arXiv subjects
Publications and source records attributed to Alexander Isaev.
Recently, I. Kossovskiy and R. Shafikov have settled the so-called Dimension Conjecture, which characterizes spherical hypersurfaces in ${\mathbb C}^2$ via the dimension of the algebra of infinitesimal automorphisms. In this note, we propose a short argument for obtaining their result.
We construct an elementary counterexample to the criterion for Kobayashi hyperbolicity for a class of tube domains in ${\mathbb C}^2$ proposed by J.-J. Loeb.
Proper group actions are ubiquitous in mathematics and have many of the attractive features of actions of compact groups. In this survey, we discuss proper actions of Lie groups on smooth manifolds. If the group dimension is sufficiently high, all proper effective actions can be explicitly determined, and our principal goal is to provide a comprehensive exposition of known classification results in the complex setting. They include a complete description of Kobayashi-hyperbolic manifolds with high-dimensional automorphism group, which is a case of special interest.
Let C(2,1) be the class of connected 5-dimensional CR-hypersurfaces that are 2-nondegenerate and uniformly Levi degenerate of rank 1. In a recent article, we proved that the CR-structures in C(2,1) are reducible to so(3,2)-valued absolute parallelisms. In the present paper, we apply this result to study tube hypersurfaces in C^3 that belong to C(2,1) and whose CR-curvature identically vanishes. Every such hypersurface is shown to be affinely equivalent to an open subset of the tube over the future light cone.
We discuss a family $M_t^n$, with $n\ge 2$, $t>1$, of real hypersurfaces in a complex affine $n$-dimensional quadric arising in connection with the classification of homogeneous compact simply-connected real-analytic hypersurfaces in ${\mathbb C}^n$ due to Morimoto and Nagano. To finalize their classification, one needs to resolve the problem of the embeddability of $M_t^n$ in ${\mathbb C}^n$ for $n=3,7$. We show that $M_t^7$ is not embeddable in ${\mathbb C}^7$ for every $t$ and that $M_t^3$ is embeddable in ${\mathbb C}^3$ for all $1<t<1+10^{-6}$. As a consequence of our analysis of a map constructed by Ahern and Rudin, we also conjecture that the embeddability of $M_t^3$ takes place for all\, $1<t<\sqrt{(2+\sqrt{2})/3}$.
It was conjectured in a recent article by M. Eastwood and the second author that all absolute classical invariants of forms of degree $m\ge 3$ on ${\mathbb C}^n$ can be extracted, in a canonical way, from those of forms of degree $n(m-2)$ by means of assigning every form with non-vanishing discriminant the so-called associated form. In that paper, this surprising conjecture was confirmed for binary forms of degree $m \le 6$ and ternary cubics. In the present article, we settle the conjecture in full generality. In addition, we propose a stronger version of this statement and obtain evidence supporting it.
Let ${\mathfrak C}_{2,1}$ be the class of connected 5-dimensional CR-hypersurfaces that are 2-nondegenerate and uniformly Levi degenerate of rank 1. We show that the CR-structures in ${\mathfrak C}_{2,1}$ are reducible to ${\mathfrak{so}}(3,2)$-valued absolute parallelisms and give applications of this result.
In an earlier article the second author introduced three families of tube domains in ${\mathbf C}^2$ with holomorphic automorphism group isomorphic to ${\mathbf R}\ltimes{\mathbf R}^2$ and envelope of holomorphy equal to ${\mathbf C}^2$. In the present paper we show that every domain in each of these families is Kobayashi-hyperbolic.
We use classical invariant theory to solve the biholomorphic equivalence problem for two families of plane curve singularities previously considered in the literature. Our calculations motivate an intriguing conjecture that proposes a way of extracting a complete set of invariants of homogeneous plane curve singularities from their moduli algebras.
To every Gorenstein algebra $A$ of finite dimension greater than 1 over a field ${\Bbb F}$ of characteristic zero, and a projection $π$ on its maximal ideal ${\mathfrak m}$ with range equal to the annihilator $\hbox{Ann}({\mathfrak m})$ of ${\mathfrak m}$, one can associate a certain algebraic hypersurface $S_π\subset{\mathfrak m}$. Such hypersurfaces possess remarkable properties. They can be used, for instance, to help decide whether two given Gorenstein algebras are isomorphic, which for ${\Bbb F}={\Bbb C}$ leads to interesting consequences in singularity theory. Also, for ${\Bbb F}={\Bbb R}$ such hypersurfaces naturally arise in CR-geometry. Applications of these hypersurfaces to problems in algebra and geometry are particularly striking when the hypersurfaces are affine homogeneous. In the present paper we establish a criterion for the affine homogeneity of $S_π$. This condition requires the automorphism group $\hbox{Aut}({\mathfrak m})$ of ${\mathfrak m}$ to act transitively on the set of hyperplanes in ${\mathfrak m}$ complementary to $\hbox{Ann}({\mathfrak m})$. As a consequence of this result we obtain the affine homogeneity of $S_π$ under the assumption that the algebra $A$ is graded.
We consider complex manifolds that admit actions by holomorphic transformations of classical simple real Lie groups and classify all such manifolds in a natural situation. Under our assumptions, which require the group at hand to be dimension-theoretically large with respect to the manifold on which it is acting, our classification result states that the manifolds which arise are described precisely as invariant open subsets of certain complex flag manifolds associated to the complexified groups.
We show that the group of holomorphic automorphisms of a Stein manifold X of dimension greater than 1 is infinite-dimensional, provided X is a homogeneous space of a holomorphic action of a complex Lie group.
We show that if the group of holomorphic automorphisms of a connected complex manifold $M$ of dimension $n$ is isomorphic as a topological group equipped with the compact-open topology to the automorphism group of the unit ball $B^n\subset\CC^n$, then $M$ is biholomorphically equivalent to either $B^n$ or $\CC\PP^n\setminus\bar{B^n}$.
We classify locally defined non-spherical real-analytic hypersurfaces in complex space whose Levi form has no more than one negative eigenvalue and for which the dimension of the group of local CR-automorphisms has the second largest value.
We classify the tube domains in C^4 with affinely homogeneous base whose boundary contains a non-degenerate affinely homogeneous hypersurface. It follows that these domains are holomorphically homogeneous and amongst them there are four new examples of unbounded homogeneous domains (that do not have bounded realisations). These domains lie to either side of a pair of Levi-indefinite hypersurface. Using the geometry of these two hypersurfaces, we find the automorphism groups of the domains.
We construct several new examples of homogeneous domains in complex space that do not have bounded realisations. They are equivalent to tubes over affinely homogeneous domains in real space and have a real-analytic everywhere Levi non-degenerate non-umbilic boundary. Using the geometry of the boundary, we determine the full automorphism groups of the domains. We also discuss some interesting examples of tube domains with everywhere umbilic boundary (i.e., boundary equivalent to the corresponding quadric).
We show that if the group of holomorphic automorphisms of a connected Stein manifold $M$ is isomorphic to that of ${\bf C}^n$ as a topological group equipped with the compact-open topology, then $M$ is biholomorphically equivalent to ${\bf C}^n$.