Soliton-type metrics associated with weighted CSCK metrics on Fano manifolds
Satoshi Nakamura
TL;DR
This work unifies two canonical metric problems on Fano manifolds by introducing a weight $g(v,w)$ from a pair of weights $(v,w)$ and proving that a $T$-invariant $(v,w)$-CSCK metric in $2\pi c_1(X)$ exists if and only if a corresponding $g(v,w)$-soliton exists, under positivity and log-concavity hypotheses. It develops the link via Futaki-type invariants and Ding/Mabuchi-type energy functionals, using a continuity method to construct the soliton and hence obtain the CSCK metric. The paper also reveals a natural Sasaki-geometric interpretation: weighted CSCK metrics and $g$-solitons on $X$ arise as quotients of transverse extremal/Mabuchi solitons on Sasaki manifolds, with explicit translation of weights. This Sasaki viewpoint supports the conjectured equivalence and provides a framework for potential moduli-theoretic applications in birational geometry and minimal model programs.
Abstract
We study weighted constant scalar curvature Kähler 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 $2πc_1(X)$ which gives rise to a $\hatξ$-transverse extremal metric on an $S^1$-bundle $N$ in the canonical bundle of a Fano manifold $X$, where $\hatξ$ is a possibly irregular Reeb field on $N$. We prove that the associated $g(v,w)$-soliton on $X$ gives rise to a $\hatξ$-transverse Mabuchi soliton on $N$.