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.