Germs of local automorphisms of real-analytic CR structures and analytic dependence on kk-jets

The topic of the paper is the study of germs of local holomorphisms $f$ between $C^n$ and $C^{n'}$ such that $f(M)\subset M'$ and $df(T^cM)=T^cM'$ for $M\subset C^n$ and $M'\subset C^{n'}$ generic real-analytic CR submanifolds of arbitrary codimensions. It is proved that for $M$ minimal and $M'$ finitely nondegenerate, such germs depend analytically on their jets. As a corollary, an analytic structure on the set of all germs of this type is obtained.Comment: 17 page

On the automorphism groups of algebraic bounded domains

Let $D$ be a bounded domain in $C^n$. By the theorem of H.~Cartan, the group $Aut(D)$ of all biholomorphic automorphisms of $D$ has a unique structure of a real Lie group such that the action $Aut(D)\times D\to D$ is real analytic. This structure is defined by the embedding $C_v\colon Aut(D)\hookrightarrow D\times Gl_n(C)$, $f\mapsto (f(v), f_{*v})$, where $v\in D$ is arbitrary. Here we restrict our attention to the class of domains $D$ defined by finitely many polynomial inequalities. The appropriate category for studying automorphism of such domains is the Nash category. Therefore we consider the subgroup $Aut_a(D)\subset Aut(D)$ of all algebraic biholomorphic automorphisms which in many cases coincides with $Aut(D)$. Assume that $n>1$ and $D$ has a boundary point where the Levi form is non-degenerate. Our main result is theat the group $Aut_a(D)$ carries a unique structure of an affine Nash group such that the action $Aut_a(D)\times D\to D$ is Nash. This structure is defined by the embedding $C_v\colon Aut_a(D)\hookrightarrow D\times Gl_n(C)$ and is independent of the choice of $v\in D$.Comment: 29 pages, LaTeX, Mathematischen Annalen, to appea

Algebraicity of local holomorphisms between real-algebraic submanifolds of complex spaces

We prove that a germ of a holomorphic map $f$ between $C^n$ and $C^{n'}$ sending one real-algebraic submanifold $M\subset C^n$ into another $M'\subset C^{n'}$ is algebraic provided $M'$ contains no complex-analytic discs and $M$ is generic and minimal. We also propose an algorithm for finding complex-analytic discs in a real submanifold.Comment: 12 pages An algorithm for finding complex-analytic discs in a real submanifold is adde

A geometric approach to Catlin's boundary systems

For a point $p$ in a smooth real hypersurface M\subset\C^n, where the Levi form has the nontrivial kernel $K^{10}_p$, we introduce an invariant cubic tensor \tau^3_p \colon \C T_p \times K^{10}_p \times \overline{K^{10}_p} \to \C\otimes (T_p/H_p), which together with Ebenfelt's tensor $\psi_3$, constitutes the full set of $3$rd order invariants of $M$ at $p$. Next, in addition, assume M\subset\C^n to be {\em (weakly) pseudoconvex}. Then $\tau^3_p$ must identically vanish. In this case we further define an invariant quartic tensor \tau^4_p \colon \C T_p \times \C T_p \times K^{10}_p\times \overline{K^{10}_p} \to \C\otimes (T_p/H_p), and for every $q=0, \ldots, n-1$, an invariant submodule sheaf of $(1,0)$ vector fields in terms of the Levi form, and an invariant ideal sheaf of complex functions generated by certain derivatives of the Levi form, such that the set of points of Levi rank $q$ is locally contained in certain real submanifolds defined by real parts of the functions in the ideal sheaf, whose tangent spaces have explicit algebraic description in terms of the quartic tensor $\tau^4$. Finally, we relate the introduced invariants with D'Angelo's finite type, Catlin's mutlitype and Catlin's boundary systems.Comment: Exposition and references are substantially expande

Domains of polyhedral type and boundary extensions of biholomorphisms

For $D$, $D'$ analytic polyhedra in $C^n$, it is proven that a biholomorphic mapping $f\colon D\to D'$ extends holomorphically to a dense boundary subset under certain condition of general position. This result is also extended to a more general class of domains with no smoothness condition on the boundary.Comment: 11 page