Steady Ricci Solitons on Complex Line Bundles

TL;DR

This paper proves the existence and uniqueness of a family of smooth, complete steady Ricci solitons on complex line bundles over Fano Kähler-Einstein bases, highlighting their non-Kähler nature except on the canonical bundle.

Contribution

It establishes the existence and uniqueness of a one-parameter family of steady Ricci solitons on complex line bundles over Fano Kähler-Einstein manifolds, expanding understanding of Ricci solitons in complex geometry.

Findings

Existence of a one-parameter family of Ricci solitons

Uniqueness of these solitons under the given conditions

Non-Kähler nature of the solitons except on the canonical bundle

Abstract

We show the existence and uniqueness of a one-parameter family of smooth complete $U(1)$-invariant gradient steady Ricci solitons on the total space of any complex line bundle over a Fano K\"ahler-Einstein base with first Chern class proportional to that of the base. These solitons are non-K\"ahler except on the total space of the canonical bundle.