Describing the group of automorphisms $\mathrm{Aut}(X)$ of a compact complex manifold $X$ is among the classical issues of complex geometry. According to the Bochner--Montgomery~[1] theorem, such groups are complex Lie groups and it is almost everything we can a priori say about them: for a majority of $X$, it is extremely complicated (or nearly impossible) to find the generating set of $\mathrm{Aut}(X)$ or some other explicit characterisation.
Therefore it is natural to investigate classification properties, i.e. such properties that the group $\mathrm{Aut}(X)$ possesses one when $X$ is a complex manifold, and does not for other $X$. It seems the Jordan property~[8] to be the most promising.
Let $G$ be a group. We say that $G$ is \textit{Jordan} (or has the \textit{Jordan property}) if there is a constant $J = J(G)\in\mathbb N$ such that for any finite subgroup $H\subset G$ there is a normal abelian subgroup $A \unlhd H$ of index at most $J(G)$.
It is known that automorphism groups of complex projective varieties~[5] and, more generally, compact Kähler manifolds~[7]
are Jordan. For non-Kähler compact complex manifolds there are only a few known results on the Jordan property for automorphism groups: for compact complex manifolds in Fujiki's class $\mathcal{C}$~[6], for compact complex surfaces~[9] and for some examples~[3,4] of non-Kähler holomorphically symplectic manifolds~[2].
Hopf manifold $\mathsf{H}_n$, i.\,e. a compact complex manifold of dimension $n\geq 2$ such that its universal cover is isomorphic to $\mathbb C^n\setminus 0$, is a natural example of non-Kähler complex manifold for studying structural properties of its automorphism group. $\mathsf{H}_n$ is realized as a quotient of $\mathbb{C}^n\setminus0$ by a free action of a group isomorphic to $\mathbb Z$, which acts on $\mathbb{C}^n\setminus0$ via biholomorphic contractions $\mathbb C^n\setminus 0\to\mathbb C^n\setminus 0$. Recently it was shown~[10] that $\mathrm{Aut}(\mathsf{H}_n)$ is Jordan. We expand on the results of~[10] proving that the group $\mathrm{Aut}(\mathsf{H}_n)/\mathrm{Aut}^0(\mathsf{H}_n)$ is finite; here $\mathrm{Aut}^0(\mathsf{H}_n)$ is the connected component of unity in $\mathrm{Aut}(\mathsf{H}_n)$. We also provide the explicit structure of mentioned biholomorphic contractions.
This is a joint research with Constantin Shramov, Steklov Mathematical Institute of RAS, Moscow.
\begin{center}
\textbf{References}\[.3cm]
\end{center}
\begin{enumerate}
\item
Bochner, S. and Montgomery, D. \textit{Groups on analytic manifolds,} Annals of Mathematics \textbf{48}, (1947), 659--669.
\item
Bogomolov, F., Kurnosov, N., Kuznetsova, A. and Yasinsky, E. \textit{Geometry and automorphisms of non-Kähler holomorphic symplectic manifolds,} Int. Math. Res. Notices IMRN 2022:6, 12302--12341.
\item
Guan, D. \textit{Examples of compact holomorphic symplectic manifolds which are not Kählerian. II,} Inventiones Mathematicae \textbf{121}:1, (1995), 135--145.
\item
Guan, D. \textit{Examples of compact holomorphic symplectic manifolds which are not Kählerian. II,} Internat. J. Math. \textbf{6}:5, (1995), 709--718.
\item Meng, S. and Zhang, D.-Q. \textit{Jordan property for nonlinear algebraic groups and projective varieties,} Amer. J. Math. \textbf{140}:4, (2018), 1133--1145.
\item
Meng, S., Perroni, F. and Zhang, D.-Q. \textit{Jordan property for automorphism groups of compact spaces in Fujiki’s class $\mathcal{C}$,} J. Topol. \textbf{15}:2, (2022), 806--814.
\item
Kim, J.\, H. \textit{Jordan property and automorphism groups of normal compact Kähler varieties,} Commun. Contemp. Math. \textbf{20}:3, (2018), 1750024.
\item
Popov, V.\,L. \textit{On the Makar-Limanov, Derksen invariants, and finite automorphism groups of algebraic varieties}, in: \textit{Affine Algebraic Geometry: The Russell Festschrift,} CRM Proceedings and Lecture Notes, Vol. 54, Amer. Math. Soc., 2011, pp. 289--311.
\item
Prokhorov, Yu. and Shramov, C. \textit{Automorphism groups of compact complex surfaces,} Int. Math. Res. Notices \textbf{2021}:14, (2021), 10490--10520.
\item Savelyeva, A. \textit{Automorphisms of Hopf manifolds,} Journal of Algebra \textbf{638}, (2024), 670--681.
\end{enumerate}