About the Calabi problem: a finite-dimensional approach
Résumé
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 \Omega−balacing flow.At the limit of the quantization,we prove that the \Omega$-balancing flow converges towards a natural flow in K\"ahler geometry, the$\Omega$-K\"ahler flow. We study the existence of the$\Omega$-K\"ahler flow and proves its long time existence and convergence towards the solution to the Calabi problem of prescribing the volume form in a given K\"ahler class. We derive some natural geometric consequences of our study.