Equivariant Kähler model for Fujiki's class (Q6624037)

A compact complex manifold is said to be in Fujiki's class \(\mathcal{C}\) if it is bimeromorphic to a compact Kähler manifold, or equivalently, it admits a big \((1,1)\)-class. It was proved by \textit{J.-P. Demailly} and \textit{M. Paun} [Ann. Math. (2) 159, No. 3, 1247--1274 (2004; Zbl 1064.32019)] that a compact complex manifold \(X\) in Fujiki's class \(\mathcal{C}\) admits a bimeromorphic holomorphic map \(\tau\colon X'\to X\) from a compact Kähler manifold \(X'\); the map \(\tau\) is a composite of blowups along smooth centers which are determined by the ideal sheaf corresponding to some Kähler current in a big \((1,1)\)-class \([\alpha]\). In general, the map \(\tau\) is not equivariant with respect to the group \(\text{Aut}(X)\) of biholomorphic automorphisms.\N\NIn the paper under review, the authors consider the subgroup \(\text{Aut}_{[\alpha]}(X)\) of automorphisms preserving a big \((1,1)\)-class \([\alpha]\) via pullback. The authors show that \(X\) admits an \(\text{Aut}_{[\alpha]}(X)\)-equivariant Kähler model (possibly different from the above \(X'\)), that is, there is a bimeromorphic holomorphic map \(\sigma\colon \widetilde{X}\to X\) from a compact Kähler manifold \(\widetilde{X}\) such that \(\text{Aut}_{[\alpha]}(X)\) lifts holomorphically via \(\sigma\); the map \(\sigma\) is obtained by a sequence of \(\text{Aut}_{[\alpha]}(X)\)-equivariant blowups along smooth centers (see Theorem 1.1). For the first application, the authors prove that the subgroup \(\text{Aut}_{[\alpha]}(X)\) is virtually contained in the identity component \(\text{Aut}_0(X)\) of \(\text{Aut}(X)\), extending a result of \N\textit{A. Fujiki} [Invent. Math. 44, 225--258 (1978; Zbl 0367.32004)] and \N\textit{D. I. Lieberman} [Lect. Notes Math. 670, 140--186 (1978; Zbl 0391.32018)] \Nto the Fujiki's class \(\mathcal{C}\) (see Corollary 1.3). For the second application, the authors show that the quotient \(\text{Aut}(X)/\text{Aut}_0(X)\) has bounded torsion subgroups (see Corollary 1.4). For the last application, the authors prove that \(\text{Aut}(X)\) has Jordan property for a compact complex space in Fujiki's class \(\mathcal{C}\), which gives an alternative proof of a theorem of \textit{S. Meng} et al. [J. Topol. 15, No. 2, 806--814 (2022; Zbl 1521.14077)] (see Corollary 1.5).