Remarks on Mukai threefolds admitting $C^{\ast }$ action

Abstract

We investigate geometric invariants of the one parameter family of Mukai threefolds that admit ${\mathbb{C}}^{\ast }$ 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 ${\mathbb{C}}^{\ast }$ action, a fact that was previously observed near the Mukai-Umemura threefold by Rollin, Simanca and Tipler. As a consequence the Kähler-Einstein manifolds in the class form an open subset in the standard topology.

20 pages

References in corpus (1)

• Kahler-Einstein metrics on symmetric Fano T-varieties

Cited by in corpus (1)

• Real Rank Geometry of Ternary Forms