Equivariant Kähler model for Fujiki's class

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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