A finite dimensional approach to Donaldson's J-flow

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Dervan, R 
Keller, J 

Consider a projective manifold with two distinct polarisations L1 and L2. From this data, Donaldson has defined a natural flow on the space of Kähler metrics in c1(L1), 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 c1(L1) for certain polarisations L2. Associated to a quantum parameter k 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 0. Defining a notion of Chow stability for linear systems, we show that this in turn implies the linear system |L2| is asymptotically Chow stable. Asymptotic Chow stability of |L2| 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 L2, and that if L2 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.

4904 Pure Mathematics, 49 Mathematical Sciences
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Communications in Analysis and Geometry
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International Press
The first author was funded by a studentship associated to an EPSRC Career Acceleration Fellowship (EP/J002062/1). The work of the second author has been carried out in the framework of the Labex Archimede (ANR-11-LABX-0033) and of the A*MIDEX project (ANR-11-IDEX- 0001-02), funded by the “Investissements d’Avenir” French Government programme managed by the French National Research Agency (ANR). The second author was also partially supported by supported by the ANR project EMARKS, decision No ANR-14-CE25-0010.