Function theory and holomorphic maps on symmetric products of planar domains

TL;DR

This paper investigates the regularity and mapping properties of holomorphic functions on symmetric products of planar domains, establishing smoothness and properness results for related holomorphic maps.

Contribution

It proves global regularity of the $ar{ ext{d}}$-problem and extends Remmert-Stein theorems to symmetric products, advancing understanding of holomorphic maps in this context.

Findings

The $ar{ ext{d}}$-problem is globally regular on symmetric products of planar domains.

Proper holomorphic maps between symmetric products are smooth up to the boundary.

Remmert-Stein type theorems are established for symmetric products.

Abstract

We show that the $\bar{\partial}$-problem is globally regular on a domain in $\mathbb{C}^n$, which is the $n$-fold symmetric product of a smoothly bounded planar domain. Remmert-Stein type theorems are proved for proper holomorphic maps between equidimensional symmetric products and proper holomorphic maps from cartesian products to symmetric products. It is shown that proper holomorphic maps between equidimensional symmetric products of smooth planar domains are smooth up to the boundary