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A. V. Isaev

Publications and source records attributed to A. V. Isaev.

At least 37 records · Page 2Linked to original sources
arXiv2008

A Remark on a Theorem by Kodama and Shimizu

We prove a characterization theorem for the unit polydisc $Δ^n\subset\CC^n$ in the spirit of a recent result due to Kodama and Shimizu. We show that if $M$ is a connected $n$-dimensional complex manifold such that (i) the group $\hbox{Aut}(M)$ of holomorphic automorphisms of $M$ acts on $M$ with compact isotropy subgroups, and (ii) $\hbox{Aut}(M)$ and $\hbox{Aut}(Δ^n)$ are isomorphic as topological groups equipped with the compact-open topology, then $M$ is holomorphically equivalent to $Δ^n$.

math.CV↗
arXiv2007

Proper actions of Lie groups of dimension $n^2+1$ on $n$-dimensional complex manifolds

In this paper we continue to study actions of high-dimensional Lie groups on complex manifolds. We give a complete explicit description of all pairs $(M,G)$, where $M$ is a connected complex manifold $M$ of dimension $n\ge 2$, and $G$ is a connected Lie group of dimension $n^2+1$ acting effectively and properly on $M$ by holomorphic transformations. This result complements a classification obtained earlier by the first author for $n^2+2\le\hbox{dim} G<n^2+2n$ and a classical result due to W. Kaup for the maximal group dimension $n^2+2n$.

math.CV↗
arXiv2007

Proper actions of high-dimensional groups on complex manifolds

We explicitly classify all pairs $(M,G)$, where $M$ is a connected complex manifold of dimension $n\ge 2$ and $G$ is a connected Lie group acting properly and effectively on $M$ by holomorphic transformations and having dimension $d_G$ satisfying $n^2+2\le d_G<n^2+2n$. These results extend -- in the complex case -- the classical description of manifolds admitting proper actions of groups of sufficiently high dimensions. They also generalize some of the author's earlier work on Kobayashi-hyperbolic manifolds with high-dimensional holomorphic automorphism group.

math.CV↗
arXiv2007

On Chern-Moser normal forms of strongly pseudoconvex hypersurfaces with high-dimensional stability group

We explicitly describe germs of strongly pseudoconvex non-spherical real-analytic hypersurfaces $M$ at the origin in $\CC^{n+1}$ for which the group of local CR-automorphisms preserving the origin has dimension $d_0(M)$ equal to either $n^2-2n+1$ with $n\ge 2$, or $n^2-2n$ with $n\ge 3$. The description is given in terms of equations defining hypersurfaces near the origin, written in the Chern-Moser normal form. These results are motivated by the classification of locally homogeneous Levi non-degenerate hypersurfaces in $\CC^3$ with $d_0(M)=1,2$ due to A. Loboda, and complement earlier joint work by V. Ezhov and the author for the case $d_0(M)\ge n^2-2n+2$.

math.CV↗
arXiv2006

Hyperbolic 2-dimensional manifolds with 3-dimensional automorphism group

In this paper we determine all Kobayashi-hyperbolic 2-dimensional complex manifolds for which the group of holomorphic automorphisms has dimension 3. This work concludes a recent series of papers by the author on the classification of hyperbolic $n$-dimensional manifolds, with automorphism group of dimension at least $n^2-1$, where $n\ge 2$.

math.CV↗
arXiv2000

Effective actions of the unitary group on complex manifolds

We classify all connected $n$-dimensional complex manifolds admitting an effective action of the unitary group $U_n$ by biholomorphic transformations. One consequence of this classification is a characterization of ${\bf C}^n$ by its automorphism group.

math.CV↗
arXiv1999

On the automorphism groups of hyperbolic manifolds

We show that there does not exist a Kobayashi hyperbolic complex manifold of dimension $n\ne 3$, whose group of holomorphic automorphisms has dimension $n^2+1$ and that, if a 3-dimensional connected hyperbolic complex manifold has automorphism group of dimension 10, then it is holomorphically equivalent to the Siegel space. These results complement earlier theorems of the authors on the possible dimensions of automorphism groups of domains in comlex space. The paper also contains a proof of our earlier result on characterizing $n$-dimensional hyperbolic complex manifolds with automorphism groups of dimension $\ge n^2+2$.

math.CV↗
arXiv1998

Invariants of elliptic and hyperbolic CR-structures of codimension 2

We reduce CR-structures on smooth elliptic and hyperbolic manifolds of CR-codimension 2 to parallelisms thus solving the problem of global equivalence for such manifolds. The parallelism that we construct is defined on a sequence of two principal bundles over the manifold, takes values in the Lie algebra of infinitesimal automorphisms of the quadric corresponding to the Levi form of the manifold, and behaves ``almost'' like a Cartan connection. The construction is explicit and allows us to study the properties of the parallelism as well as those of its curvature form. It also leads to a natural class of ``semi-flat'' manifolds for which the two bundles reduce to a single one and the parallelism turns into a true Cartan connection. In addition, for real-analytic manifolds we describe certain local normal forms that do not require passing to bundles, but in many ways agree with the structure of the parallelism.

math.CV↗
arXiv1998

Canonical isomorphism of two Lie algebras arising in CR-geometry

We show that the maximal prolongation of a certain algebra associated with a non-degenerate Hermitian form on ${\Bbb C}^n\times{\Bbb C}^n$ with values in ${\Bbb R}^k$ is canonically isomorphic to the Lie algebra of infinitesimal holomorphic automorphisms of the corresponding quadric in ${\Bbb C}^{n+k}$. This fact creates a link between different approaches to the equivalence problem for Levi-nondegenerate strongly uniform CR-manifolds.

math.CV↗
arXiv1997

Domains with Non-Compact Automorphism Group: A Survey

We survey results arising from the study of domains in C^n with non-compact automorphism group. Beginning with a well-known characterization of the unit ball, we develop ideas toward a consideration of weakly pseudoconvex (and even non-pseudoconvex) domains with particular emphasis on characterizations of (i) smoothly bounded domains with non-compact automorphism group and (ii) the Levi geometry of boundary orbit accumulation points. Particular attention will be paid to results derived in the past ten years.

math.CV↗
arXiv1996

Finitely smooth Reinhardt domains with non-compact automorphism group

We give a complete description of bounded Reinhardt domains of finite boundary smoothness that have non-compact automorphism group. As part of this program, we show that the classification of domains with non-compact automorphism group and having only finite boundary smoothness is considerably more complicated than the classification of such domains that have infinitely smooth boundary.

math.CV↗
arXiv1996

On the boundary orbit accumulation set for a domain with non-compact automorphism group

For a smoothly bounded pseudoconvex domain $D\subset{\Bbb C}^n$ of finite type with non-compact holomorphic automorphism group $\text{Aut}(D)$, we show that the set $S(D)$ of all boundary accumulation points for $\text{Aut}(D)$ is a compact subset of $\partial D$ and, if $S(D)$ contains at least three points, it is connected and thus has the power of the continuum. We also discuss how $S(D)$ relates to other invariant subsets of $\partial D$ and show in particular that $S(D)$ is always a subset of the Šilov boundary.

math.CV↗