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Consider injective homolomorphic functions $f:\mathbb D\to \mathbb C$ on the unit disk $|z|\leq 1$, normalized by the conditions $f(0)=0$ and $f'(0)=1$. Thus for $|z|\leq 1$ we have $ f(z)=\sum …

Consider a complex power series $\sum a_n z^n \in \mathbb C[[z]]$ with radius of convergence $0\lt r\lt\infty$ and suppose that for every $w$ with $\mid w\mid =r$ the series $\sum a_n w^n $ converges …

Yes, every holomorphic vector bundle of any rank is trivial on the punctured disk $\dot{\Delta}$ . Indeed, since $\dot{\Delta}$ is a Stein manifold ( like any non-compact Riemann surface ! ) the Oka m …

First let me note that your definition of coherent sheaf is misleading: it implies that $\mathcal O_X$ is coherent by definition, whereas in reality coherence of $\mathcal O_X$ is a very deep theor …

Dear kaddar, here is a partial answer. According to a theorem of Douady, a flat map $f:X\to S$ between complex analytic spaces is always open . So if you assume that the fibers of $f$ are reduced an …

Dear David, I think there might be a slightly simpler proof of the analytic Quillen-Suslin theorem. Given a holomorphic vector bundle $E$ on $\mathbb C^n$, it has a holomorphic connection since its A …

let's call Runge an open subset $\Omega \subset \mathbb C $ such that polynomials are dense in $\mathcal H(\Omega)$ . A hole of $\Omega$ is a compact connected component of $ \mathbb C \setminus \Omeg …

There exist infinitely many holomorphically non-isomorphic complex structures on the unit ball of R^4 (or more generally R^2n): this is a beautiful theorem of Burns, Shnider, Wells ( Inventiones Math. …

Dear unknown, here is a sketch of proof of your question ( which I have modified to make it more accurate, as explained in my comments to your original post .) Statement If $V=V_1 \cup V_2$ with $V …

Consider a connected holomorphic manifold $X$ and its ring of holomorphic functions $\mathcal O(X).$ My general question is simply: in which cases is the Krull dimension $\dim \mathcal O(X)$ known? …

A justly celebrated theorem by Ehresmann states that a proper smooth submersion $\pi: X\to S$ between smooth manifolds is locally trivial in the sense that every point $s\in S$ downstairs has a neigh …

Dear Gunnar, let $f:X\to Y$ be a proper flat morphism of smooth varieties. This does not imply that $f$ has smooth fibres either in the algebraic or in the analytic case. For example, any non-constant …

Dear Colin , for $X$ a holomorphic connected manifold, denote by $\mathcal M (X)$ its field of meromorphic functions. A) It is not true that a germ of holomorphic function $f_x\in \mathcal O_{X,x}$ …

Here are a few points to guide you into the beautiful subject you had the good taste to choose. 1) Hartogs extension phenomenon :given two concentric balls in $ \mathbb C^n$, any holomorphic funct …

1) There is a great book From Holomorphic Functions to Complex Manifolds by Fritzsche-Grauert. It is very geometric and gives you the fundamentals on complex manifolds, including specialized topics, f …