Fujiki类的等变Kähler模型
Equivariant Kähler model for Fujiki's class
摘要：
设$X$是Fujiki类$\mathcal{C}$中的紧致复流形，即，承认一个大的$(1,1)$级$[α]$。考虑$\text{Aut}(X)$为双全纯自同构群，$\text{Aut}_{[α]}(X)$为通过拉回保持类$[α]$的自同构子群。证明了$X$允许$\text{Aut}_{[α]}(X)$ -等变Kähler模型：存在一个来自Kähler流形$\widetilde{X}$的双纯全纯映射$σ\colon \widetilde{X}\to X$，使得$\text{Aut}_{[α]}(X)$通过$σ$全纯提升.有几种应用。我们证明了$\text{Aut}_{[α]}(X)$是一个只具有100个分支的李群。这推广了利伯曼和藤木在凯勒案例中的早期结果。我们还证明了$\text{Aut}(X)$的每个扭子群几乎是阿贝尔群，并且如果$\text{Aut}(X)$是扭群，则它是有限的.
英文摘要：
Let $X$ be a compact complex manifold in Fujiki's class $\mathcal{C}$, i.e., admitting a big $(1,1)$-class $[α]$. Consider $\text{Aut}(X)$ the group of biholomorphic automorphisms and $\text{Aut}_{[α]}(X)$ the subgroup of automorphisms preserving the class $[α]$ via pullback. We show that $X$ admits an $\text{Aut}_{[α]}(X)$-equivariant Kähler model: there is a bimeromorphic holomorphic map $σ\colon \widetilde{X}\to X$ from a Kähler manifold $\widetilde{X}$ such that $\text{Aut}_{[α]}(X)$ lifts holomorphically via $σ$. There are several applications. We show that $\text{Aut}_{[α]}(X)$ is a Lie group with only finitely many components. This generalizes an early result of Lieberman and Fujiki on the Kähler case. We also show that every torsion subgroup of $\text{Aut}(X)$ is almost abelian, and $\text{Aut}(X)$ is finite if it is a torsion group.
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