arXiv subjects

Viêt-Anh Nguyên

Publications and source records attributed to Viêt-Anh Nguyên.

7 recordsLinked to original sources

arXiv2024

Generic singularities of holomorphic foliations by curves on $\mathbb{P}^n$

Let $\mathcal{F}_d(\mathbb{P}^n)$ be the space of all singular holomorphic foliations by curves on $\mathbb{P}^n$ ($n \geq 2$) with degree $d \geq 1.$ We show that there is subset $\mathcal{S}_d(\mathbb{P}^n)$ of $\mathcal{F}_d(\mathbb{P}^n)$ with full Lebesgue measure with the following properties: 1. for every $\mathcal{F} \in \mathcal{S}_d(\mathbb{P}^n),$ all singular points of $\mathcal{F}$ are linearizable hyperbolic. 2. If, moreover, $d \geq 2,$ then every $\mathcal{F}$ does not possess any invariant algebraic curve.

math.CV↗

arXiv2024

Zeros of random holomorphic sections of big line bundles with continuous metrics

Let $X$ be a compact normal complex space, $L$ be a big holomorphic line bundle on $X$ and $h$ be a continuous Hermitian metric on $L$. We consider the spaces of holomorphic sections $H^0(X, L^{\otimes p})$ endowed with the inner product induced by $h^{\otimes p}$ and a volume form on $X$, and prove that the corresponding sequence of normalized Fubini-Study currents converge weakly to the curvature current $c_1(L,h_{\mathrm{eq}})$ of the equilibrium metric $h_{\mathrm{eq}}$ associated to $h$. We also show that the normalized currents of integration along the zero divisors of random sequences of holomorphic sections converge almost surely to $c_1(L,h_{\mathrm{eq}})$, for very general classes of probability measures on $H^0(X, L^{\otimes p})$.

math.CV↗

arXiv2023

Restricted spaces of holomorphic sections vanishing along subvarieties

Let $X$ be a compact normal complex space of dimension $n$ and $L$ be a holomorphic line bundle on $X$. Suppose that $Σ=(Σ_1,\ldots,Σ_\ell)$ is an $\ell$-tuple of distinct irreducible proper analytic subsets of $X$, $τ=(τ_1,\ldots,τ_\ell)$ is an $\ell$-tuple of positive real numbers, and let $H^0_0(X,L^p)$ be the space of holomorphic sections of $L^p:=L^{\otimes p}$ that vanish to order at least $τ_jp$ along $Σ_j$, $1\leq j\leq\ell$. If $Y\subset X$ is an irreducible analytic subset of dimension $m$, we consider the space $H^0_0 (X|Y, L^p)$ of holomorphic sections of $L^p|_Y$ that extend to global holomorphic sections in $H^0_0(X,L^p)$. Assuming that the triplet $(L,Σ,τ)$ is big in the sense that $\dim H^0_0(X,L^p)\sim p^n$, we give a general condition on $Y$ to ensure that $\dim H^0_0(X|Y,L^p)\sim p^m$. When $L$ is endowed with a continuous Hermitian metric, we show that the Fubini-Study currents of the spaces $H^0_0(X|Y,L^p)$ converge to a certain equilibrium current on $Y$. We apply this to the study of the equidistribution of zeros in $Y$ of random holomorphic sections in $H^0_0(X|Y,L^p)$ as $p\to\infty$.

math.CV↗

arXiv2022

Positive plurisubharmonic currents: Generalized Lelong numbers and Tangent theorems

Dinh--Sibony theory of tangent and density currents is a recent but powerful tool to study positive closed currents. Over twenty years ago, Alessandrini and Bassanelli initiated the theory of the Lelong number of a positive plurisubharmonic current in $\mathbb{C}^k$ along a linear subspace. Although the latter theory is intriguing, it has not yet been explored in-depth since then. Introducing the concept of the generalized Lelong numbers and studying these new numerical values, we extend both theories to a more general class of positive plurisubharmonic currents and in a more general context of ambient manifolds. More specifically, in the first part of our article, we consider a positive plurisubharmonic current $T$ of bidegree $(p,p)$ on a complex manifold $X$ of dimension $k,$ and let $V\subset X$ be a Kähler submanifold of dimension $l$ and $B$ a relatively compact piecewise $\mathcal{C}^2$-smooth open subset of $V.$ We define the notion of the $j$-th Lelong number of $T$ along $B$ for every $j$ with $\max(0,l-p)\leq j\leq \min(l,k-p)$ and prove their existence as well as their basic properties. Our method relies on some Lelong-Jensen formulas for the normal bundle to $V$ in $X,$ which are of independent interest. The second part of our article is devoted to geometric characterizations of the generalized Lelong numbers. As a consequence of this study, we show that the top degree Lelong number of $T$ along $B$ is totally intrinsic. This is a generalization of the fundamental result of Siu (for positive closed currents) and of Alessandrini--Bassanelli (for positive plurisubharmonic currents) on the independence of Lelong numbers at a single point on the choice of coordinates.

math.CV↗

arXiv2019

Holomorphic sections of line bundles vanishing along subvarieties

Let $X$ be a compact normal complex space of dimension $n$, and $L$ be a holomorphic line bundle on $X$. Suppose $Σ=(Σ_1,\ldots,Σ_\ell)$ is an $\ell$-tuple of distinct irreducible proper analytic subsets of $X$, $τ=(τ_1,\ldots,τ_\ell)$ is an $\ell$-tuple of positive real numbers, and consider the space $H^0_0 (X, L^p)$ of global holomorphic sections of $L^p:=L^{\otimes p}$ that vanish to order at least $τ_{j}p$ along $Σ_{j}$, $1\leq j\leq\ell$. We find necessary and sufficient conditions which ensure that $\dim H^0_0(X,L^p)\sim p^n$, analogous to Ji-Shiffman's criterion for big line bundles. We give estimates of the partial Bergman kernel, investigate the convergence of the Fubini-Study currents and their potentials, and the equilibrium distribution of normalized currents of integration along zero divisors of random holomorphic sections in $H^0_0 (X, L^p)$ as $p\to\infty$. Regularity results for the equilibrium envelope are also included.

math.CV↗

arXiv2017

Hölder singular metrics on big line bundles and equidistribution

We show that normalized currents of integration along the common zeros of random $m$-tuples of sections of powers of $m$ singular Hermitian big line bundles on a compact Kähler manifold distribute asymptotically to the wedge product of the curvature currents of the metrics. If the Hermitian metrics are Hölder with singularities we also estimate the speed of convergence.

math.CV↗

arXiv2016

Approximation and equidistribution results for pseudo-effective line bundles

We study the distribution of the common zero sets of $m$-tuples of holomorphic sections of powers of $m$ singular Hermitian pseudo-effective line bundles on a compact Kähler manifold. As an application, we obtain sufficient conditions which ensure that the wedge product of the curvature currents of these line bundles can be approximated by analytic cycles.

math.CV↗