From Kähler Ricci solitons to Calabi-Yau Kähler cones

TL;DR

The paper unifies Kähler–Ricci solitons and Calabi–Yau cone structures through the framework of $v$-solitons on Fano manifolds. It proves an openness result for the set of weight functions admitting a soliton, uses Sasaki/cone machinery to connect transversal and base data, and derives a pathway from KRS on a Fano base to Calabi–Yau cone structures on the associated canonical cone via taking products with large projective spaces. It also establishes a Lichnerowicz-type obstruction for transversal KRS solitons and a weighted Fujita volume bound, providing rigorous a priori constraints on existence and stability data. Together, these results give an asymptotic route to Calabi–Yau cones from KRS and deepen the link between weighted canonical geometry and cone metrics.

Abstract

We show that if $X$ is a smooth Fano manifold which caries a Kähler 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 Kähler Ricci soliton and a Fujita type volume bound for the existence of a $v$-soliton.

Theorems & Definitions (57)

• Theorem 1.1
• Remark 1.1
• Conjecture 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Remark 2.1
• Definition 2.1: $v$-soliton
• Definition 2.2: $v$-Futaki invariant