Hermitian structures and harmonic morphisms in higher dimensional Euclidean spaces

TL;DR

This paper constructs new complex-valued harmonic morphisms in higher-dimensional Euclidean spaces using Hermitian structures, providing the first global examples for dimensions greater than four that are not derived from Kähler structures.

Contribution

It introduces novel harmonic morphisms from Euclidean spaces based on Hermitian structures, expanding known examples beyond Kähler-based constructions for dimensions greater than four.

Findings

First global examples of such morphisms for n > 4

Examples do not originate from Kähler structures

No such examples exist for n ≤ 4

Abstract

We construct new complex-valued harmonic morphisms from Euclidean spaces from functions which are holomorphic with respect to Hermitian structures. In particular, we give the first global examples of complex-valued harmonic morphisms from ${\bf R}^n$ for each $n>4$ which do not arise from a K\"ahler structure; it is known that such examples do not exist for $n \leq 4$.