# A finite dimensional approach to Donaldson's J-flow

11 Aug 2017

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\$ → +∞, 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 |\$L_2\$| is asymptotically Chow stable. Asymptotic Chow stability of |\$L_2\$| implies an analogue of K-semistability for the J-flow introduced by Lejmi-Székelyhidi, which we call J-semistability. We prove also that Jstability 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.