arXiv subjects

Duc-Bao Nguyen

Publications and source records attributed to Duc-Bao Nguyen.

5 recordsLinked to original sources

arXiv2026

Singularity of non-pluripolar cohomology classes

We establish a relation between Lelong numbers and the full mass property of relative non-pluripolar products. We use this relation to prove that if the restricted volume of a big class $α$ along an effective divisor $D$ is of full mass, then the Lelong numbers of the non-pluripolar class $\langle α^{n-1}\rangle$ at every point in the support of $D$ are zero. In particular, we obtain that on projective manifolds, the Lelong numbers of the non-pluripolar class $\langle α^{n-1}\rangle$ of a big class $α$ are zero.

math.CV↗

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↗

arXiv2026

Non-collapsing volume estimate for local Kähler metrics in big cohomology classes

We prove a uniform local non-collapsing volume estimate for a large family of singular metrics in the big cohomology classes, which are Kähler on an open Euclidean subset of the manifold. The key ingredient is a generalization of a mixed energy estimate for functions in the complex Sobolev space to the setting of big cohomology classes.

math.DG↗

arXiv2025

Higher complex Sobolev spaces on complex manifolds

We study higher complex Sobolev spaces and their corresponding functional capacities. In particular, we prove the Moser-Trudinger inequality for these spaces and discuss some relationships between these spaces and the complex Monge-Ampère equation.

math.CV↗

arXiv2024

Uniform diameter estimates for Kaehler metrics in big cohomology classes

We generalize previous diameter estimates and local non-vanishing of volumes for Kaehler metrics to the case of big cohomology classes. In our proof, among other things, we will prove a uniform diameter estimate for a family of smooth Kaehler metrics only involving an integrability condition. We also have to use fine stability properties of complex Monge-Ampere equations with prescribed singularities.

math.DG↗