arXiv subjects

Sahil Gehlawat

Publications and source records attributed to Sahil Gehlawat.

12 recordsLinked to original sources

arXiv2026

A Canonical Positive Definite Kernel Associated with the $ξ$-Bergman Kernel

Let $Ω\subset \mathbb{C}^{n}$ and $ξ\in \ell^{1}$. The $ξ$-Bergman kernel $K_{ξ, Ω}$, introduced by Bao and Guan, generalizes the classical Bergman kernel by replacing the point evaluation functional with a functional determined by sequence $ξ$. While this kernel inherits several important extremal and plurisubharmonic properties, it is intrinsically an on-diagonal object and therefore lacks the two-variable reproducing kernel structure that lies at the heart of the classical Bergman theory. The purpose of this paper is to associate a canonical Hermitian positive-definite kernel with the $ξ$-Bergman kernel and to investigate its analytic and geometric properties. Our construction is based on the family of Riesz representatives corresponding to the $ξ$-evaluation functionals. More precisely, we introduce a Hermitian kernel obtained as the Gram kernel of these representatives and show that it is positive definite and for $z \in Ω$ satisfies \[ B_{ξ,Ω}(z,z)=K_{ξ,Ω}(z), \] thereby recovering the $ξ$-Bergman kernel as its diagonal restriction. As a consequence, we prove that the $ξ$-Bergman kernel is real analytic on $Ω$. We also establish biholomorphic transformation laws, and obtain a representation of the $ξ$-Bergman kernel in terms of derivatives of the classical Bergman kernel. Furthermore, we obtain explicit formulas for the $ξ$-Bergman kernel on the upper half-plane $\mathbb{H}$ corresponding to several classes of sequences $ξ$, establish corresponding $ξ$-Lu Qi-Keng results, and derive precise boundary asymptotics. These examples illustrate how the choice of the differential functional influences both the zero set and the boundary growth of the associated kernel.

math.CV↗

arXiv2026

Stability of the Monomial Basis Kernel of Reinhardt domains

On a pseudoconvex Reinhardt domain $Ω\subset\mathbb{C}^n$ the $p$-Bergman space $A^p(Ω)$ admits a canonical basis of monomials indexed by a subset $S_p(Ω)\subset\mathbb{Z}^n$. The corresponding $p$-Monomial Basis Kernel (or $p$-MBK) is defined by a series involving these monomials and their norms. This article records stability properties of the $p$-MBK and of the index set $S_p(Ω)$ with respect to the parameter $p$. First, under mild hypotheses, the $p$-MBK depends continuously on $p\in[1,\infty)$, and a Ramadanov-type theorem holds for $p$-MBK for an increasing sequence of pseudoconvex Reinhardt domains. Second, for certain special classes of monomial polyhedra, we explicitly compute the index set and the associated Threshold exponents. Finally, these explicit models are used to illustrate structural properties of the index sets under finite unions, intersections, and products.

math.CV↗

arXiv2025

Riemann surface foliations with non-discrete singular set

Let $\mathcal{F}$ be a singular Riemann surface foliation on a complex manifold $M$, such that the singular set $E \subset M$ is non-discrete. We study the behavior of the foliation near the singular set $E$, particularly focusing on singular points that admit invariant submanifolds (locally) passing through them. Our primary focus is on the singular points that are removable singularities for some proper subfoliation. We classify singular points based on the dimension of their invariant submanifold and, consequently, establish that for hyperbolic foliations $\mathcal{F}$, the presence of such singularities ensures the continuity of the leafwise Poincaré metric on $M \setminus E$.

math.CV↗

arXiv2024

Generic singularities of holomorphic foliations by curves on $\mathbb{P}^n$

Let $\mathcal{F}_d(\mathbb{P}^n)$ be the space of all singular holomorphic foliations by curves on $\mathbb{P}^n$ ($n \geq 2$) with degree $d \geq 1.$ We show that there is subset $\mathcal{S}_d(\mathbb{P}^n)$ of $\mathcal{F}_d(\mathbb{P}^n)$ with full Lebesgue measure with the following properties: 1. for every $\mathcal{F} \in \mathcal{S}_d(\mathbb{P}^n),$ all singular points of $\mathcal{F}$ are linearizable hyperbolic. 2. If, moreover, $d \geq 2,$ then every $\mathcal{F}$ does not possess any invariant algebraic curve.

math.CV↗

arXiv2024

A Note on Kernel Functions of Dirichlet Spaces

For a planar domain $Ω$, we consider the Dirichlet spaces with respect to a base point $ζ\inΩ$ and the corresponding kernel functions. It is not known how these kernel functions behave as we vary the base point. In this note, we prove that these kernel functions vary smoothly. As an application of the smoothness result, we prove a Ramadanov-type theorem for these kernel functions on $Ω\timesΩ$. This extends the previously known convergence results of these kernel functions. In fact, we have made these observations in a more general setting, that is, for weighted kernel functions and their higher-order counterparts.

math.CV↗

arXiv2023

The Reduced Bergman Kernel and its Properties

In this article, we study some properties of the $n$-th order weighted reduced Bergman kernels for planar domains, $n\geq 1$. Specifically, we look at Ramadanov type theorems, localization, and boundary behaviour of the weighted reduced Bergman kernel and its higher-order counterparts. We also give a transformation formula for these kernels under biholomorphisms.

math.CV↗

arXiv2023

Few remarks on the Poincaré metric on a singular holomorphic foliation

Let $\mathcal{F}$ be a Riemann surface foliation on $M \setminus E$, where $M$ is a complex manifold and $E \subset M$ is a closed set. Assume that $\mathcal{F}$ is hyperbolic, i.e., all leaves of the foliation $\mathcal{F}$ are hyperbolic Riemann surface. Fix a hermitian metric $g$ on $M$. We will consider the Verjovsky's modulus of uniformization map $η$, which measures the largest possible derivative in the class of holomorphic maps from the unit disk into the leaves of $\mathcal{F}$. Various results are known to ensure the continuity of the map $η$ along the transverse directions, with suitable conditions on $M$, $\mathcal{F}$ and $E$. For a domain $U \subset M$, let $\mathcal{F}_{U}$ be the holomorphic foliation given by the restriction of $\mathcal{F}$ to the domain $U$, i.e., $\mathcal{F}\vert_{U}$. We will consider the modulus of uniformization map $η_{U}$ corresponding to the foliation $\mathcal{F}_{U}$, and study its variation when the corresponding domain $U$ varies in the Caratheodory kernel sense, motivated by the work of Lins Neto--Martins.

math.CV↗

arXiv2023

Regularity of the leafwise Poincare metric on singular holomorphic foliations

Let $\mathcal F$ be a smooth Riemann surface foliation on $M \setminus E$, where $M$ is a complex manifold and the singular set $E \subset M$ is an analytic set of codimension at least two. Fix a hermitian metric on $M$ and assume that all leaves of $\mathcal F$ are hyperbolic. Verjovsky's modulus of uniformization $η$ is a positive real function defined on $M \setminus E$ defined in terms of the family of holomorphic maps from the unit disc $\mathbb D$ into the leaves of $\mathcal F$ and is a measure of the largest possible derivative in the class of such maps. Various conditions are known that guarantee the continuity of $η$ on $M \setminus E$. The main question that is addressed here is its continuity at points of $E$. To do this, we adapt Whitney's $C_4$-tangent cone construction for analytic sets to the setting of foliations and use it to define the tangent cone of $\mathcal F$ at points of $E$. This leads to the definition of a foliation that is of {\it transversal type} at points of $E$. It is shown that the map $η$ associated to such foliations is continuous at $E$ provided that it is continuous on $M \setminus E$ and $\mathcal F$ is of transversal type. We also present observations on the locus of discontinuity of $η$. Finally, for a domain $U \subset M$, we consider $\mathcal F_U$, the restriction of $\mathcal F$ to $U$ and the corresponding positive function $η_U$. Using the transversality hypothesis leads to strengthened versions of the results of Lins Neto--Martins on the variation $U \mapsto η_U$.

math.CV↗

arXiv2022

Transformation formula for the Reduced Bergman kernel and its Application

In this article, we prove the transformation formula for the reduced Bergman kernels under proper holomorphic correspondences between bounded domains in the complex plane. As a corollary, we obtain the transformation formula for the reduced Bergman kernels under proper holomorphic maps. We also establish the transformation formula for the weighted reduced Bergman kernels under proper holomorphic maps. Finally, we provide an application of this transformation formula.

math.CV↗

arXiv2021

Non-negative divisors and the Grauert metric

Grauert showed that it is possible to construct complete Kähler metrics on the complement of complex analytic sets in a domain of holomorphy. In this note, we study the holomorphic sectional curvatures of such metrics on the complement of a principal divisor in $\mathbb{C}^n$, $n \ge 1$. In addition, we also study how this metric and its holomorphic sectional curvature behaves when the corresponding principal divisors vary continuously.

math.CV↗

arXiv2020

Two remarks on the Poincaré metric on a singular Riemann surface foliation

Let $\mathcal{F}$ be a smooth Riemann surface foliation on $M \setminus E$, where $M$ is a complex manifold and $E \subset M$ is a closed set. Fix a hermitian metric $g$ on $M \setminus E$ and assume that all leaves of $\mathcal{F}$ are hyperbolic. For each leaf $L \subset \mathcal{F}$, the ratio of $g | L$, the restriction of $g$ to $L$, and the Poincaré metric $λ_L$ on $L$ defines a positive function $η$ that is known to be continuous on $M \setminus E$ under suitable conditions on $M, E$. For a domain $U \subset M$, we consider $\mathcal{F}_U$, the restriction of $\mathcal{F}$ to $U$ and the corresponding positive function $η_U$ by considering the ratio of $g$ and the Poincaré metric on the leaves of $\mathcal{F}_U$. First, we study the variation of $η_U$ as $U$ varies in the Hausdorff sense motivated by the work of Lins Neto-Martins. Secondly, Minda had shown the existence of a domain Bloch constant for a hyperbolic Riemann surface $S$, which in other words shows that every holomorphic map from the unit disc into $S$, whose distortion at the origin is bounded below, must be locally injective in some hyperbolic ball of uniform radius. We show how to deduce a version of this Bloch constant for $\mathcal{F}$

math.CV↗

arXiv2020

On Grauert's examples of complete Kähler metrics

Grauert showed that the existence of a complete Kähler metric does not characterize domains of holomorphy by constructing such metrics on the complements of complex analytic sets in a domain of holomorphy. In this note, we study the holomorphic sectional curvatures of such metrics in two prototype cases namely, $\mathbb{C}^n \setminus \{0\}, n \ge 2$ and $\mathbb{B}^N \setminus A$, $N \ge 2$ and $A \subset \mathbb{B}^N$ is a hyperplane of codimension at least two. This is done by computing the Gaussian curvature of its restriction to the leaves of a suitable holomorphic foliation of these two examples. We also examine this metric on the punctured plane $\mathbb{C}^{\ast}$ and show that it behaves very differently in this case.

math.CV↗