Communications in Analysis and Geometry
Volume 27 (2019)
A finite dimensional approach to Donaldson’s J‑flow
Pages: 1025 – 1085
Consider a projective manifold with two distinct polarisations $L_1$ and $L_2$. From this data, Donaldson has defined a natural flow on the space of Kähler metrics in $c_1 (L_1)$, 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 $c_1 (L_1)$ for certain polarisations $L_2$.
Associated to a quantum parameter $k \gg 0$, we define a flow over Bergman type metrics, which we call the J‑balancing flow. We show that in the quantum limit $k \to + \infty$, 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 $k \gg 0$. Defining a notion of Chow stability for linear systems, we show that this in turn implies the linear system $\lvert L_2 \rvert$ is asymptotically Chow stable. Asymptotic Chow stability of $\lvert L_2 \rvert$ 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 $L_2$, and that if $L_2$ 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.
Received 27 July 2015
Accepted 12 May 2017
Published 12 November 2019