Kodaira dimension of holomorphic singular foliations

We introduce numerical invariants of holomorphic singular foliations under bimeromorphic transformations of surfaces. The basic invariant is a foliated version of the Kodaira dimension of compact complex manifolds.The author was supported by CNPq-Brazil in 1998 and Conseil Régional de Bourgogne in 1999.     

Sampling Theorem and Discrete Fourier Transform on the Riemann Sphere

Using coherent-state techniques, we prove a sampling theorem for Majorana’s (holomorphic) functions on the Riemann sphere and we provide an exact reconstruction formula as a convolution product of N samples and a given reconstruction kernel (a sinc-type function). We also discuss the effect of over- and under-sampling. Sample points are roots of unity, a fact which allows explicit inversion formulas for resolution and overlapping kernel operators through the theory of Circulant Matrices and Rectangular Fourier Matrices. The case of band-limited functions on the Riemann sphere, with spins up to J, is also considered. The connection with the standard Euler angle picture, in terms of spherical harmonics, is established through a discrete Bargmann transform.      

CLASSIFICATION OF A CLASS OF HYPERBOLIC REINHARDT DOMAINS

In this article, the author discusses the dimension of holomorphic automorphism groups on hyperbolic Reinhardt domains, and classifies those hyperbolic Reinhardt domains whose automorphism group has prescribed dimension n²-2 (where n is the dimension of domain).     

函数与其导数具有公共值的全纯函数族的正规性

设F为区域G上的全纯函数族, a,b(≠0)为两有穷复数,n为正整数,本文推广了Miranda定则,证明了:若对任意的f∈F,(a,b)为f与f(n)在G上的IM分担数组,且当f=a时, f'=f(n+1)=b,则F在G中正规.     

Weighted Composition Operators from the Bloch Spaces to Weighted Hardy Spaces on Bounded Symmetric Domains*

Let B_E be a bounded symmetric domain realized as the unit open ball of JB*-triples.The authors will characterize the bounded weighted composition operator from the Bloch space B(BE) to weighted Hardy space H_v^∞ in terms of Kobayashi distance.The authors also give a sufficient condition for the compactness,and also give the upper bound of its essential norm.As a corollary,they show that the boundedness and compactness are equivalent for composition operator fromB(B...     

Bolzano's Theorems for Holomorphic Mappings

The existence of a zero for a holomorphic functions on a ball or on a rectangle under some sign conditions on the boundary generalizing Bolzano's ones for real functions on an interval is deduced in a very simple way from Cauchy's theorem for holomorphic functions.A more complicated proof,using Cauchy's argument principle,provides uniqueness of the zero,when the sign conditions on the boundary are strict.Applications are given to corresponding Brouwer fixed point theorems for holomorphic functions.Extensions to holomorphic mappings from Cn to Cn are obtained using Brouwer degree.     

Lagrangian surfaces in the complex hyperquadric Q2

In this paper, we introduce a one-form $\Phi$ associated to a Lagrangian immersion $f$ from Riemann surface $M$ into the complex hyperquadric $Q_2$. It is proved that $f$ is minimal if and only if $\Phi$ is vanishing and $f$ is H-minimal if and only if $\Phi$ is holomorphic. As an application, a family of $S^1$-equivariant minimal Lagrangian tori in $Q_2$ are obtained.     

A Criterion of Normality Concerning Holomorphic Functions Whose Derivative Omits a Function

The authors discuss the normality concerning holomorphic functions and get the following result. Let F be a family of holomorphic functions on a domain D ⊂ ℂ, all of whose zeros have multiplicity at least k, where k ≥ 2 is an integer. And let h(z) ≢ 0 be a holomorphic function on D. Assume also that the following two conditions hold for every f ∈ F: (a) f(z) = 0 ⇒ |f ^(k)(z)| < |h(z)|; (b) f ^(k)(z) ≠ h(z). Then F is normal on D.     

一般Hopf曲面上向量丛的结构

得到了非主Hopf曲面上连续复向量丛全纯结构的存在性及可滤性问题的充要条件.