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Let $\mu $ be a probability measure on $\mathrm {GL}_d(\mathbb {R})$ , and denote by $S_n:= g_n \cdots g_1$ the associated random matrix product, where $g_j$ are i.i.d. with law $\mu $ . Under the assumptions that $\mu $ has a finite exponential moment and generates a proximal and strongly irreducible semigroup, we prove a Berry–Esseen bound with the optimal rate $O(1/\sqrt n)$ for the coefficients of $S_n$ , settling a long-standing question considered since the fundamental work of Guivarc’h and Raugi. The local limit theorem for the coefficients is also obtained, complementing a recent partial result of Grama, Quint and Xiao.

We study zero entropy automorphisms of a compact K\"ahler manifold $X$. Our goal is to bring to light some new structures of the action on the cohomology of $X$, in terms of the so-called dynamical filtrations on $H^{1,1}(X, {\mathbb R})$. Based on these filtrations, we obtain the first general upper bound on the polynomial growth of the iterations $(g^m)^* \, {\circlearrowleft} \, H^2(X, {\mathbb C})$ where $g$ is a zero entropy automorphism, in terms of ${\rm dim} \, X$ only. We also give an upper bound for the (essential) derived length $\ell_{\rm ess}(G, X)$ for every zero entropy subgroup $G$, again in terms of the dimension of $X$ only. We propose a conjectural upper bound for the essential nilpotency class $c_{\rm ess}(G,X)$ of a zero entropy subgroup $G$. Finally, we construct examples showing that our upper bound of the polynomial growth (as well as the conjectural upper bound of $c_{\rm ess}(G,X)$) are optimal., Comment: Geometric and Functional Analysis (2022), to appear; Title shortened a bit; Part of the content is moved to another paper in progress

Let f be a meromorphic correspondence on a compact Kahler manifold X of dimension k. Assume that its topological degree is larger than the dynamical degree of order k - 1. We obtain a quantitative regularity of the equilibrium measure of f in terms of its super-potentials. Classification AMS 2020: 32H, 37F05, 32U40.

We show, among other things, that for each integer $$n \ge 3$$ , there is a smooth complex projective rational variety of dimension n, with discrete non-finitely generated automorphism group and with infinitely many mutually non-isomorphic real forms. Our result is inspired by the work of Lesieutre and the work of Dinh and Oguiso.

We study local positive d d c dd^{c} -closed currents directed by a foliation by Riemann surfaces near a hyperbolic singularity which have no mass on the separatrices. A theorem of Nguyên says that the Lelong number of such a current at the singular point vanishes. We prove that this property is sharp: one cannot have any better mass estimate for this current near the singularity.

We present a list of open questions in the theory of holomorphic foliations, possibly with singularities. Some problems have been around for a while, others are very accessible.

Let $X$ be a compact Kähler manifold of dimension $n$ and $ω$ a Kähler form on $X$. We consider the complex Monge-Ampère equation $(dd^c u+ω)^n=μ$, where $μ$ is a given positive measure on $X$ of suitable mass and $u$ is an $ω$-plurisubharmonic function. We show that the equation admits a Hölder continuous solution {\it if and only if} the measure $μ$, seen as a functional on a complex Sobolev space $W^*(X)$, is Hölder continuous. A similar result is also obtained for the complex Monge-Ampère equations on domains of $\mathbb{C}^n$., 16 pages. Final version, to appear in Bulletin of the London Mathematical Society

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., 28 pages; dedicated to Professor Le Tuan Hoa on the occasion of his sixtieth birthday

Let L L be a positive line bundle over a projective complex manifold X X , L p L^p its tensor power of order p p , H 0 ( X , L p ) H^0(X,L^p) the space of holomorphic sections of L p L^p , and N p N_p the complex dimension of H 0 ( X , L p ) H^0(X,L^p) . The determinant of a basis of H 0 ( X , L p ) H^0(X,L^p) , together with some given probability measure on a weighted compact set in X X , induces naturally a β \beta -ensemble, i.e., a random N p N_p -point process on the compact set. Physically, depending on X X and the value of β \beta , this general setting corresponds to a gas of free or interacting fermions on X X and may admit an interpretation in terms of some random matrix models. The empirical measures, associated with such β \beta -ensembles, converge almost surely to an equilibrium measure when p p goes to infinity. We establish a large deviation theorem (LDT) with an effective speed of convergence for these empirical measures. Our study covers a large class of β \beta -ensembles on a compact subset of the unit sphere S n ⊂ R n + 1 \mathbb {S}^n\subset \mathbb {R}^{n+1} or of the Euclidean space R n \mathbb {R}^n .

Consider a foliation in the projective plane admitting a projective line as the unique invariant algebraic curve. Assume that the foliation is generic in the sense that its singular points are hyperbolic. We show that there is a unique positive $${dd^c}$$ -closed (1, 1)-current of mass 1 which is directed by the foliation and this is the current of integration on the invariant line. A unique ergodicity theorem for the distribution of leaves follows: for any leaf L, appropriate averages of L converge to the current of integration on the invariant line. The result uses an extension of our theory of densities for currents. Foliations on compact Kahler surface with one or more invariant curves are also considered.

Let f be a rational self-map on a projective manifold. In general, the set of periodic points of f may have a positive dimension. We will discuss some estimates for the number of isolated periodic points. Although the question is of algebraic nature, the techniques we used are based on some recent progress in complex analysis of several variables. In this report, we will present the main results and approaches in this direction and try to give a guide for non-experts in the reading of the original articles., 於 城崎国際アートセンター(2017年10月24日-10月27日), 平成29年度科学研究費補助金 基盤研究(S)(課題番号25220701, 代表 向井 茂), 平成29年度科学研究費補助金 基盤研究(S)(課題番号17H06127, 代表 齋藤 政彦), 平成29年度科学研究費補助金 基盤研究(S)(課題番号15H05738, 代表 金銅 誠之), Date : Oct. 24, 2017 (Tue) — Oct. 27, 2017 (Fri). Venue : Kinosaki International Arts Center., Kinosaki algebraic geometry symposium 2017 is partially supported by Grantin-Aid for Scientific Research (S) 25220701, (S) 15H05738, and (S) 17H06127, We are grateful to the support. Organizers: T. Fujisawa (Tokyo Denki), T. Okada (Saga), T. Sano(Kobe)

Working within the polynomial quadratic family, we introduce a new point of view on bifurcations which naturally allows to see the seat of bifurcations as the projection of a Julia set of a complex dynamical system in dimension three. We expect our approach to be extendable to other holomorphic families of dynamical systems. MSC 2010: 37Fxx, 32H50.

We give a deviation estimate for the empirical spectral distribution of random covariance matrices whose entries are independent random variables with mean 0, variance 1, and controlled fourth moments. We also give some new properties of Laguerre polynomials.

We establish the equidistribution of zeros of random holomorphic sections of powers of a semipositive singular Hermitian line bundle, with an estimate of the convergence speed.

In this paper, we prove the equidistribution of saddle periodic points for Henon-type automorphisms of \(\mathbb {C}^k\) with respect to its equilibrium measure. A general strategy to obtain equidistribution properties in any dimension is presented. It is based on our recent theory of densities for positive closed currents. Several fine properties of dynamical currents are also proved.

We study the global dynamics of holomorphic correspondences [Formula: see text] on a compact Riemann surface [Formula: see text] in the case, so far not well understood, where [Formula: see text] and [Formula: see text] have the same topological degree. In the absence of a mild and necessary obstruction that we call weak modularity, [Formula: see text] admits two canonical probability measures [Formula: see text] and [Formula: see text] which are invariant by [Formula: see text] and [Formula: see text] respectively. If the critical values of [Formula: see text] (respectively, [Formula: see text]) are not periodic, the backward (respectively, forward) orbit of any point [Formula: see text] equidistributes towards [Formula: see text] (respectively, [Formula: see text]), uniformly in [Formula: see text] and exponentially fast.

Let G be a group of automorphisms of a compact K\"ahler manifold X of dimension n and N(G) the subset of null-entropy elements. Suppose G admits no non-abelian free subgroup. Improving the known Tits alternative, we obtain that, up to replace G by a finite-index subgroup, either G/N(G) is a free abelian group of rank < n-1, or G/N(G) is a free abelian group of rank n-1 and X is a complex torus, or G is a free abelian group of rank n-1. If the last case occurs, X is G-equivariant birational to the quotient of an abelian variety provided that X is a projective manifold of dimension n > 2 and is not rationally connected. We also prove and use a generalization of a theorem by Fujiki and Lieberman on the structure of Aut(X)., Comment: Proof of Theorem 4.1(1) is slightly changed

We prove that the Julia set of a Henon type automorphism on C^2 is very rigid: it supports a unique positive ddc-closed current of mass 1. A similar property holds for the cohomology class of the Green current associated with an automorphism of positive entropy on a compact Kaehler surface. Relations between this phenomenon, several quantitative equidistribution properties and the theory of value distribution will be discussed. We also survey some rigidity properties of Henon type maps on C^k and of automorphisms of compact Kaehler manifolds., Comment: 56 pages, to appear in Journal of Modern Dynamics, Volume 8, No. 3, 2014

We show that there is a smooth complex projective variety, of any dimension greater than or equal to two, whose automorphism group is discrete and not finitely generated. Moreover, this variety admits infinitely many real forms which are mutually non-isomorphic over the real number field. Our result is inspired by the work of Lesieutre and answers questions by Dolgachev, Esnault and Lesieutre., final form to appear in Duke Math. J

Let -Delta+V be the Schrodinger operator acting on L^2(R^d,C) with d odd larger than 2. Here V is a bounded real- or complex-valued function vanishing outside the closed ball of center 0 and radius a. If V belongs to the class of potentials introduced by Christiansen, we show that when r goes to infinity, the resonances of -Delta+V, scaled down by the factor r, are asymptotically distributed, with respect to an explicit probability distribution on the closed lower unit half-disc of the complex plane. The rate of convergence is also considered for subclasses of potentials., 19 pages

We show that the complex Monge–Ampere equation on a compact Kahler manifold ( X , ω ) of dimension n admits a Holder continuous ω-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 2 n / p − 2 + ϵ , p ( X ) or to the Besov space B ∞ , ∞ ϵ − 2 ( X ) for some ϵ > 0 and p > 1 .

We give abstract versions of the large deviation theorem for the distribution of zeros of polynomials and apply them to the characteristic polynomials of Hermitian random matrices. We obtain new estimates related to the local semi-circular law for the empirical spectral distribution of these matrices when the 4th moments of their entries are controlled., 19 pages

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., 15 pages

We show that any dominant meromorphic self-map f of a compact Kaehler manifold X is an Artin-Mazur map. More precisely, if P_n(f) is the number of its isolated periodic points of period n (counted with multiplicity), then P_n(f) grows at most exponentially fast with respect to n and the exponential rate is at most equal to the algebraic entropy of f. Further estimates are given when X is a surface. Among the techniques introduced in this paper, the h-dimension of the density between two arbitrary positive closed currents on a compact Kaehler surface is obtained., Comment: 18 pages

We consider N-tensor powers of a positive Hermitian line bundle L over a non-compact complex manifold X. In the compact case, B. Shiffman and S. Zelditch proved that the zeros of random sections become asymptotically uniformly distributed with respect to the natural measure coming from the curvature of L, as N tends to infinity. Under certain boundedness assumptions on the curvature of the canonical line bundle of X and on the Chern form of L we prove a non-compact version of this result. We give various applications, including the limiting distribution of zeros of cusp forms with respect to the principal congruence subgroups of SL2(Z) and to the hyperbolic measure, the higher dimensional case of arithmetic quotients and the case of orthogonal polynomials with weights at infinity. We also give estimates for the speed of convergence of the currents of integration on the zero-divisors., 25 pages; v.2 is a final update to agree with the published paper

We study holomorphic automorphisms on compact Kähler manifolds having simple actions on the Hodge cohomology ring. We show for such automorphisms that the main dynamical Green currents admit complex laminar structures (woven currents) and the Green measure is the unique invariant probability measure of maximal entropy.

We introduce a notion of super-potential (canonical function) associated to positive closed ( p , p ) (p,p) -currents on compact Kähler manifolds and we develop a calculus on such currents. One of the key points in our study is the use of deformations in the space of currents. As an application, we obtain several results on the dynamics of holomorphic automorphisms: regularity and uniqueness of the Green currents. We also get regularity, entropy, ergodicity and hyperbolicity of the equilibrium measures.

We prove some properties of analytic multiplicative and sub-multiplicative cocycles. The results allow to construct natural invariant analytic sets associated to complex dynamical systems., 11 pages

Let f be a meromorphic correspondence on a compact Kahler manifold. We show that the topological entropy of f is bounded from above by the logarithm of its maximal dynamical degree. An analogous estimate for the entropy on subvarieties is given. We also discuss a notion of Julia and Fatou sets.

Let f be a holomorphic endomorphism of ℙ k having an attracting setA. We construct an attracting current and an equilibrium measure associated toA. The attracting current is weakly laminar and extremal in the cone of invariant currents. The equilibrium measure is mixing and has maximal entropy onA.

Let K be the closure of a bounded open set with smooth boundary in C^n. A Fekete configuration of order p for K is a finite subset of K maximizing the Vandermonde determinant associated with polynomials of degree at most p. A recent theorem by Berman, Boucksom and Witt Nystrom implies that Fekete configurations for K are asymptotically equidistributed with respect to a canonical equilibrium measure, as p tends to infinite. We give here an explicit estimate for the speed of convergence. The result also holds in a general setting of Fekete points associated with an ample line bundle over a projective manifold. Our approach requires a new estimate on Bergman kernels for line bundles and quantitative results in pluripotential theory which are of independent interest., 33 pages, minor modifications, to appear in Annales de l'Ecole Normale Superieure

We prove the Hodge–Riemann bilinear relations, the hard Lefschetz theorem and the Lefschetz decomposition for compact Kahler manifolds in the mixed situation.

We find a class of manifolds whose ‘pseudoconcave holes’ can be filled in, even in dimension two. To cite this article: G. Marinescu, T.-C. Dinh, C. R. Acad. Sci. Paris, Ser. I 342 (2006).

Let f be a dominant meromorphic self-map of large topological degree on a compact Kahler manifold. We give a new construction of the equilibrium measure μ of f and prove that μ is exponentially mixing. As a consequence, we get the central limit theorem in particular for Holder-continuous observables, but also for noncontinuous observables. © 2005 Wiley Periodicals, Inc.

We show that the pseudoconcave holes of some naturally arising class of manifolds, called hyperconcave ends, can be filled in, including the case of complex dimension 2 . As a consequence we obtain a stronger version of the compactification theorem of Siu-Yau and extend Nadel's theorems to dimension 2., Comment: 13 pages, AMSLaTeX, short version accepted for publication in Inventiones Math

In this paper we give an introduction to the notion of meromorphic transform. We describe some equidistribution problems and their solution, using the ddc-method. In particular, we give some statistical properties of the equilibrium measure for meromorphic maps on compact Kahler manifolds: K-mixing, exponential decay of correlations and central limit theorem.

We construct the Green current for a random iteration of horizontal-like mappings in Open image in new window. This is applied to the study of a polynomial map Open image in new window with the following properties

Let Fn: X1 → X2 be a sequence of (multivalued) meromorphic maps between compact Kahler manifolds. We study the asymptotic distribution of preimages of points by Fn and, for multivalued self-maps of a compact Riemann surface, the asymptotic distribution of repelling fixed points.

We develop the study of some spaces of currents of bidegree ( p , p ) . As an application we construct the equilibrium measure for a large class of birational maps of P k , as intersection of Green currents. We show that these currents are extremal and that the corresponding measure is mixing.

Let X be a complex projective manifold and let f be a dominating rational map from X onto X. We show that the topological entropy h(f) of f is bounded from above by the logarithm of its maximal dynamical degree.

Nous construisons pour toute correspondance polynomiale F d'exposant de Lojasiewicz l > 1 une mesure d'equilibre μ. Nous montrons que μ est approximable par les preimages d'un point generique et que les points periodiques repulsifs sont equidistribues sur le support de μ, En utilisant ces resultats, nous donnons une caracterisation des ensembles d'unicite pour les polynomes.

Letffbe a holomorphic automorphism of a compact Kähler manifold(X,ω)(X,\omega )of dimensionk≥2k\geq 2. We study the convex cones of positive closed(p,p)(p,p)-currentsTpT_p, which satisfy a functional relation\[f∗Tp=λTp, λ>1,f^* T_p=\lambda T_p,\ \ \lambda >1,\]and some regularity condition (PB, PC). Under appropriate assumptions on dynamical degrees we introduce closed finite dimensional cones, not reduced to zero, of such currents. In particular, when the topological entropyh(f)\mathrm {h}(f)offfis positive, then for somem≥1m\geq 1, there is a positive closed(m,m)(m,m)-currentTmT_mwhich satisfies the relation\[f∗Tm=exp⁡(h(f))Tm.f^* T_m=\exp (\mathrm {h}(f)) T_m.\]Moreover, every quasi-p.s.h. function is integrable with respect to the trace measure ofTmT_m. When the dynamical degrees offfare all distinct, we construct an invariant measureμ\muas an intersection of closed currents. We show that this measure is mixing and gives no mass to pluripolar sets and to sets of small Hausdorff dimension.