Summary: 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 \(\vert L_2 \vert\) is asymptotically Chow stable. Asymptotic Chow stability of \(\vert L_2 \vert\) 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.