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arXiv · 1506.03286

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

Slawomir Dinew·Grzegorz Kapustka·Michal Kapustka
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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.

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Slawomir Dinew, Grzegorz Kapustka, Michal Kapustka. 2015-06-10. Remarks on Mukai threefolds admitting $C^{*}$ action. https://arxiv.org/abs/1506.03286

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