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Some QCH Kahler surfaces with zero scalar curvature

Włodzimierz Jelonek

TL;DR

This work situates scalar-flat Kähler toric surfaces within the framework of quasi-constant holomorphic curvature (QCH Kähler) by exploiting an integrable opposite Hermitian structure $I$ and a degenerate half-Weyl tensor $W^-$. It demonstrates that Weber’s scalar-flat Kähler surfaces are QCH and that the Burns metric is a Calabi-type QCH surface, while the generalized Taub-NUT family is orthotoric for $k\in(-1,1)$ and Calabi type for $k=\pm1$, with the exceptional half-plane also Calabi type. The analysis hinges on toric Kähler geometry with two commuting holomorphic Killing fields, expressing metrics in orthotoric/Calabi-type forms via Volumetric coordinates and moment maps, and deriving explicit relations among parameters to realize isometries between Taub-NUT data and orthotoric representations. Overall, the paper unifies known zero-scalar-curvature Kähler examples under the QCH, Calabi/orthotoric taxonomy, clarifying their Weyl and Ricci structures and providing explicit metric forms and integrability results that enhance geometric and potentially physical interpretations.

Abstract

In this paper we prove that some well known Kähler surfaces with zero scalar curvature are QCH Kähler. We prove that family of generalized Taub-Nut Kähler surfaces parametrized by $k\in[-1,1]$ is of orthotoric type for $k\in(-1,1)$ and of Calabi type for $k\in\{-1,1\}$ and the Burn's metric is of Calabi type.

Some QCH Kahler surfaces with zero scalar curvature

TL;DR

This work situates scalar-flat Kähler toric surfaces within the framework of quasi-constant holomorphic curvature (QCH Kähler) by exploiting an integrable opposite Hermitian structure and a degenerate half-Weyl tensor . It demonstrates that Weber’s scalar-flat Kähler surfaces are QCH and that the Burns metric is a Calabi-type QCH surface, while the generalized Taub-NUT family is orthotoric for and Calabi type for , with the exceptional half-plane also Calabi type. The analysis hinges on toric Kähler geometry with two commuting holomorphic Killing fields, expressing metrics in orthotoric/Calabi-type forms via Volumetric coordinates and moment maps, and deriving explicit relations among parameters to realize isometries between Taub-NUT data and orthotoric representations. Overall, the paper unifies known zero-scalar-curvature Kähler examples under the QCH, Calabi/orthotoric taxonomy, clarifying their Weyl and Ricci structures and providing explicit metric forms and integrability results that enhance geometric and potentially physical interpretations.

Abstract

In this paper we prove that some well known Kähler surfaces with zero scalar curvature are QCH Kähler. We prove that family of generalized Taub-Nut Kähler surfaces parametrized by is of orthotoric type for and of Calabi type for and the Burn's metric is of Calabi type.
Paper Structure (6 sections, 5 theorems, 34 equations)

This paper contains 6 sections, 5 theorems, 34 equations.

Table of Contents

  1. Introduction
  2. Toric Kähler surfaces.
  3. Generalized Taub-NUT surfaces
  4. Exceptional generalized Taub-NUT.
  5. Exceptional half plane.
  6. The Burns metric.

Key Result

Theorem 1.1

Let $(M,g,J)$ be a Kähler surface. If $(M,g,J)$ is a QCH manifold, then $W^-=c(\frac{1}{6}\Pi-\Phi+\Psi)$ and $W^-$ is degenerate. The 2-form $\overline{\omega}$ is an eigenvector of $W^-$ corresponding to a simple eigenvalue of $W^-$ and $\overline{J}$ preserves the Ricci tensor. On the other hand

Theorems & Definitions (12)

  • Theorem 1.1
  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Theorem 3.1
  • proof
  • Theorem 4.1
  • proof
  • Theorem 5.1
  • proof
  • ...and 2 more