Equivariant Kähler model for Fujiki’s class

Let X be a compact complex manifold in Fujiki’s class C, i.e., admitting a big (1,1)-class [α]. Consider Aut(X) the group of biholomorphic automorphisms and Aut_[α](X) the subgroup of automorphisms preserving the class [α] via pullback. We show that X admits an Aut_[α](X)-equivariant Kähler model: there is a bimeromorphic holomorphic map σ:X~→X from a Kähler manifold X~ such that Aut_[α](X) lifts holomorphically via σ. There are several applications. We show that Aut_[α](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 Aut(X) is almost abelian, and Aut(X) is finite if it is a torsion group.