About the Calabi problem: a finite dimensional approach

Let us consider a projective manifold and $\Omega$ a volume form. We define the gradient flow associated to the problem of $\Omega$-balanced metrics in the quantum formalism, the \Omega$-balacing flow. At the limit of the quantization, we prove that the $\Omega$-balacing 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.