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On a Modification of the Twistor Space

Anna Fino, Gueo Grantcharov, Alberto Pipitone Federico

Abstract

In the paper we construct a modification $S(M)$ of the twistor space of a Kähler scalar flat surface $M$ and study its complex-geometric and metric properties. In particular, we construct complete balanced metrics on $S(M)$ and show that $S(M)$ can not be Kähler when $M$ is a compact simple hyperkähler manifold.

On a Modification of the Twistor Space

Abstract

In the paper we construct a modification of the twistor space of a Kähler scalar flat surface and study its complex-geometric and metric properties. In particular, we construct complete balanced metrics on and show that can not be Kähler when is a compact simple hyperkähler manifold.
Paper Structure (6 sections, 12 theorems, 45 equations)

This paper contains 6 sections, 12 theorems, 45 equations.

Table of Contents

  1. Introduction
  2. Construction and basic properties of the modified twistor space $S(M)$
  3. Construction of $S(M)$
  4. Necessary and sufficient conditions for the existence of $f$
  5. Properties of $Tw(M)$ and $S(M)$ for a Kähler scalar-flat surface $M$
  6. Modified hyperkähler twistor space

Key Result

Lemma 2.2

For an oriented Riemannian 4-manifold $M$, the horizontal space $\mathcal{H}$ of the induced connection on $\Lambda_+^2 M$ is tangent to $Tw(M)$. If, in addition, $(M,g, I, \omega)$ is Kähler, the horizontal space $\mathcal{H}$ is also tangent to $S(M)$.

Theorems & Definitions (26)

  • Definition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • Lemma 2.5
  • proof
  • Remark 2.6
  • Theorem 2.7
  • ...and 16 more