Generalized orthotoric Kahler surfaces

TL;DR

This paper classifies and describes generalized orthotoric K"ahler surfaces, providing structure equations, integrating special cases, and exploring hyperk"ahler structures with degenerate Weyl tensor.

Contribution

It introduces a framework for generalized orthotoric K"ahler surfaces, derives structure equations, and characterizes hyperk"ahler surfaces with degenerate Weyl tensor.

Findings

Derived structure equations for generalized orthotoric K"ahler surfaces.

Integrated equations to recover orthotoric K"ahler surfaces.

Proved existence of Calabi or orthotoric structures in certain hyperk"ahler cases.

Abstract

In this paper we describe QCH K\"ahler surfaces $(M,g,J)$ of generalized orthotoric type. We introduce a distinguished orthonormal frame on $(M,g)$ and give the structure equations for $(M,g,J)$. In the case when $I$ is conformally K\"ahler and $(M,g,J)$ is not hyperk\"ahler we integrate these structure equations and rediscover the orthotoric K\"ahler surfaces. We also investigate the hyperk\"aler case. We prove in a simple way that if $(M,g,J)$ is a hyperk\"ahler surface with a degenerate Weyl tensor $W^-$ (i.e. QCH hyperk\"ahler surface) then among all hyperk\"aler structures on $(M,g)$ there exists a K\"ahler structure $J_0$ such that $(M,g,J_0)$ is of Calabi type or of orthotoric type.