Plurisubharmonic polynomials and bumping

We wish to study the problem of bumping outwards a pseudoconvex, finite-type domain \Omega\subset C^n in such a way that pseudoconvexity is preserved and such that the lowest possible orders of contact of the bumped domain with bdy(\Omega), at the site of the bumping, are explicitly realised. Generally, when \Omega\subset C^n, n\geq 3, the known methods lead to bumpings with high orders of contact -- which are not explicitly known either -- at the site of the bumping. Precise orders are known for h-extendible/semiregular domains. This paper is motivated by certain families of non-semiregular domains in C^3. These families are identified by the behaviour of the least-weight plurisubharmonic polynomial in the Catlin normal form. Accordingly, we study how to perturb certain homogeneous plurisubharmonic polynomials without destroying plurisubharmonicity.Comment: 24 pages; corrected typos, fixed errors in Lemma 3.3; accepted for publication in Math.

Bergman kernel and complex singularity exponent

We give a precise estimate of the Bergman kernel for the model domain defined by $\Omega_F=\{(z,w)\in \mathbb{C}^{n+1}:{\rm Im}w-|F(z)|^2>0\},$ where $F=(f_1,...,f_m)$ is a holomorphic map from $\mathbb{C}^n$ to $\mathbb{C}^m$, in terms of the complex singularity exponent of $F$.Comment: to appear in Science in China, a special issue dedicated to Professor Zhong Tongde's 80th birthda

An Integral Kernel for Weakly Pseudoconvex Domains

A new explicit construction of Cauchy-Fantappi\'e kernels is introduced for an arbitrary weakly pseudoconvex domain with smooth boundary. While not holomorphic in the parameter, the new kernel reflects the complex geometry and the Levi form of the boundary. Some estimates are obtained for the corresponding integral operator, which provide evidence that this kernel and related constructions give useful new tools for complex analysis on this general class of domains

Weights of holomorphic extension and restriction

AbstractLet D ⊂⊂Cn be a domain and D′ ⊂ D a closed complex submanifold. A normalized weight function ϕ on D′ is called weight of restriction, if the restriction of any L2-holomorphic function f on D to D′ is contained in L2(D′, ϕ), and it is called a weight of extension, if any holomorphic function in L2(D′, ϕ) can be extended to a L2-holomorphic function on D. Properties of the families of weights of restriction and weights of extension and relations between them are studied in this article. An application to the boundary behavior of the Bergman metric is given

Super-resolution microscopy: a brief history and new avenues.

Super-resolution microscopy (SRM) is a fast-developing field that encompasses fluorescence imaging techniques with the capability to resolve objects below the classical diffraction limit of optical resolution. Acknowledged with the Nobel prize in 2014, numerous SRM methods have meanwhile evolved and are being widely applied in biomedical research, all with specific strengths and shortcomings. While some techniques are capable of nanometre-scale molecular resolution, others are geared towards volumetric three-dimensional multi-colour or fast live-cell imaging. In this editorial review, we pick on the latest trends in the field. We start with a brief historical overview of both conceptual and commercial developments. Next, we highlight important parameters for imaging successfully with a particular super-resolution modality. Finally, we discuss the importance of reproducibility and quality control and the significance of open-source tools in microscopy. This article is part of the Theo Murphy meeting issue 'Super-resolution structured illumination microscopy (part 2)'

On the growth of the Bergman kernel near an infinite-type point

We study diagonal estimates for the Bergman kernels of certain model domains in $\mathbb{C}^2$ near boundary points that are of infinite type. To do so, we need a mild structural condition on the defining functions of interest that facilitates optimal upper and lower bounds. This is a mild condition; unlike earlier studies of this sort, we are able to make estimates for non-convex pseudoconvex domains as well. This condition quantifies, in some sense, how flat a domain is at an infinite-type boundary point. In this scheme of quantification, the model domains considered below range -- roughly speaking -- from being ``mildly infinite-type'' to very flat at the infinite-type points.Comment: Significant revisions made; simpler estimates; very mild strengthening of the hypotheses on Theorem 1.2 to get much stronger conclusions than in ver.1. To appear in Math. An

Pluripolarity of Graphs of Denjoy Quasianalytic Functions of Several Variables

In this paper we prove pluripolarity of graphs of Denjoy quasianalytic functions of several variables on the spanning se

On a hyperconvex manifold without non-constant bounded holomorphic functions

An example is given of a hyperconvex manifold without non-constant bounded holomorphic functions, which is realized as a domain with real-analytic Levi-flat boundary in a projective surface.Comment: 10 pages, final version, to appear in "Geometric Complex Analysis", Springer Proceedings in Mathematics & Statistic