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Divisorial properties and special metrics on hypercomplex twistor spaces

Alberto Pipitone Federico

Abstract

We prove that the general fiber of a compact hypercomplex twistor space with a Kähler 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 Kähler and not even pluriclosed metrics.

Divisorial properties and special metrics on hypercomplex twistor spaces

Abstract

We prove that the general fiber of a compact hypercomplex twistor space with a Kähler 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 Kähler and not even pluriclosed metrics.

Paper Structure

This paper contains 6 sections, 10 theorems, 9 equations.

Table of Contents

  1. Main definitions and divisorial properties
  2. Hypercomplex manifolds
  3. Twistor spaces and ample rational curves
  4. Divisorial properties of the fibers
  5. Exotic stuctures
  6. Metric properties

Key Result

Theorem 1.3

Let $\pi: \mathcal{Z} \to \mathbb{P}^1$ be the twistor projection. If $(X, \lambda)$ admits a Kähler metric, for some $\lambda\in \mathbb{P}^1$, then the general fiber contains no curves nor divisors.

Theorems & Definitions (23)

  • Definition 1.1
  • Remark 1.2
  • Theorem 1.3
  • proof
  • Theorem 1.4
  • proof
  • Corollary 1.5
  • Remark 1.6
  • Definition 1.7
  • Conjecture 1.8
  • ...and 13 more