Multiplier Hermitian-Einstein metrics on Fano manifolds of KSM-type

TL;DR

This paper investigates the existence of multiplier Hermitian-Einstein metrics on KSM-manifolds, providing a criterion based on KSM-data and constructing explicit examples connecting known special metrics.

Contribution

It establishes a new criterion for the existence of multiplier Hermitian-Einstein metrics on KSM-manifolds using KSM-data and constructs explicit examples linking different special metrics.

Findings

Established a criterion for existence of multiplier Hermitian-Einstein metrics on KSM-manifolds.

Constructed explicit examples of KSM-manifolds with a family of such metrics.

Connected K"ahler-Ricci solitons and Mabuchi solitons via a continuous path.

Abstract

In this article we focus on multiplier Hermitian-Einstein metrics introduced by Mabuchi which include K\"ahler-Einstein metrics, K\"ahler-Ricci solitons and Mabuchi solitons as special cases. We also focus on KSM-manifolds, which are introduced by the first author as toric bundles, to establish a criterion for the existence of multiplier Hermitian-Einstein metrics in terms of KSM-data. An explicit example for a KSM-manifold admitting a family of multiplier Hermitian-Einstein metrics is constructed by using a continuous path connecting a K\"ahler-Ricci soliton and a Mabuchi soliton.