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arXiv · 2309.12522

K-stability of Casagrande-Druel varieties

Ivan Cheltsov·Tiago Duarte Guerreiro·Kento Fujita·Igor Krylov·Jesus Martinez-Garcia
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Abstract

We introduce a new subclass of Fano varieties (Casagrande-Druel varieties), that are $n$-dimensional varieties constructed from Fano double covers of dimension $n-1$. We conjecture that a Casagrande-Druel variety is K-polystable if the double cover and its base space are K-polystable. We prove this for smoothable Casagrande-Druel threefolds, and for Casagrande-Druel varieties constructed from double covers of $\mathbb{P}^{n-1}$ ramified over smooth hypersurfaces of degree $2d$ with $n>d>\frac{n}{2}>1$. As an application, we describe the connected components of the K-moduli space parametrizing smoothable K-polystable Fano threefolds in the families 3.9 and 4.2 in the Mori-Mukai classification.

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Ivan Cheltsov, Tiago Duarte Guerreiro, Kento Fujita, Igor Krylov, Jesus Martinez-Garcia. 2023-09-21. K-stability of Casagrande-Druel varieties. https://arxiv.org/abs/2309.12522

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