From Calabi's extremal metrics to scalar-flat K\"ahler cones

TL;DR

The paper demonstrates that high-dimensional cones over polarized manifolds with extremal K"ahler metrics admit scalar-flat K"ahler cone metrics, extending to Sasaki joins with spheres of large dimension.

Contribution

It establishes the existence of scalar-flat K"ahler cone metrics on certain high-dimensional cones derived from extremal K"ahler manifolds, answering an open asymptotic question.

Findings

High-dimensional cones over extremal K"ahler manifolds admit scalar-flat K"ahler cone metrics.

Unweighted Sasaki joins with large-dimensional spheres admit constant scalar curvature Sasaki metrics.

Provides an affirmative answer to an asymptotic question by Boyer et al.

Abstract

We prove that for any smooth polarized complex $n$-dimensional manifold $(X, L_X)$ which admits an extremal K\"ahler metric in $c_1(L_X)$, and for any integer $k$ large enough (in terms of a bound depending on $(X, L_X)$), the $(n+k+1)$-dimensional complex cone $\mathcal{Y}:= \overline{(L_X \otimes \mathcal{O}_{\mathbb{P}^k}(1))^{\times}}$ with section $X \times \mathbb{P}^k$ admits a scalar-flat K\"ahler cone metric. Equivalently, the unweighted Sasaki join of a smooth compact quasi-regular extremal Sasaki manifold with a regular Sasaki sphere $\mathbb{S}^{2k+1}$ of sufficiently large dimension $(2k+1)$ admits a Sasaki metric of constant (positive) scalar curvature. This gives an affirmative answer to an asymptotic version of a question raised by Boyer--Huang--Legendre--T{\o}nnesen-Friedman in arXiv:1906.04827.