Equivariant Kähler model for Fujiki’s class

Let \(X\) $X$ be a compact complex manifold in Fujiki’s class \(\mathcal {C}\) $\mathcal{C}$ , i.e., admitting a big \((1,1)\) $(1,1)$ -class \([\alpha ]\) $[\alpha ]$ . Consider \({{\,\textrm{Aut}\,}}(X)\) $\text{Aut}(X)$ the group of biholomorphic automorphisms and \({{\,\textrm{Aut}\,}}_{[\alpha ]}(X)\) ${\text{Aut}}_{[\alpha ]}(X)$ the subgroup of automorphisms preserving the class \([\alpha ]\) $[\alpha ]$ via pullback. We show that \(X\) $X$ admits an \({{\,\textrm{Aut}\,}}_{[\alpha ]}(X)\) ${\text{Aut}}_{[\alpha ]}(X)$ -equivariant Kähler model: there is a bimeromorphic holomorphic map \(\sigma :{\widetilde{X}}\rightarrow X\) $\sigma :\tilde{X}\to X$ from a Kähler manifold \({\widetilde{X}}\) $\tilde{X}$ such that \({{\,\textrm{Aut}\,}}_{[\alpha ]}(X)\) ${\text{Aut}}_{[\alpha ]}(X)$ lifts holomorphically via \(\sigma \) $\sigma$ . There are several applications. We show that \({{\,\textrm{Aut}\,}}_{[\alpha ]}(X)\) ${\text{Aut}}_{[\alpha ]}(X)$ is a Lie group with only finitely many components. This generalizes an early result of Fujiki and Lieberman on the Kähler case. We also show that every torsion subgroup of \({{\,\textrm{Aut}\,}}(X)\) $\text{Aut}(X)$ is almost abelian, and \({{\,\textrm{Aut}\,}}(X)\) $\text{Aut}(X)$ is finite if it is a torsion group.