1
vote
0answers
78 views

Colimit and push forward of quasi-coherent sheaves in the analytic setting

I'm currently working on my master thesis and have some unresolved questions about quasi-coherent sheaves. Since I'm new to algebraic geometry, they might be rather trivial. I'm working in the ...
1
vote
1answer
61 views

Prove that each convex open subset of $\mathbb C^n$ is a domain of holomorphy. [closed]

Prove that each convex open subset of $\mathbb C^n$ is a domain of holomorphy. Please help. Thanks in advance!
0
votes
1answer
103 views

Why is the set of points where a complex polynomial does not vanish is connected?

Let $p$ be a complex multivariate polynomial. Let $C$ be the set of those complex tuples where $p$ is nonzero. Then, $C$ is connected.
1
vote
0answers
36 views

“Center of mass” of a complex hypersurface

Background: Suppose first that $f \in \mathbb{C}[z]$ and consider the "analytic hypersurfaces" $V(f)$ and $V(df)$, which are of course just sets of points with multiplicities. (That is, we think of ...
2
votes
2answers
340 views

Generalization of Cauchy Residue theorem to Multi-dimensional holomorphic functions

We know Cauchy Residue theorem from the Complex analysis. however I wonder if there is a kind of Generalization of Cauchy integral and Residue theorem to the complex multidimensional holomorphic ...
5
votes
0answers
59 views

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 ...
7
votes
2answers
587 views

Where to learn algebraic analysis

I have been studying categories, sheaf cohomology and complex analysis (the basics since I know just a little). Then recently I tried to find out more about algebraic analysis and these microlocal ...
3
votes
0answers
180 views

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 ...