Mathematics

We have a survey to Nevanlinna-type theory of holomorphic mappings from a complete and stochastically complete Kähler manifold into compact complex manifolds with a positive line bundle. When some energy and Ricci curvature conditions are imposed, the Nevanlinna-type defect relations based on heat diffusions are obtained.

It is known that the Cauchy's argument principle, applied to an holomorphic function f , requires that f has no zeros on the curve of integration. In this short note, we give a generalization of such a principle which covers the case when f has zeros on the curve, as well as an application.

Let M be a complete Kähler manifold, whose universal covering is biholomorphic to a ball B m ( R 0 ) in C m ( 0< R 0 ≤+∞ ). Our first aim in this paper is to study the algebraic dependence problem of differentiably meromorphic mappings. We will show that if k differentibility nondegenerate meromorphic mappings f 1 ,…, f k of M into P n (C) (n≥2) satisfying the condition ( C ρ ) and sharing few hyperplanes in subgeneral position regardless of multiplicity then f 1 ∧⋯∧ f k ≡0 . For the second aim, we will show that there are at most two different differentiably nondegenerate meromorphic mappings of M into P n (C) sharing q (q∼2N−n+3+O(ρ)) hyperplanes in N− subgeneral position regardless of multiplicity. Our results generalize previous finiteness and uniqueness theorems for differentiably meromorphic mappings of C m and extend some previous results for the case of mappings on Kähler manifold.

Our main result introduces a new way to characterize two-dimensional finite ball quotients by algebraicity of their Bergman kernels. This characterization is particular to dimension two and fails in higher dimensions, as is illustrated by a counterexample in dimension three constructed in this paper. As a corollary of our main theorem, we prove, e.g., that a smoothly bounded strictly pseudoconvex domain G in C 2 has rational Bergman kernel if and only if there is a rational biholomorphism from G to the 2-dimensional unit ball.

We present an Almansi-type decomposition for polynomials with Clifford coefficients, and more generally for slice-regular functions on Clifford algebras. The classical result by Emilio Almansi, published in 1899, dealt with polyharmonic functions, the elements of the kernel of an iterated Laplacian. Here we consider polynomials of the form P(x)= ∑ d k=0 x k a k , with Clifford coefficients a k ∈ R n , and get an analogous decomposition related to zonal polyharmonics. We show the relation between such decomposition and the Dirac (or Cauchy-Riemann) operator and extend the results to slice-regular functions.

Given a compact Kähler manifold, Geometric Invariant Theory is applied to construct analytic GIT-quotients that are local models for a classifying space of (poly)stable holomorphic vector bundles containing the coarse moduli space of stable bundles as an open subspace. For local models invariant generalized Weil-Petersson forms exist on the parameter spaces, which are restrictions of symplectic forms on smooth ambient spaces. If the underlying Kähler manifold is of Hodge type, then the Weil-Petersson form on the moduli space of stable vector bundles is known to be the Chern form of a certain determinant line bundle equipped with a Quillen metric. It gives rise to a holomorphic line bundle on the classifying GIT space together with a continuous hermitian metric.

In this work we prove an FBI characterization for Gevrey vectors on hypo-analytic structures, and we analyze the main differences of Gevrey regularity and hypo-analyticity concerning the FBI transform. We end with an application of this characterization on a propagation of Gevrey singularities result, for solutions of the non-homogeneous system associated with the hypo-analytic structure, for analytic structures of tube type.

In this paper, we derive an analytic characterization of the symmetric extension of a Herglotz-Nevanlinna function in several variables. Here, the main tools used are the so-called variable non-dependence property and the symmetry formula satisfied by Herglotz-Nevanlinna function and Cauchy-type functions. We also provide an extension of the Stieltjes inversion formula for Cauchy-type functions.

We continue our study on variability regions in \cite{Ali-Vasudevarao-Yanagihara-2018}, where the authors determined the region of variability V j Ω ( z 0 ,c)={ ∫ z 0 0 z j (g(z)−g(0))dz:g(D)⊂Ω,( P −1 ∘g)(z)= c 0 + c 1 z+⋯+ c n z n +⋯} for each fixed z 0 ∈D , j=−1,0,1,2,… and c=( c 0 , c 1 ,…, c n )∈ C n+1 , when Ω⊊C is a convex domain, and P is a conformal map of the unit disk D onto Ω . In the present article, we first show that in the case n=0 , j=−1 and c=0 , the result obtained in \cite{Ali-Vasudevarao-Yanagihara-2018} still holds when one assumes only that Ω is starlike with respect to P(0) . Let CV(Ω) be the class of analytic functions f in D with f(0)= f ′ (0)−1=0 satisfying 1+z f ′′ (z)/ f ′ (z)∈Ω . As applications we determine variability regions of log f ′ ( z 0 ) when f ranges over CV(Ω) with or without the conditions f ′′ (0)=λ and f ′′′ (0)=μ . Here λ and μ are arbitrarily preassigned values. By choosing particular Ω , we obtain the precise variability regions of log f ′ ( z 0 ) for other well-known subclasses of analytic and univalent functions.

We study the family of Fourier-Laplace transforms F α,β (z)=F.p. ∫ ∞ 0 t β exp(i t α −izt)dt,Imz<0, for α>1 and β∈C , where Hadamard finite part is used to regularize the integral when Reβ≤−1 . We prove that each F α,β has analytic continuation to the whole complex plane and determine its asymptotics along any line through the origin. We also apply our ideas to show that some of these functions provide concrete extremal examples for the Wiener-Ikehara theorem and a quantified version of the Ingham-Karamata theorem, supplying new simple and constructive proofs of optimality results for these complex Tauberian theorems.