Local description of Bochner-flat (pseudo-)K\"ahler metrics

TL;DR

This paper provides local normal forms for (pseudo-)K"ahler manifolds with vanishing Bochner tensor, characterizing their structure through curvature operators and c-projectively equivalent metrics, including K"ahler-Einstein cases.

Contribution

It introduces new local descriptions of Bochner-flat (pseudo-)K"ahler metrics and classifies associated symmetric spaces and c-projectively equivalent metrics.

Findings

Derived local normal forms for Bochner-flat (pseudo-)K"ahler metrics.

Characterized symmetric spaces via curvature operators.

Described K"ahler-Einstein metrics with c-projective equivalence.

Abstract

The Bochner tensor is the K\"ahler analogue of the conformal Weyl tensor. In this article, we derive local (i.e., in a neighbourhood of almost every point) normal forms for a (pseudo-)K\"ahler manifold with vanishing Bochner tensor. The description is pined down to a new class of symmetric spaces which we describe in terms of their curvature operators. We also give a local description of weakly Bochner-flat metrics defined by the property that the Bochner tensor has vanishing divergence. Our results are based on the local normal forms for c-projectively equivalent metrics. As a by-product, we also describe all K\"ahler-Einstein metrics admitting a c-projectively equivalent one.