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

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

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

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

Large deviation theorem for random covariance matrices

We establish a large deviation theorem for the empirical spectral distribution of random covariance matrices whose entries are independent random variables with mean 0, variance 1 and having controlled forth moments. Some new properties of Laguerre polynomials are also given.Comment: 21 page

Super-potentials, densities of currents and number of periodic points for holomorphic maps

We prove that if a positive closed current is bounded by another one with bounded, continuous or Hoelder continuous super-potentials, then it inherits the same property. There are two different methods to define wedge-products of positive closed currents of arbitrary bi-degree on compact Kaehler manifolds using super-potentials and densities. When the first method applies, we show that the second method also applies and gives the same result. As an application, we obtain a sharp upper bound for the number of isolated periodic points of holomorphic maps on compact Kaehler manifolds whose actions on cohomology are simple. A similar result still holds for a large class of holomorphic correspondences.Comment: 28 pages; dedicated to Professor Le Tuan Hoa on the occasion of his sixtieth birthda

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

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