The extension of holomorphic functions on a non-pluriharmonic locus

TL;DR

This paper proves that holomorphic functions defined near certain complex subvarieties in a hyperconvex domain can be extended to the entire domain, expanding the understanding of extension phenomena in complex analysis.

Contribution

It introduces new extension results for holomorphic functions on hyperconvex domains, specifically related to the support of powers of the complex Hessian of a plurisubharmonic function.

Findings

Holomorphic functions near the support of (i∂∂̄ϕ)^{n-3} extend to the whole domain.

Extension results hold in bounded hyperconvex domains in ℂ^{n} for n≥4.

The work generalizes previous extension theorems to non-pluriharmonic loci.

Abstract

Let $n \geq 4$ and let $\Omega$ be a bounded hyperconvex domain in $\mathbb{C}^{n}$. Let $\varphi$ be a negative exhaustive smooth plurisubharmonic function on $\Omega$. We show that any holomorphic function defined on a connected open neighborhood of the support of $(i\partial \overline{\partial}\varphi)^{n-3}$ can be extended to the holomorphic function on $\Omega$.