25-30 November 2024
Saint-Petersburg University
Europe/Moscow timezone

Automorphisms of Hopf manifolds of dimension $n\geq2$

30 Nov 2024, 12:35
35m
Saint-Petersburg University

Saint-Petersburg University

Department of Mathematics and Computer Sciences, Saint-Petersburg University, Saint Petersburg, 14 line V.O., 29B Yandex maps link: https://yandex.ru/maps/-/CDw2mFl9 Google maps link: https://maps.app.goo.gl/L1Nrzf81wahREKop6 ZOOM streaming at: TBA

Speaker

Elijah Lopatin (Steklov Mathematical Institute of RAS, Moscow)

Description

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.

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\textbf{References}\[.3cm]
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\end{enumerate}

Primary author

Elijah Lopatin (Steklov Mathematical Institute of RAS, Moscow)

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