In this paper we consider the dynamical system involved by the Ricci operator on the space of Kähler metrics. A. Nadel has defined an iteration scheme given by the Ricci operator for Fano manifold and asked whether it has some nontrivial periodic points. First, we prove that no such periodic points can exist. We define the inverse of the Ricci operator and consider the dynamical behaviour of its iterates for a Fano Kähler-Einstein manifold. In particular we show that the iterates do converge to the Kähler-Ricci soliton for toric manifolds. Finally, we define a finite dimensional procedure to give an approximation of Kähler-Einstein metrics using this iterative procedure and apply it for P 2 blown up in 3 points.

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.Contents metrics. In the present work we study a flow on the space of Bergman metrics, which we call the J-balancing flow. Critical points of the J-balancing flow are called J-balanced metrics, and these fit naturally into a finite dimensional moment map picture. Our main result is that in the quantum limit k → +∞, the J-balancing flow converges to the J-flow.Theorem 1.1. Fix T > 0 and let ω k (t) be the solution of the J-balancing flow, for t ∈ [0, T ]. Then as k → ∞, the sequence ω k (t) converges in C ∞ to the solution of the J-flow as k → ∞. Furthermore, the convergence is C 1 in the variable t. Assuming there is a critical point of the J-flow, the convergence holds for all t > 0.By showing J-balanced metrics are unique, we obtain the following Corollary, which is an analogue of Donaldson's quantisation proof of uniqueness of cscK metrics [Don01]. Corollary 1.2. Critical metrics of the J-flow are unique.This was first proven by Chen using the strict convexity of the I µ J functional along certain geodesics in the space of Kähler metrics on L 1 [Che04]. Similarly, we recover the fact that J-balanced metrics achieve the absolute minimum of an associated functional. Corollary 1.3. Critical metrics of the J-flow achieve the absolute minimum of the I µ J functional.This is analogous to Donaldson's proof that cscK metrics achieve the absolute minimum of the Mabuchi functional [Don05].Our next results relate the existence of a critical metric to algebro-geometric notions of stability. We define a notion of Chow stability for a linear system on a polarised manifold, and by relating this notion of stabilit...

We introduce a simple and very fast algorithm to compute Weil-Petersson metrics on moduli spaces of Calabi-Yau varieties. Additionally, we introduce a second algorithm to approximate the same metric using Donaldson's quantization link between infinite and finite dimensional Geometric Invariant Theoretical (GIT) quotients that describe moduli spaces of varieties. Although this second algorithm is slower and more sophisticated, it can also be used to compute similar metrics on other moduli spaces (e.g. moduli spaces of vector bundles on Calabi-Yau varieties). We study the convergence properties of both algorithms and provide explicit computer implementations using a family of Calabi-Yau quintic hypersurfaces in P 4 . Also, we include discussions on: the existing methods that are used to compute this class of metrics, the background material that we use to build our algorithms, and how to extend the second algorithm to the vector bundle case.