Conformally K\"ahler structures

TL;DR

This paper establishes a correspondence between conformal K"ahler metrics and certain parallel sections of a vector bundle, providing new invariants and obstructions for conformal K"ahler structures in higher dimensions.

Contribution

It introduces a novel vector bundle framework linking conformal K"ahler metrics to parallel sections, leading to explicit algebraic conditions and invariants in conformal geometry.

Findings

Derived obstructions for metrics to be conformal to K"ahler

Explicit algebraic condition for Weyl tensor with conformal Killing-Yano tensor

Invariant characterization of algebraically special metrics of type D

Abstract

We establish a one-to-one correspondence between K\"ahler metrics in a given conformal class and parallel sections of a certain vector bundle with conformally invariant connection, where the parallel sections satisfy a set of non--linear algebraic constraints that we describe. The vector bundle captures 2-form prolongations and is isomorphic to $\Lambda^3(\cT)$, where ${\cT}$ is the tractor bundle of conformal geometry, but the resulting connection differs from the normal tractor connection by curvature terms. Our analysis leads to a set of obstructions for a Riemannian metric to be conformal to a K\"ahler metric. In particular we find an explicit algebraic condition for a Weyl tensor which must hold if there exists a conformal Killing-Yano tensor, which is a necessary condition for a metric to be conformal to K\"ahler. This gives an invariant characterisation of algebraically special Riemannian metrics of type $D$ in dimensions higher than four.