K-polystability of the first secant varieties of rational normal curves

Abstract

The first secant variety $\Sigma$ of a rational normal curve of degree $d\ge 3$ is known to be a $\mathbf{Q}$-Fano threefold. In this paper, we prove that $\Sigma$ is K-polystable, and hence, $\Sigma$ admits a weak Kähler-Einstein metric. We also show that there exists a $(-K_{\Sigma })$-polar cylinder in $\Sigma$.

17 pages