Characterization of Monge-Ampere measures with Holder continuous potentials

We show that the complex Monge-Ampere equation on a compact Kaehler manifold (X,\omega) of dimension n admits a Holder continuous omega-psh solution if and only if its right-hand side is a positive measure with Holder continuous super-potential. This property is true in particular when the measure has locally Holder continuous potentials or when it belongs to the Sobolev space W^{2n/p-2+epsilon,p}(X) or to the Besov space B^{epsilon-2}_{\infty,\infty}(X) for some epsilon>0 and p>1.Comment: 17 page

On the Lefschetz and Hodge-Riemann theorems

We give an abstract version of the hard Lefschetz theorem, the Lefschetz decomposition and the Hodge-Riemann theorem for compact Kaehler manifolds.Comment: 12 page

Comparison of dynamical degrees for semi-conjugate meromorphic maps

Let f be a dominant meromorphic self-map on a projective manifold X which preserves a meromorphic fibration pi: X --> Y of X over a projective manifold Y. We establish formulas relating the dynamical degrees of f, the dynamical degrees of f relative to the fibration and the dynamical degrees of the self-map g on Y induced by f. Applications are given.Comment: 23 page

Large deviations principle for some beta-ensembles

Let L be a positive line bundle over a projective complex manifold X. Consider the space of holomorphic sections of the tensor power of order p of L. The determinant of a basis of this space, together with some given probability measure on a weighted compact set in X, induces naturally a beta-ensemble, i.e., a random point process on the compact set. Physically, this general setting corresponds to a gas of free fermions in X and may admit some random matrix models. The empirical measures, associated with such beta-ensembles, converge almost surely to an equilibrium measure when p goes to infinity. We establish a large deviations principle (LDP) with an effective speed of convergence for these empirical measures. Our study covers the case of some beta-ensembles on a compact subset of a real sphere or of a real Euclidean space.Comment: 19 pages. arXiv admin note: text overlap with arXiv:1505.0805

The mixed Hodge-Riemann bilinear relations for compact Kahler manifolds

We prove the Hodge-Riemann bilinear relations, the hard Lefschetz theorem and the Lefschetz decomposition for compact Kahler manifolds in the mixed situation.Comment: 11 pages, to appear in GAF

Heat equation and ergodic theorems for Riemann surface laminations

We introduce the heat equation relative to a positive dd-bar-closed current and apply it to the invariant currents associated with Riemann surface laminations possibly with singularities. The main examples are holomorphic foliations by Riemann surfaces in projective spaces. We prove two kinds of ergodic theorems for such currents: one associated to the heat diffusion and one close to Birkhoff's averaging on orbits of a dynamical system. The heat diffusion theorem with respect to a harmonic measure is also developed for real laminations.Comment: 44 page

Unique Ergodicity for foliations on compact K\"ahler surfaces

Let \Fc be a holomorphic foliation by Riemann surfaces on a compact K\"ahler surface X. Assume it is generic in the sense that all the singularities are hyperbolic and that the foliation admits no directed positive closed (1,1)-current. Then there exists a unique (up to a multiplicative constant) positive \ddc-closed (1,1)-current directed by \Fc. This is a very strong ergodic property of \Fc. Our proof uses an extension of the theory of densities to a class of non-\ddc-closed currents. A complete description of the cone of directed positive \ddc-closed (1,1)-currents is also given when \Fc admits directed positive closed currents.Comment: Main results improved, proofs simplified, presentation changed. 50 page

On the asymptotic behavior of Bergman kernels for positive line bundles

Let L be a positive line bundle on a projective complex manifold. We study the asymptotic behavior of Bergman kernels associated with the tensor powers L^p of L as p tends to infinity. The emphasis is the dependence of the uniform estimates on the positivity of the Chern form of the metric on L. This situation appears naturally when we approximate a semi-positive singular metric by smooth positively curved metrics.Comment: 15 page

Exponential estimates for plurisubharmonic functions and stochastic dynamics

We prove exponential estimates for plurisubharmonic functions with respect to Monge-Ampere measures with Holder continuous potential. As an application, we obtain several stochastic properties for the equilibrium measures associated to holomorphic maps on projective spaces. More precisely, we prove the exponential decay of correlations, the central limit theorem for general d.s.h. observables, and the large deviations theorem for bounded d.s.h. observables and Holder continuous observables.Comment: 24 pages, theorem and references adde

Entropy for hyperbolic Riemann surface laminations I

We develop a notion of entropy, using hyperbolic time, for laminations by hyperbolic Riemann surfaces. When the lamination is compact and transversally smooth, we show that the entropy is finite and the Poincare metric on leaves is transversally Holder continuous. A notion of metric entropy is also introduced for harmonic measures.Comment: 27 pages, Part 1. The article is adapted for the use we need in the second part of our study of hyperbolic entropy for singular foliation