arXiv subjects

Yun-Heng Du

Publications and source records attributed to Yun-Heng Du.

5 recordsLinked to original sources

arXiv2026

Analytic Construction of Rational Curves on Fano Manifolds

Inspired by methods for constructing entire curves in Oka geometry, we give an analytic construction of rational curves on a complex Fano manifold $X$. Yau's theorem provides a Kähler metric with positive Ricci curvature. Using this curvature to guide deformations of holomorphic discs, we construct maps from discs of radii tending to infinity with uniformly bounded area. A central point is to preserve the derivative normalization through the limiting process. This yields a nonconstant entire map $f:\mathbb C\rightarrow X$ of finite area. This map extends across infinity to a nonconstant holomorphic map $\mathbb P^1\to X$. Combined with algebraic arguments in characteristic zero, the construction yields proofs of the rational connectedness of Fano manifolds and of Hartshorne's conjecture on ample tangent bundles.

math.CV↗

arXiv2026

Analytic and Algebraic Oka-1 Approximation for Smooth Projective Morphisms with Rationally Connected Fibers

Let $π:Z\rightarrow Y$ be a smooth projective morphism of complex manifolds with connected rationally connected fibers. We prove holomorphic approximation on arbitrary compact sets and finite-jet interpolation on arbitrary closed discrete sets for continuous liftings defined on open Riemann surfaces and holomorphic near those sets. For smooth projective morphisms of smooth complex algebraic varieties and algebraic base maps from smooth affine curves, the approximating liftings can be chosen algebraic, with interpolation on any finite set. In both cases the resulting lifting is homotopic to the initial one through continuous liftings of the fixed base map. For connected smooth projective complex manifolds, this gives the equivalence between the algebraic Oka-1 property and rational connectedness. Every rationally connected smooth projective complex manifold is also Oka-1.

math.CV↗

arXiv2026

Some Examples and Counterexamples in Oka Theory

This paper gives examples and counterexamples in Oka theory: two about deleting closed sets from complex Euclidean space, one about blowing up along a connected Oka center, and one about deforming continuous maps to regular maps. Two positive results show that $\mathbb{C}^3\setminus S$ is Oka for every closed set $S\subset\mathbb{R}^3$, and that the complement of the closed Hartogs triangle in $\mathbb{C}^2$ is Oka. In contrast, for every $n\ge3$ there is a proper holomorphic embedding $\mathbb{C}\hookrightarrow\mathbb{C}^n$ whose image $A$ is a closed connected smooth curve biholomorphic to $\mathbb{C}$, but whose blow-up $Bl_A\mathbb{C}^n$ is Brody volume hyperbolic and hence not Oka. Finally, for every $n\geq 2$, there exist a smooth connected affine algebraic variety $X$ and a continuous map $X\to\mathbb{C}^n\setminus\{0\}$ that is not homotopic to any regular map $X\to\mathbb{C}^n\setminus\{0\}$; equivalently, $\mathbb{C}^n\setminus\{0\}$ fails the algebraic basic Oka property (aBOP).

math.CV↗

arXiv2026

Sharp Summability of Nevanlinna Defects for Finite-Lower-Order Holomorphic Curves

For a countable family of hyperplanes $H_j\subset \mathbb{P}^m$, $j\in\mathbb{N}$, in general position and a linearly nondegenerate holomorphic curve $f\colon \mathbb{C}\to \mathbb{P}^m$ of finite lower order, we prove that the Nevanlinna defects $δ_f(H_j)$ satisfy $$ \sum_{j=1}^{\infty}δ_f(H_j)^{1/3}<\infty. $$ This resolves a long-standing open problem in Nevanlinna theory and extends Weitsman's celebrated scalar endpoint theorem (the case $m=1$) as well as Krutin's results for exponents strictly greater than $1/3$. The same uniform finite-family estimate yields the corresponding endpoint theorem for divisors cut out on a projective variety by ambient hypersurfaces of uniformly bounded degree, assuming that the divisors are in general position with respect to the variety and that the curve is not contained in the support of any divisor.

math.CV↗

arXiv2026

Hole Phenomenon of Gaussian Analytic Functions with Power-exponential Weights

We establish the \emph{hole phenomenon} for the Gaussian analytic function \[ F_β(z)=\sum_{n=0}^{\infty}\frac{ξ_{n}}{\sqrt{Γ\bigl(\frac{2}β(n+1)\bigr)}}\,z^{n}, \] associated with the power-exponential weight $e^{-|z|^β}$ on $\mathbb{C}$, where $β>0$. Under the condition that $F_β(z)$ has no zeros in $D(0,r)$, the scaled zero counting measure converges to a limiting measure $μ_{0}^β$ vaguely in distribution. This limit exhibits a \emph{forbidden region} \[ \bigl\{1<|z|<e^{1/β}\bigr\}, \] which zeros asymptotically avoid. This generalizes the remarkable discovery of Ghosh and Nishry for the Gaussian entire function (the case $β=2$), who first revealed this striking conditional convergence and the emergence of a hole. Our analysis extends their phenomenon to the entire family of power-exponential weights.

math.CV↗