Let $D$ be a ${𝒞}^d$$q$-convex intersection, $d\ge 2$, $0\le q\le n-1$, in a complex manifold $X$ of complex dimension $n$, $n\ge 2$, and let $E$ be a holomorphic vector bundle of rank $N$ over $X$. In this paper, ${𝒞}^k$-estimates, $k=2,3,\cdots ,\infty$, for solutions to the $\overline{\partial }$-equation with small loss of smoothness are obtained for $E$-valued $(0,s)$-forms on $D$ when $n-q\le s\le n$. In addition, we solve the $\overline{\partial }$-equation with a support condition in ${𝒞}^k$-spaces. More...

We derive conditions under which a holomorphic mapping of a taut Riemann surface must be an automorphism. This is an analogue involving invariant distances of a result of H. Cartan. Using similar methods we prove an existence result for 1-dimensional holomorphic retracts in a taut complex manifold.

Let $E$ be a holomorphic Banach bundle over a compact complex manifold, which can be defined by a cocycle of holomorphic transition functions with values of the form $\mathrm{id}+K$ where $K$ is compact. Assume that the characteristic fiber of $E$ has the compact approximation property. Let $n$ be the complex dimension of $X$ and $0\le q\le n$. Then: If $V\to X$ is a holomorphic vector bundle (of finite rank) with $H^q(X,V)=0$, then $dim{H}^q(X,V\otimes E)\<\infty$. In particular, if $dim{H}^q(X,𝒪)=0$, then $dim{H}^q(X,E)\<\infty$.

We introduce a new invariant Kähler metric on relatively compact domains in a complex manifold, which is the Bergman metric of the L² space of holomorphic sections of the pluricanonical bundle equipped with the Hermitian metric introduced by Narasimhan-Simha.

We show that the symmetrized bidisc 𝔾₂ = {(λ₁+λ₂,λ₁λ₂):|λ₁|,|λ₂| < 1} ⊂ ℂ² cannot be exhausted by domains biholomorphic to convex domains.

We show a relation between the Kobayashi pseudodistance of a holomorphic fiber bundle and the Kobayashi pseudodistance of its base. Moreover, we prove that a holomorphic fiber bundle is taut iff both the fiber and the base are taut.

Let $\Omega$ an open set in ${\mathbf{C}}^4$ near $z_0\in \partial \Omega$, $\lambda$ a suitable holomorphic function near $z_0$. If we know that we can solve the following problem (see [M. Derridj, Annali. Sci. Norm. Pisa, Série IV, vol. IX (1981)]) : $\bar{\partial }u=\lambda f$, ($f$ is a $(0,1)$ form, $\bar{\partial }$ closed in $U\left(z_0\right)$ in $U\left(z_0\right)$ with supp$\left(u\right)\subset \bar{\Omega }\cap U(z_0)$, then we deduce an extension result for $C.R.$ functions on $\partial \Omega \cap U\left(z_0\right)$, as holomorphic fonctions in $\Omega \cap V\left(z_0\right)$.

Let $B$ be the open unit ball of a Banach space $E$, and let $f:B\to B$ be a holomorphic map with $f\left(0\right)=0$. In this paper, we discuss a condition whereby $f$ is a linear isometry on $E$.