Extremal metrics on del Pezzo threefolds

TL;DR

This paper establishes the existence of Kahler-Einstein metrics on specific algebraic varieties, including certain Grassmannian sections and hypersurfaces, and computes a key invariant for the Mukai--Umemura variety.

Contribution

It proves the existence of Kahler-Einstein metrics on new classes of algebraic varieties and determines the global log canonical threshold for the Mukai--Umemura variety.

Findings

Existence of Kahler-Einstein metrics on a Grassmannian section and a degree 6 hypersurface.

Calculation of the global log canonical threshold of the Mukai--Umemura variety as 1/2.

Abstract

We prove the existence of Kahler-Einstein metrics on a nonsingular section of the Grassmannian $\mathrm{Gr}(2, 5)\subset\mathbb{P}^9$ by a linear subspace of codimension 3, and the Fermat hypersurface of degree 6 in $\mathbb{P}(1,1,1,2,3)$. We also show that a global log canonical threshold of the Mukai--Umemura variety is equal to 1/2.