A finite dimensional approach to Donaldson's J-flow

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.