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Tien-Cuong Dinh

Publications and source records attributed to Tien-Cuong Dinh.

At least 19 recordsLinked to original sources
arXiv2026

Siu's analyticity theorem for positive pluriharmonic currents

Let $T$ be a positive $dd^c$-closed current of bidimension $(q,q)$ on a compact Kähler manifold $X$. For every $c>0$, let $E_c(T)$ be the set of points of $X$ where the Lelong number of $T$ is larger or equal to $c$. We show that $E_c(T)$ is an analytic subset of dimension at most $q$ of $X$. Moreover, the following Siu decomposition holds $$T=\sum_{i\in I} λ_i[V_i] +T_0,$$ where $\{V_i\}_{i\in I}$ is a (possibly empty) finite or countable family of compact analytic subsets of dimension $q$ in $X$, $λ_i\in\mathbb{R}^+$, and $T_0$ is a positive $dd^c$-closed current such that $E_c(T_0)$ is an analytic subset of dimension at most $q-1$ of $X$ for every $c>0$. The proof relies on the duality between the pseudoeffective cone and the movable cone of a compact Kähler manifold, recently obtained by Tosatti (2026), together with a theorem of Vigny (2009) and the theory of density currents for positive $dd^c$-closed currents developed by Sibony, the first and third authors.

math.CV↗
arXiv2026

A rational surface with discrete and non-finitely generated automorphism group

We construct a smooth complex projective rational surface whose automorphism group is discrete and not finitely generated. It is obtained by blowing up a point on a branch curve of a rational quotient of a product Kummer surface. Every point with transcendental coordinate gives such a surface. The proof uses leading jets along the branch curve, the complete family of blowups and the cubic intersection of the family, which are not being considered in relevant earlier works.

math.AG↗
arXiv2026

Quantitative Dynamics of complex Hénon maps

Let $f$ be a Hénon map of $\mathbb C^2$. We provide quantitative versions of several results involving dynamical objects associated to $f$. Our results can be interpreted as a quantitative version of Pesin theory with geometric control. As an application we show that the perdiodic points of period $n$ of $f$ equidistribute towards its equilibrium measure exponentially fast as $n$ tends to infinity.

math.DS↗
arXiv2026

Quantitative hyperbolicity for complex manifolds via numerical invariants

We introduce numerical invariants called hyperbolic indices, which measure the hyperbolicity of compact Kähler manifolds using directed positive closed currents. We prove that if a manifold $X$ has positive hyperbolic indices, then $X$ is Kobayashi hyperbolic; and if $X$ satisfies Demailly's condition of negative jet curvature, then it has positive hyperbolic indices. In particular, by combining the method of jet differentials and density currents, we can prove that for a general hypersurface $X_d$ of degree $d$ in $\mathbb{P}^{n+1}$, the hyperbolic indices of $X_d$ grows to $\infty$ with at least linear growth in $d$. Finally, we discuss an analytic approach to the Kobayashi conjecture.

math.CV↗
arXiv2025

Lower Bounds for Galois Orbits of periodic points for polarized endomorphisms

Let $K$ be a number field, $X$ a smooth projective variety over $K$ and $f: X \to X$ a polarized endomorphism of degree $d \geq 2$. We prove an exponential lower bound on $[K(\Per_n):K]$, where $\Per_n$ is the set of $n$-periodic points, extending results of [Yap24] to higher dimensions. We also prove a quantitative rate of equidistribution for $\Per_n$ to the equilibrium measure.

math.NT↗
arXiv2025

Gaussian fluctuations for spin systems and point processes: near-optimal rates via quantitative Marcinkiewicz's theorem

We establish asymptotically Gaussian fluctuations for functionals of a large class of spin models and strongly correlated random point fields, achieving near-optimal rates. For spin models, we demonstrate Gaussian asymptotics for the magnetization for a wide class of ferromagnetic spin systems on Euclidean lattices, in particular those with continuous spins. Specific applications include, in particular, the celebrated XY and Heisenberg models under ferromagnetic conditions, and more broadly, systems with very general rotationally invariant spins in arbitrary dimensions. We address both the setting of free boundary conditions and a large class of ferromagnetic boundary conditions, and our CLTs are endowed with near-optimal rate. Our approach leverages the classical Lee-Yang theory for the zeros of partition functions, and subsumes as a special case results of Lebowitz, Ruelle, Pittel and Speer on CLTs in discrete statistical mechanical models for which we obtain sharper convergence rates. In a different direction, we obtain CLTs for linear statistics of a wide class of point processes known as $α$-determinantal point processes which interpolate between negatively and positively associated random point fields. We contribute a unified approach to CLTs in such models; significantly, our approach is able to analyse such processes in dimensions $\ge 3$, where structural alternatives such as connections to random matrix theory are not available. A key ingredient of our approach is a broad, quantitative extension of the classical Marcinkiewicz Theorem that holds under the limited condition that the characteristic function is non-vanishing only on a bounded disk. In spite of the general applicability of the results, our rates for the CLT match the classic Berry-Esseen bounds for independent sums up to a log factor.

math.PR↗
arXiv2025

On the virtual invariants of zero entropy groups of compact Kähler manifolds

Let $X$ be a compact Kähler manifold. We study subgroups $G \le \mathrm{Aut}(X)$ of biholomorphic automorphisms of zero entropy when $\mathrm{Aut}^0(X)$ is compact (e.g. when $\mathrm{Aut}^0(X)$ is trivial). We show that the virtual derived length $\ell_{\mathrm{vir}}(G)$ of $G$ satisfies $\ell_{\mathrm{vir}}(G) \le \dim X -κ(X)$, where $κ(X)$ is the Kodaira dimension of $X$. Modulo the main conjecture of our previous work concerning the essential nilpotency class, we obtain the same upper bound $c_{\mathrm{vir}}(G) \le \dim X -κ(X)$ for the virtual nilpotency class $c_{\mathrm{vir}}(G)$, together with a geometric description of the $G$-action on $X$ when the equality holds.

math.AG↗
arXiv2025

Exponential equidistribution of periodic points for endomorphisms of $\mathbb P^k$

Let $f$ be a holomorphic endomorphism of $\mathbb P^k$ of algebraic degree $d\geq 2$. We show that the periodic points of $f$ of period $n$ equidistribute towards the equilibrium measure of $f$ exponentially fast as $n$ tends to infinity. This quantifies a theorem of Lyubich for $k=1$ and of Briend-Duval for $k\geq 2$. A byproduct of our proof is the existence of a large number of periodic cycles in the small Julia set with large multipliers.

math.DS↗
arXiv2024

Hölder continuity and laminarity of the Green currents for Hénon-like maps

Under a natural assumption on the dynamical degrees, we prove that the Green currents associated to any Hénon-like map in any dimension have Hölder continuous super-potentials, i.e., give Hölder continuous linear functionals on suitable spaces of forms and currents. As a consequence, the unique measure of maximal entropy is the Monge-Ampère of a Hölder continuous plurisubharmonic function and has strictly positive Hausdorff dimension. Under the same assumptions, we also prove that the Green currents are woven. When they are of bidegree $(1,1)$, they are laminar. In particular, our results generalize results known until now only in algebraic settings, or in dimension 2.

math.CV↗
arXiv2024

Hole event for random holomorphic sections on compact Riemann surfaces

Let $X$ be a compact Riemann surface and $\mathcal L$ be a positive line bundle on it. We study the conditional zero expectation of all the holomorphic sections of $\mathcal L^n$ which do not vanish on $D$ for some fixed open subset $D$ of $X$. We prove that as $n$ tends to infinity, the zeros of these sections are equidistributed outside $D$ with respect to a probability measure $ν$. This gives rise to a surprising forbidden set.

math.CV↗
arXiv2023

Periodic points for meromorphic self-maps of Fujiki varieties

Let $f\colon X\to X$ be a dominant meromorphic self-map of a compact complex variety $X$ in the Fujiki class $\mathcal{C}$. If the topological degree of $f$ is strictly larger than the other dynamical degrees of $f$, we show that the number of isolated $f$-periodic points grows exponentially fast similarly to the topological degrees of the iterates of $f$; in particular, we give a positive answer to a conjecture of Shou-Wu Zhang. In the general case, we show that the exponential growth of the number of isolated $f$-periodic points is at most the algebraic entropy of $f$.

math.DS↗
arXiv2023

Monotonicity of dynamical degrees for H{é}non-like and polynomial-like maps

We prove that, for every invertible horizontal-like map (i.e., H{é}non-like map) in any dimension, the sequence of the dynamical degrees is increasing until that of maximal value, which is the main dynamical degree, and decreasing after that. Similarly, for polynomial-like maps in any dimension, the sequence of dynamical degrees is increasing until the last one, which is the topological degree. This is the first time that such a property is proved outside of the algebraic setting. Our proof is based on the construction of a suitable deformation for positive closed currents, which relies on tools from pluripotential theory and the solution of the $d$, $\bar \partial$, and $dd^c$ equations on convex domains.

math.CV↗
arXiv2023

Exponential mixing of all orders and CLT for automorphisms of compact K{ä}hler manifolds

We consider the unique measure of maximal entropy of an automorphism of a compact K{ä}hler manifold with simple action on cohomology. We show that it is exponentially mixing of all orders with respect to H{ö}lder observables. It follows that the Central Limit Theorem (CLT) holds for these observables. In particular, our result applies to all automorphisms of compact K{ä}hler surfaces with positive entropy.

math.CV↗
arXiv2023

Random walks on SL_2(C): spectral gap and limit theorems

We obtain various new limit theorems for random walks on SL_2(C) under low moment conditions. For non-elementary measures with a finite second moment, we prove a Local Limit Theorem for the norm cocycle, yielding the optimal version of a theorem of E. Le Page. For measures with a finite third moment, we obtain the Local Limit Theorem for the matrix coefficients, improving a recent result of Grama-Quint-Xiao and the authors, and Berry-Esseen bounds with optimal rate $O(1 / \sqrt n)$ for the norm cocycle and the matrix coefficients. The main tool is a detailed study of the spectral properties of the Markov operator and its purely imaginary perturbations acting on different function spaces. We introduce, in particular, a new function space derived from the Sobolev space $W^{1,2}$ that provides uniform estimates.

math.PR↗
arXiv2023

Regularity of the equilibrium measure for meromorphic correspondences

Let $f$ be a meromorphic correspondence on a compact Kähler 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.

math.CV↗
arXiv2023

Smooth projective surfaces with infinitely many real forms

The aim of this paper is twofold. First of all, we confirm a few basic criteria of the finiteness of real forms of a given smooth complex projective variety, in terms of the Galois cohomology set of the discrete part of the automorphism group, the cone conjecture and the topological entropy. We then apply them to show that a smooth complex projective surface has at most finitely many non-isomorphic real forms unless it is either rational or a non-minimal surface birational to either a K3 surface or an Enriques surface. In the second part of the paper, we construct an Enriques surface whose blow-up at one point admits infinitely many non-isomorphic real forms. This answers a question of Kondo to us and also shows the three exceptional cases really occur.

math.AG↗