About the Calabi problem: a finite dimensional approach

Let us consider a projective manifold and $Ω$ a volume form. We define the gradient flow associated to the problem of $Ω$-balanced metrics in the quantum formalism, the Ω$-balacing flow. At the limit of the quantization, we prove that the$Ω$-balacing flow converges towards a natural flow in Kähler geometry, the$Ω$-Kähler flow. We study the existence of the$Ω$-Kähler flow and proves its long time existence and convergence towards the solution to the Calabi problem of prescribing the volume form in a given Kähler class. We derive some natural geometric consequences of our study.

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