Remarks on Mukai threefolds admitting $C^{*}$ action

TL;DR

This paper studies Mukai threefolds with a C* action, identifying invariants, bounds on thresholds, and symmetries, revealing their geometric structure and implications for Kähler-Einstein metrics.

Contribution

It provides an explicit description of Mukai threefolds with C* symmetry, including invariant divisors and additional symmetries, advancing understanding of their geometric properties.

Findings

Identified invariant divisors in the anticanonical system.

Established bounds on log canonical thresholds.

Discovered an additional symmetry that anticommutes with the C* action.

Abstract

We investigate geometric invariants of the one parameter family of Mukai threefolds that admit $\mathbb C^{*}$ action. In particular we find the invariant divisors in the anticanonical system, and thus establish a bound on the log canonical thresholds. Furthermore we find an explicit description of such threefolds in terms of the quartic associated to the variety-of-sum-of-powers construction. This yields that any such threefold admits an additional symmetry which anticommutes with the $\mathbb C^{*}$ action, a fact that was previously observed near the Mukai-Umemura threefold by Rollin, Simanca and Tipler. As a consequence the K\"ahler-Einstein manifolds in the class form an open subset in the standard topology.