Soliton-type metrics associated with weighted CSCK metrics on Fano manifolds

TL;DR

This paper establishes a link between weighted constant scalar curvature Kähler metrics and soliton-type metrics on Fano manifolds, introducing a new weight function and demonstrating their equivalence under certain conditions, with connections to Sasaki geometry.

Contribution

It introduces a new weight function $g(v,w)$ and proves the equivalence between $(v,w)$-CSCK metrics and $g(v,w)$-solitons on Fano manifolds, extending the understanding of these metrics and their geometric origins.

Findings

Existence of $(v,w)$-CSCK metric is equivalent to $g(v,w)$-soliton under positivity and log-concavity.

A $g(v,w)$-soliton naturally arises from Sasaki geometry.

The $g(v,w)$-soliton on Fano manifolds induces a transverse Mabuchi soliton on associated Sasaki structures.

Abstract

We study weighted constant scalar curvature K\"ahler metrics, introduced by Lahdili as $(v,w)$-CSCK metrics, on Fano manifolds and their relationship with soliton-type metrics. In this paper, we introduce a weight function $g(v,w)$ associated with a pair of weight functions $(v,w)$. Assuming that $v$ and $g(v,w)$ are positive and log-concave on the moment polytope, we prove that the existence of a $(v,w)$-CSCK metric in the first Chern class is equivalent to the existence of a $g(v,w)$-soliton. We also explain that a $g(v,w)$-soliton arises naturally from Sasaki geometry. More precisely, let $(v,w)$ be the weight functions defining a weighted CSCK metric in $2\pi c_1(X)$ which gives rise to a $\hat{\xi}$-transverse extremal metric on an $S^1$-bundle $N$ in the canonical bundle of a Fano manifold $X$, where $\hat{\xi}$ is a possibly irregular Reeb field on $N$. We prove that the associated $g(v,w)$-soliton on $X$ gives rise to a $\hat{\xi}$-transverse Mabuchi soliton on $N$.