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

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 $\mathcal{Y}:=\stackrel{‾}{(L_X\otimes {\mathcal{O}}_{{\mathbb{P}}^k}(1){)}^{\times }}$ 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.