arXiv subjects

Duc-Viet Vu

Publications and source records attributed to Duc-Viet Vu.

45 records · Page 3Linked to original sources

arXiv2017

Entire holomorphic curves into projective spaces intersecting a generic hypersurface of high degree

In this note, we establish the following Second Main Theorem type estimate for every entire non-algebraically degenerate holomorphic curve $f\colon\mathbb{C}\rightarrow\mathbb{P}^n(\mathbb{C})$, in present of a {\sl generic} hypersuface $D\subset\mathbb{P}^n(\mathbb{C})$ of sufficiently high degree $d\geq 15(5n+1)n^n$: \[ T_f(r) \leq \,N_f^{[1]}(r,D) + O\big(\log T_f(r) + \log r \big)\parallel, \] where $T_f(r)$ and $N_f^{[1]}(r,D)$ stand for the order function and the $1$-truncated counting function in Nevanlinna theory. This inequality quantifies recent results on the logarithmic Green--Griffiths conjecture.

math.AG↗

arXiv2017

Meromorphic maps of Kahler manifolds with trivial canonical bundles

Let M be a (bounded or not) domain of C^n which is complete with respect to a Kähler metric, or more generally, a complete Kähler manifold with trivial canonical bundle. Let f be a linearly nondegenerate meromorphic map from M to the complex projective space P^m. Under an assumption on the positivity of the pull-back by f of the Fubini-Study form on P^m, we prove that f can not omit a certain number of hyperplanes in subgeneral position in P^m. This is deduced directly from a non-integrated defect relation for such f which generalizes that obtained by Fujimoto in the case where M is a ball.

math.CV↗

arXiv2017

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.

math.DS↗

arXiv2017

Families of Monge-Ampere measures with Holder continuous potentials

Let X be a compact Kahler manifold. Let F be a family of probability measures on X whose superpotentials are Holder continuous with uniform Holder exponent and Holder constant. We prove that the corresponding family of solutions of the complex Monge-Ampere equations whose right-hand sides are measures in F is Holder continuous.

math.CV↗

arXiv2017

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.

math.CV↗

arXiv2016

Complex Monge-Ampère equation for measures supported on real submanifolds

Let $(X,ω)$ be a compact $n$-dimensional Kähler manifold on which the integral of $ω^n$ is $1$. Let $K$ be an immersed real $\mathcal{C}^3$ submanifold of $X$ such that the tangent space at any point of $K$ is not contained in any complex hyperplane of the (real) tangent space at that point of $X.$ Let $μ$ be a probability measure compactly supported on $K$ with $L^p$ density for some $p>1.$ We prove that the complex Monge-Ampère equation $(dd^c φ+ ω)^n=μ$ has a Hölder continuous solution.

math.CV↗

arXiv2015

Equidistribution rate for Fekete points on some real manifolds

Let L be a positive line bundle over a compact complex projective manifold X and K be a compact subset of X which is regular in a sense of pluripotential theory. A Fekete configuration of order k is a finite subset of K maximizing a Vandermonde type determinant associated with the power L^k of L. Berman, Boucksom and Witt Nyström proved that the empirical measure associated with a Fekete configuration converges to the equilibrium measure of K as k tends to infinity. Dinh, Ma and Nguyen obtained an estimate for the rate of convergence. Using techniques from Cauchy-Riemann geometry, we show that the last result holds when K is a real nondegenerate C^5-piecewise submanifold of X such that its tangent space at any regular point is not contained in a complex hyperplane of the tangent space of X at that point. In particular, the estimate holds for Fekete points on some compact sets in R^n or the unit sphere in R^{n+1}.

math.CV↗

arXiv2015

Intersection of positive closed currents of higher bidegree

Let $X$ be a compact Kähler manifold of dimension $n.$ Let $T$ and $S$ be two positive closed currents on $X$ of bidegree $(p,p)$ and $(q,q)$ respectively with $p+q\le n.$ Assume that $T$ has a continuous super-potential. We prove that the wedge product $T \wedge S,$ defined by Dinh and Sibony, is a positive closed current.

math.DS↗

arXiv2012

Asymptotic number of scattering resonances for generic Schrodinger operators

Let -Delta+V be the Schrodinger operator acting on L^2(R^d,C) with d>2 odd. Here V is a bounded real or complex function vanishing outside the closed ball of center 0 and of radius a. We show for generic potentials V that the number of resonances of -Delta+V with modulus less than r is approximatively equal to a constant times a^dr^d.

math.SP↗