polarisation involves the canonical bundle.

New K-stable polarisations are identified for surfaces of general type.

Abstract

We define a quantisation of the J-flow over a projective complex manifold. As corollaries, we obtain new proofs of uniqueness of critical points of the J-flow and 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 certain canonical metrics, that we call J-balanced metrics. We define a notion of Chow stability for linear systems and relate it to the existence of J-balanced metrics. We also relate the asymptotic Chow stability of a linear system to an analogue of K-semistability that was introduced by Lejmi-Sz\'ekelyhidi, which we call J-semistability. Then, we relate J-semistability to K-stability when one of the polarisation is the canonical bundle. Eventually, this gives new K-stable polarisations of surfaces of general type.