# Tagged Questions

Use this tag for questions related to the study of analytic functions of strictly more than one complex variables. For the single complex dimension case, use (complex-analysis).

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### Methods for “recognizing” a polynomial of several variables.

If $f(z)$ is an entire function of a single complex variable, then the following are indirect methods for recognizing that $f$ is a polynomial. 1) Show that $f^{(n)}\equiv0$ for some $n\geq0$. 2) ...
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### Branch locus along a smooth curve

Let $f : S \to X$ be a dominant morphism of smooth complex surfaces. Let $C \subset S$ be a smooth curve such that $df$ is of rank $1$ along $C$ and that in a neighborhood of $C$, $df_x$ is an ...
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### entension of a holomorphic map on projective algebraic manifolds

I'm reading a paper.It says if $M$ is a projective algebraic variety and $f:N-S\rightarrow M$ is a holomorphic map,where $N$ is a connected complex manifold and $S\subset N$ is a proper analytic ...
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### Openness of a subset in complex 2-plane

Let $U$ be a subset of $\mathbb{C}^{2}$ containing the origin $0$. Assume that for any curve $C$ (an affine variety of dimension 1, maybe singular) passing through $0$ we have $U \cap C$ is ...
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### What's the difference between analytic singularity and algebraic singularity?

Let $f$ be a holomorphic function defined in $U\in \mathbb{C}^n$, and $f=0$ gives an analytic variety. Suppose $f$ is irreducible as a germ of holomorphic function in $\mathcal{A}_z$, here $z$ is a ...
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### vanishing theorem in algebraic geometry

This is a general question: As we know there are lot of vanishing theorem like Fujita vanishing, kodaira Nakano vanishing, vanishing for big nef line bundle, Kollár vanishing, etc. Those just for ...
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### When a current is actually a holomorphic form?

If a current $f$ of bidegree $(p,0)$ (acting on forms of bidegree $(n-p,n)$) satisfies $\bar{d}f=0$, is it true that $f$ is a holomorphic differential form? In general, do we have any standard ...
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### $(P,H)$ Euclidean Hartogs figure

Let $(P,H)$ be a Euclidean Hartogs figure in $\mathbb{C}^n$ and $f:H\to \mathbb{C}^n$ a holomorphic injective map, then we know that $f$ extend holomorphically to polidisc $P$ (i.e. there is a ...
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### Multivariate Residue Theorem

Let $G(s,t)$ be a complex valued function in two variables that converges absolutely for $Re(s), Re(t)>1$. Suppose we can analytically continue $G$ in such a way that $$G(s,t) = f(s)g(t)H(s,t)$$ ...
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### Analytic variety

Let $\Omega$ be a convex domain in $\mathbb{C^{n}}$ , $\Delta$ be unit disk in $\mathbb{C}$ and f: $\Delta \rightarrow b\Omega$ be a non-constant holomorphic map.Then the convex hull of f($\Delta$) ...
Let $W\subset \mathbb C^n$ be a domain such that $W$ is the domain of convergence for a certain power series at the origin. How to show that the interior of $W$ is a complete Reinhardt domain?