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I would say that Mandelbrot set (like similar fractal objects coming from complex dynamics) can be seen as a "pathological" object, at least from the point of view of regularity (the boundary is nowhe …

The answer is no. In fact, Kallin has shown in [Kal64] that the union of three disjoint closed balls is polynomially convex, but the union of three disjoint closed polydisks needs not to be polynomi …

Are you also looking for holomorphic manifolds with $\dim \mathcal O=\infty$? In that case, in the paper by Sasane On the Krull Dimension of Rings of Transfer Functions [Acta Applicandae Mathematicae …

Your question is not very clear. However, I guess you are asking for the branch points of the cover of $\mathbb{C}$ defined by $f(z, \alpha)=0$. In this case, let us assume for the sake of symplicity …

If we work in the category of holomorphic functions, then we can give the following definition, that generalises to the complex-analytic setting the classical intersection multiplicity used in algebra …

The answer is no, because of the following general result. Theorem (Grauert's contractibility criterion). Let $X$ be a smooth complex surface and let $E \subset X$ be a connected curve in $X$, wit …

One reference is Proposition 14.7 in Remmert's paper Local Theory of complex analytic spaces, Several complex variable VII, Encyclopaedia of Math. Sci. vol 74. For the reader's convenience I will rest …

The answer is yes: the equivalence class of the covering is detected by the monodromy representation of the fundamental group of the base minus the branch locus, up to conjugacy. More precisely, let …

Every finitely presented group is the fundamental group of a compact complex manifold of dimension $3$. This is proven in the book by Amoros, Burger, Corlette, Kotschick and Toledo Fundamental group …

Compact Riemann surfaces of genus at least $2$ are uniformized by the unit disk, hence they do not admit any non-constant holomorphic map from $\mathbb{C}$.

Over $\mathbf{C}$, algebraic normalization and analytic normalization are equivalent concepts. See N. Kuhlmann: Die Normalisierung komplexer Räume, Math. Ann. 144 (1961), 110-125, ZBL0096.27801. Qu …

Let me give a proof using the deformation theory of holomorphic maps developed by Horikawa [Journal Math. Soc. Japan 25]. This can be seen as a purely analytic proof in the spirit of Kodaira's "defor …

Some standard examples are Hopf surfaces, obtained as quotients of the form $X=\mathbb{C}^2 - \{0\} /G$, where $G$ is the subgroup generated by the homothety $(z_1, z_2) \to (\alpha_1 z_1, \alpha …

There are actually counterexamples in real dimension $4$. The first examples of compact almost complex $4$-manifolds admitting no complex structure were produced by Van de Ven in his paper "On the Ch …

I think that the answer is no. Take a complex torus $X$ without any holomorphic curve. If the result you are asking for were true, it would imply (in the empty sense) that every subset $A \subset X$ i …