From K\"ahler Ricci solitons to Calabi-Yau K\"ahler cones

TL;DR

This paper demonstrates that the canonical cone of a product of a Fano manifold with a complex projective space becomes a Calabi-Yau cone if the Fano manifold admits a K"ahler Ricci soliton, linking solitons to Calabi-Yau geometries.

Contribution

It establishes a connection between K"ahler Ricci solitons on Fano manifolds and Calabi-Yau cones on their products, extending previous conjectures.

Findings

The canonical cone of the product becomes Calabi-Yau under certain conditions.

Openness of the set of weight functions for which a v-soliton exists.

Obstructions and volume bounds related to K"ahler Ricci solitons.

Abstract

We show that if $X$ is a smooth Fano manifold which caries a K\"ahler Ricci soliton, then the canonical cone of the product of $X$ with a complex projective space of sufficiently large dimension is a Calabi--Yau cone. This can be seen as an asymptotic version of a conjecture by Mabuchi and Nikagawa. This result is obtained by the openness of the set of weight functions $v$ over the momentum polytope of a given smooth Fano manifold, for which a $v$-soliton exists. We discuss other ramifications of this approach, including a Licherowicz type obstruction to the existence of a K\"ahler Ricci soliton and a Fujita type volume bound for the existence of a $v$-soliton.