Function theoretic properties of symmetric powers of complex manifolds

TL;DR

This paper explores how symmetric powers of complex manifolds retain certain function theoretic properties, providing detailed analysis especially for planar domains, including hyperbolicity and completeness characteristics.

Contribution

It offers new insights into the preservation of function theoretic properties under symmetric powers and provides a complete description for planar domains.

Findings

Symmetric powers preserve (quasi) c-finite compactness and peak functions.

Complete characterization of hyperbolicity and Kobayashi completeness for symmetric products of planar domains.

Detailed analysis of function theoretic properties in symmetric powers of complex manifolds.

Abstract

In the paper we study properties of symmetric powers of complex manifolds. We investigate a number of function theoretic properties (e. g. (quasi) $c$-finite compactness, existence of peak functions) that are preserved by taking the symmetric power. The case of symmetric products of planar domains is studied in a more detailed way. In particular, a complete description of the Carath\'eodory and Kobayashi hyperbolicity and Kobayashi completeness in that class of domains is presented.