We show that any bounded balanced domain of holomorphy is an $L{²}_h$-domain of holomorphy.

A nonlinear generalization of convergence sets of formal power series, in the sense of Abhyankar-Moh [J. Reine Angew. Math. 241 (1970)], is introduced. Given a family $y={\phi }_s(t,x)=sb₁\left(x\right)t+b₂\left(x\right)t²+\cdots$ of analytic curves in ℂ × ℂⁿ passing through the origin, $Con{v}_{\phi }\left(f\right)$ of a formal power series f(y,t,x) ∈ ℂ[[y,t,x]] is defined to be the set of all s ∈ ℂ for which the power series $f({\phi }_s(t,x),t,x)$ converges as a series in (t,x). We prove that for a subset E ⊂ ℂ there exists a divergent formal power series f(y,t,x) ∈ ℂ[[y,t,x]] such that $E=Con{v}_{\phi }\left(f\right)$ if and only if...

A complex analytic space is said to have the D*-extension property if and only if any holomorphic map from the punctured disk to the given space extends to a holomorphic map from the whole disk to the same space. A Hartogs domain H over the base X (a complex space) is a subset of X x C where all the fibers over X are disks centered at the origin, possibly of infinite radius. Denote by φ the function giving the logarithm of the reciprocal of the radius of the fibers, so that, when X is pseudoconvex,...

In the present paper, we associate the techniques of the Lewy-Pinchuk reflection principle with the Behnke-Sommer continuity principle. Extending a so-called reflection function to a parameterized congruence of Segre varieties, we are led to studying the envelope of holomorphy of a certain domain covered by a smooth Levi-flat “hat”. In our main theorem, we show that every ${𝒞}^{\infty }$-smooth CR diffeomorphism $h:M\to M^{\text{'}}$ between two globally minimal real analytic hypersurfaces in ${ℂ}^n$ ($n\ge 2$) is real analytic at every point...

Let $V$ be an algebraic variety in $C^n$ and when $k\ge 0$ is an integer then $\mathrm{Pol}(V,k)$ denotes all holomorphic functions $f\left(z\right)$ on $V$ satisfying $\left|f\left(z\right)\right|\le C_f(1+|z|{)}^k$ for all $z\in V$ and some constant $C_f$. We estimate the least integer $ϵ(V,k)$ such that every $f\in \mathrm{Pol}(V,k)$ admits an extension from $V$ into $C^n$ by a polynomial $P(z_1,...,z_n)$, of degree $k+ϵ(V,k)$ at most. In particular ${lim}_{k\>\infty }ϵ(V,k)$ is related to cohomology groups with coefficients in coherent analytic sheaves on $V$. The existence of the finite integer $ϵ(V,k)$ is for example an easy consequence of Kodaira’s Vanishing Theorem.

We study the extension problem of holomorphic maps $\sigma :H\to X$ of a Hartogs domain $H$ with values in a complex manifold $X$. For compact Kähler manifolds as well as various non-Kähler manifolds, the maximal domain ${\Omega }_{\sigma }$ of extension for $\sigma$ over $\Delta$ is contained in a subdomain of $\Delta$. For such manifolds, we define, in this paper, an invariant Hex${}_n\left(X\right)$ using the Hausdorff dimensions of the singular sets of $\sigma$’s and study its properties to deduce informations on the complex structure of $X$.

We present various characterizations of n-circled domains of holomorphy $G\subset {ℂ}^n$ with respect to some subspaces of ${\mathscr{H}}^{\infty }\left(G\right)$.

To a pair of a Lie group $G$ and an open elliptic convex cone $W$ in its Lie algebra one associates a complex semigroup $S=G\mathrm{Exp}\left(iW\right)$ which permits an action of $G\times G$ by biholomorphic mappings. In the case where $W$ is a vector space $S$ is a complex reductive group. In this paper we show that such semigroups are always Stein manifolds, that a biinvariant domain $D\subseteq S$ is Stein is and only if it is of the form $G\mathrm{Exp}\left(D_h\right)$, with $Dh\subseteq iW$ convex, that each holomorphic function on $D$ extends to the smallest biinvariant Stein domain containing $D$,...

Let $M=G/H$ be a real symmetric space and $𝔤=𝔥+𝔮$ the corresponding decomposition of the Lie algebra. To each open $H$-invariant domain $D_{𝔮}\subseteq i𝔮$ consisting of real ad-diagonalizable elements, we associate a complex manifold $\Xi \left(D_{𝔮}\right)$ which is a curved analog of a tube domain with base $D_{𝔮}$, and we have a natural action of $G$ by holomorphic mappings. We show that $\Xi \left(D_{𝔮}\right)$ is a Stein manifold if and only if $D_{𝔮}$ is convex, that the envelope of holomorphy is schlicht and that $G$-invariant plurisubharmonic functions correspond to convex $H$-invariant...