A compactification and the Euler characteristic of the spaces of real meromorphic functions

Meromorphic limits of automorphisms

Let $X$ be a compact complex manifold in the Fujiki class $\mathscr{C}$. We study the compactification of $\operatorname{Aut}^0(X)$ given by its closure in Barlet cycle space. The boundary points give rise to non-dominant meromorphic self-maps of $X$. Moreover convergence in cycle space yields convergence of the corresponding meromorphic maps. There are analogous compactifications for reductive subgroups acting trivially on $\operatorname{Alb} X$. If $X$ is Kähler, these compactifications are projective. Finally we give applications to the action of $\operatorname{Aut}(X)$ on the set of probab…

欧拉乘积的显式解析延拓

Explicit Analytic Continuation of Euler Products

本文介绍了一种通过因子法对欧拉乘积进行显式解析延拓的方法，并提供了相关技术的深入阐述和研究成果。

Order of Meromorphic Maps and Rationality of the Image Space

Let $ι: \C^2 \hookrightarrow S$ be a compactification of the two dimensional complex space $\C^2$. By making use of Nevanlinna theoretic methods and the classification of compact complex surfaces K. Kodaira proved in 1971 (\cite{ko71}) that $S$ is a rational surface. Here we deal with a more general meromorphic map $f: \C^n \to X$ into a compact complex manifold $X$ of dimension $n$, whose differential $df$ has generically rank $n$. Let $ρ_f$ denote the order of $f$. We will prove that if $ρ_f<2$, then every global symmetric holomorphic tensor must vanish; in particular, {\it if $\dim X=2$ and…

Carlitz 模与 Euler gamma 函数的微分 Ax-Lindemann-Weierstrass 定理

The Carlitz module and a differential Ax-Lindemann-Weierstrass theorem for the Euler gamma function

本文证明了 Euler gamma 函数的 Ax-Lindemann-Weierstrass 型微分超越结果：对于模 Z 两两不同且代数于亚纯 1-周期函数域 k 的亚纯函数 ζ_i，函数 Γ(ν-ζ_i(ν)) 在 k(ν) 上微分独立。

Hypertranscendency of Perturbations of Hypertranscendental Functions

Inspired by the work of Bank on the hypertranscendence of $Γe^h$ where $Γ$ is the Euler gamma function and $h$ is an entire function, we investigate when a meromorphic function $fe^g$ cannot satisfy any algebraic differential equation over certain field of meromorphic functions, where $f$ and $g$ are meromorphic and entire on the complex plane, respectively. Our results (Theorem 1 and 2) give partial solutions to Bank's Conjecture (1977) on the hypertranscendence of $Γe^h$. We also give some sufficient conditions for hypertranscendence of meromorphic function of the form $f+g$, $f\cdot g$ and …

Euler--Mellin integrals and A-hypergeometric functions

We consider integrals that generalize both the Mellin transforms of rational functions of the form 1/f and the classical Euler integrals. The domains of integration of our so-called Euler--Mellin integrals are naturally related to the coamoeba of f, and the components of the complement of the closure of the coamoeba give rise to a family of these integrals. After performing an explicit meromorphic continuation of Euler--Mellin integrals, we interpret them as A-hypergeometric functions and discuss their linear independence and relation to Mellin--Barnes integrals.