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Min Ru

Publications and source records attributed to Min Ru.

13 recordsLinked to original sources
arXiv2025

Campana's orbifold conjecture for numerically equivalent divisors

We prove the following version of the Campana's orbifold conjecture: Let $X$ be a complex non-singular projective variety of dimension $n$. Let $D_1,\ldots,D_{n+1}$ be $\mathbb Z$-linearly independent effective divisors in ${\rm Div}(X)$ and $D:=D_1+\cdots+D_{n+1}$ be a normal crossing divisor of $X$. Assume furthermore that they are numerically parallel. Let $\Delta=\sum_{i=1}^{n+1} (1-m_i^{-1}) D_i$ and let $f:\mathbb C\to (X,\Delta) $ be an orbifold entire curve. Then, there exists a positive integer $\ell$ such that, the orbifold $ (X,\Delta_{\ell}) $ is of general type, where $\Delta_{\ell}=\sum_{i=1}^{n+1} (1-\frac1{\ell})D_i$, and if $f$ has multiplicity at least $\ell$ along $D_i$, $1\le i\le n+1$, then $f$ must be algebraically degenerate.

math.CV↗
arXiv2025

A non-integrated defect relation for holomorphic maps into algebraic varieties

In 1983, relating to the study of value distribution of the Guass maps of complete minimal surfaces in ${\Bbb R}^m$, H. Fujimoto introduced the notion of the non-integrated defect for holomorphic maps of an open Riemann surface into $\mathbb{P}^n(\mathbb{C})$ and obtained some results analogous to the Nevanlinna-Cartan defect relation. This paper establishes the non-integrated defect relation for holomorphic maps into projective varieties.

math.CV↗
arXiv2024

Vojta's abc conjecture for entire curves in toric varieties highly ramified over the boundary

We prove Vojta's abc conjecture for projective space ${\Bbb P}^n({\Bbb C})$, assuming that the entire curves in ${\Bbb P}^n({\Bbb C})$ are highly ramified over the coordinate hyperplanes. This extends the results of Guo Ji and the second-named author for the case $n=2$ (see \cite{GW22}). We also explore the corresponding results for projective toric varieties. Consequently, we establish a version of Campana's orbifold conjecture for finite coverings of projective toric varieties.

math.CV↗
arXiv2024

Defect relation of $n+1$ components through the GCD method]

This paper studies the defect relation through the GCD method. In particular, among other results, we extend the defect relation result of Chen, Huynh, Sun and Xie to moving targets. The truncated defect relation is also studied. Furthermore, we obtain the degeneracy locus, which can be determined effectively and is independent of the maps under the consideration.

math.CV↗
arXiv2021

The Ru-Vojta result for subvarieties

In their recent article, Min Ru and Paul Vojta, among other things, proved the so-called general theorem (arithmetic part) which can be viewed as an extension of Schmidt's subspace theorem. In this note, we extend their result by replacing the divisors by closed subschemes.

math.NT↗
arXiv2021

Nevanlinna Pair and Algebraic Hyperbolicity

We introduce the notion of the $\textit{Nevanlinna pair}$ for a pair $(X, D)$, where $X$ is a projective variety and $D$ is an effective Cartier divisor on $X$. This notion links and unifies the Nevanlinna theory, the complex hyperbolicity (Brody and Kobayashi hyperbolicity), the big Picard type extension theorem (more generally the Borel hyperbolicity), as well as the algebraic hyperbolicity. The key is to use the Nevanlinna theory on parabolic Riemann surfaces recently developed by P\v{a}un and Sibony.

math.AG↗
arXiv2020

An Evertse-Ferretti Nevanlinna constant and its consequences

In this paper, we introduce the notion of an Evertse-Ferretti Nevanlinna constant and compare it with the birational Nevanlinna constant introduced by the authors in a recent joint paper. We then use it to recover several previously known results. This includes a 1999 example of Faltings from his Baker's Garden article.

math.NT↗
arXiv2019

A generalized subspace theorem for closed subschemes in subgeneral position

In this paper, we extend the recent theorem of G. Heier and A. Levin [arXiv:1712.02456] on the generalization of Schmidt's subspace theorem and Cartan's Second Main Theorem in Nevanlinna theory to closed subschemes located in $l$-subgeneral position, using the generic linear combination technique due to Quang.

math.NT↗
arXiv2016

A birational Nevanlinna constant and its consequences

The purpose of this paper is to modify the notion of the Nevanlinna constant $\operatorname{Nev}(D)$, recently introduced by the first author, for an effective Cartier divisor on a projective variety $X$. The modified notion is called the birational Nevanlinna constant and is denoted by $\operatorname{Nev}_{\text{bir}}(D)$. By computing $\operatorname{Nev}_{\text{bir}}(D)$ using the filtration constructed by Autissier in 2011, we establish a general result (see the General Theorem in the Introduction), in both the arithmetic and complex cases, which extends to general divisors the 2008 results of Evertse and Ferretti and the 2009 results of the first author. The notion $\operatorname{Nev}_{\text{bir}}(D)$ is originally defined in terms of Weil functions for use in applications, and it is proved later in this paper that it can be defined in terms of local effectivity of Cartier divisors after taking a proper birational lifting. In the last two sections, we use the notion $\operatorname{Nev}_{\text{bir}}(D)$ to recover the proof of an example of Faltings from his 2002 Baker's Garden article.

math.NT↗
arXiv2010

On essentially large divisors

Motivated by the classical Theorems of Picard and Siegel and their generalizations, we define the notion of an {\it essentially large} effective divisor and derive some of its geometric and arithmetic consequences. We then prove that on a nonsingular projective variety $X$ whose codimension is no greater than $\dim X-2$, every effective divisor with $\dim X +2$ or more components in general position is essentially large.

math.AG↗