Divisorial properties and special metrics on hypercomplex twistor spaces

TL;DR

This paper investigates the geometric properties of hypercomplex twistor spaces, showing they lack divisors, curves, and certain metrics, and establishing constraints on their meromorphic function fields.

Contribution

It proves that hypercomplex twistor spaces with Kähler fibers have no divisors or curves and cannot admit Kähler or pluriclosed metrics, revealing new geometric restrictions.

Findings

No divisors or curves in the fibers.

Transcendental degree of meromorphic functions is one.

Spaces admit no Kähler or pluriclosed metrics.

Abstract

We prove that the general fiber of a compact hypercomplex twistor space with a K\"{a}hler fiber has no divisors nor curves. This is first used to prove that, under the same assumption, the trascendental degree of the field of meromoprhic functions is one. The same result allows to prove that these spaces admit no K\"{a}hler and not even pluriclosed metrics.