Characterization of Cn{\bf C}^n by its automorphism group

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 ${\bf C}^n$.Comment: to appear in Proc. Steklov Inst. mat

Homogeneous Kobayashi-hyperbolic manifolds with high-dimensional group of holomorphic automorphisms

We determine all connected homogeneous Kobayashi-hyperbolic manifolds of dimension $n\ge 2$ whose holomorphic automorphism group has dimension $n^2-2$. This result complements an existing classification for automorphism group dimension $n^2-1$ and greater obtained without the homogeneity assumption

On necessary and sufficient conditions for the Kobayashi hyperbolicity of tube domains in C2{\mathbb C}^2

This note concerns tube domains in ${\mathbb C}^2$ with the envelope of holomorphy not equal to the entire space. We construct examples showing that for such domains the sufficient condition for Kobayashi hyperbolicity due to M. Jarnicki and P. Pflug cannot be replaced by its weaker "affine" variant, which is known to be a necessary condition for hyperbolicity. Thus, we arrive at the somewhat unexpected conclusion that the obstructions for a domain in the above class to be Kobayashi hyperbolic are not just "affine"

A combinatorial proof of the smoothness of catalecticant schemes associated to complete intersections

For zero-dimensional complete intersections with homogeneous ideal generators of equal degrees over an algebraically closed field of characteristic zero, we give a combinatorial proof of the smoothness of the corresponding catalecticant schemes along an open subset of a particular irreducible component.Comment: To appear in the Annals of Combinatoric

On the Kobayashi hyperbolicity of tube domains in C2{\mathbb C}^2

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

Homogeneous Kobayashi-hyperbolic manifolds with automorphism group of subcritical dimension

We determine all connected homogeneous Kobayashi-hyperbolic manifolds of dimension $n\ge 2$ whose holomorphic automorphism group has dimension $n^2-3$. This result complements existing classifications for automorphism group dimension $n^2-2$ (which is in some sense critical) and greater.Comment: arXiv admin note: substantial text overlap with arXiv:1709.0305

On the Classification of Homogeneous Hypersurfaces in Complex Space

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\, $1<t<\sqrt{(2+\sqrt{2})/3}$

Further steps towards classifying homogeneous Kobayashi-hyperbolic manifolds with high-dimensional automorphism group

We determine all connected homogeneous Kobayashi-hyperbolic manifolds of dimension $n\ge 4$ whose group of holomorphic automorphisms has dimension either $n^2-4$, or $n^2-5$, or $n^2-6$. This paper continues a series of articles that achieve classifications for automorphism group dimension $n^2-3$ and greater.Comment: arXiv admin note: text overlap with arXiv:1709.03052, arXiv:1709.0704

Characterization of the unit ball in Cn{\bf C}^n among complex manifolds of dimension nn

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}.Comment: J. Geometric Analysis 14(2004), 697-700; erratum, to appear in J. Geometric Analysis 18(2008), no.

On the Affine Homogeneity of Algebraic Hypersurfaces Arising from Gorenstein Algebras

To every Gorenstein algebra $A$ of finite dimension greater than 1 over a field ${\Bbb F}$ of characteristic zero, and a projection $\pi$ 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 $S_{\pi}\subset{\mathfrak m}$. 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 $S_{\pi}$. 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 $S_{\pi}$ under the assumption that the algebra $A$ is graded.Comment: 13 page