The Tanno-Theorem for K\"ahlerian metrics with arbitrary signature

TL;DR

This paper extends Tanno's theorem, originally for positive-definite K"ahler metrics, to K"ahler metrics with arbitrary signature, showing the manifold's structure under certain solutions of the Tanno equation.

Contribution

The paper provides a proof of Tanno's theorem for K"ahler metrics with arbitrary signature, broadening the understanding of the manifold's structure beyond positive-definite cases.

Findings

Manifold can be finitely covered by complex projective space with scaled Fubini-Study metric

Extension of Tanno's theorem to indefinite K"ahler metrics

Structural characterization of solutions to the Tanno equation

Abstract

Considering a non-constant smooth solution $f$ of the Tanno equation on a closed, connected K\"ahler manifold $(M,g,J)$ with positively definite metric $g$, Tanno showed that the manifold can be finitely covered by $(\mathbb{C}P(n),\mbox{const}\cdot g_{FS})$, where $g_{FS}$ denotes the Fubini-Study metric of constant holomorphic sectional curvature equal to $1$. The goal of this paper is to give a proof of Tannos Theorem for K\"ahler metrics with arbitrary signature.