$K$-polystability of smooth Fano $\text{SL}_2$-threefolds

TL;DR

This paper establishes the $K$-polystability of certain smooth Fano threefolds with an $ ext{SL}_2$ action, confirming the existence of Kähler-Einstein metrics for these varieties within the Mori-Mukai classification.

Contribution

It proves $K$-polystability for all smooth Fano threefolds with an $ ext{SL}_2$ action, extending the understanding of their geometric properties.

Findings

Proves $K$-polystability for specific Fano threefolds

Confirms existence of Kähler-Einstein metrics on these varieties

Completes classification for varieties with $ ext{SL}_2$ symmetry

Abstract

We prove the $K$-polystability of all smooth complex Fano threefolds admitting an effective action of $\text{SL}_2$ but not of a 2-torus or 3-torus. In particular, the existence of K\"{a}hler-Einstein metrics on varieties in the families (1.10), (1.15), (1.16), (1.17), (2.21), (2.27), (2.32), (3.13), (3.17), (3.25) and (4.6) of the Mori-Mukai classification of smooth Fano threefolds is proved.