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Blake J. Boudreaux

Publications and source records attributed to Blake J. Boudreaux.

6 recordsLinked to original sources
arXiv2024

Hypersurface Convexity and Extension of Kähler Forms

The following generalization of a result of S. Nemirovski is proved: if $X$ is either a projective or a Stein manifold and $K\subset X$ is a compact sublevel set of a strictly plurisubharmonic function $φ$ defined in a neighborhood of $K$, then $X\setminus K$ is a union of positive divisors if and only if $dd^cφ$ extends to a Hodge form on $X$. For an arbitrary compact subset $K\subsetneq X$, this gives that $X\setminus K$ is a union of positive divisors if and only if $K$ admits a neighbourhood basis of sublevel sets of strictly plurisubharmonic functions with the $dd^c$-extension property.

math.CV↗
arXiv2024

Relationships Between the Bergman Kernels of Hartogs Domains and Their Base

We explore the relationship between the Bergman kernel of a Hartogs domain and weighted Bergman kernels over its base domain. In particular we develop a representation of the Bergman kernel of a Hartogs domain as a series involving weighted Bergman kernels over its base, as well as a "transformation" formula for some weighted Bergman kernels. Other relationships of this type are presented.

math.CV↗
arXiv2024

$T$-polynomial convexity and holomorphic convexity

We compare the $T$-polynomial convexity of Guedj with holomorphic convexity away from the support of $T$. In particular we show an Oka--Weil theorem for $T$-polynomial convexity, as well as present a situation when the notions of $T$-polynomial convexity and holomorphic convexity of $X\setminus\text{Supp }T$ coincide in the context of complex projective algebraic manifolds.

math.CV↗
arXiv2023

On Rational Convexity of Totally Real Sets

Under a mild technical assumption, we prove a necessary and sufficient condition for a totally real compacdt set in $\mathbb{C}^n$ to be rationally convex. This generalizes a classical result of Duval-Sibony

math.CV↗
arXiv2018

Equivalent Bergman Spaces with Inequivalent Weights

We give a proof that every space of weighted square-integrable holomorphic functions admits an equivalent weight whose Bergman kernel has zeroes. Here the weights are equivalent in the sense that they determine the same space of holomorphic functions. Additionally, a family of radial weights in $L^1(\mathbb{C})$ whose associated Bergman kernels have infinitely many zeroes is exhibited.

math.CV↗