Questions tagged [several-complex-variables]
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213 questions
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What´s essential to learn about complex spaces and several complex variables for an algebraic geometer?
Hi, I don´t know if this question is suitable for this site. The field of several complex variables is too broad, so I would like to know what´s essential to learn about complex spaces and several ...
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How does the complex convex set look like?
The usual convex set is the real linear convex set, if we change the real linear map into complex linear map, we can get the complex convex set. A system way to do this is in the several complex ...
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Extension of pluriharmonic functions
Suppose $M$ is a complex manifold and $\Omega$ a (edit: bounded) pseudoconvex domain in $M$. Let $u:M\setminus\Omega\to\mathbb{R}$ be a pluriharmonic function. Is it true that $u$ has a pluriharmonic ...
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Another proof of the bidisc and the ball are biholomorphically inequivalent?
Does this outline of a proof work?
Consider the ball and the bidisc in $\mathbb{C}^2$. Give each space its Bergman metric. To show that the ball and the bidisc are not holomorphic, it is enough to ...
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Fundamental motivation for several complex variables [closed]
I have 3 general abstract reasons to care about complex analysis in a single variable:
The laplacian is, up to a constant multiple, the only isometry invariant PDO in the plane, and so it is ...
5
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A sequence that tell us if a holomorphic function of several variables is identically zero
Is there any sequence $ \{ Z_{\nu} \}_{\nu \in \mathbb{N}}$ in $\mathbb{C}^{n}$, $Z_{\nu} \rightarrow 0$, such that any holomorphic function in $\mathbb{C}^{n}$ which vanishes in $Z_{\nu}$ for all $\...
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Plurisubharmonic function
Let $\Omega$ be a pseudo convex domain. Let $r$ be any $C^2$ function $r: \mathbb C^2\to \mathbb R$. Let $\Omega: \{z: r(z)<0\}$. Then we know that $\psi: -log(-r)+\lambda |z|^2$ is a ...
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Analytic extension across the boundary.
Let $Q=[0,\infty)\times [0,\infty)\subset \mathbb C$ and $f: Q\times Q\to Q\times Q$ be a diffeomorphism.
such that $f$ is holomorphic in the interior of $Q\times Q$. Can we extend this map ...
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How does pseudoconvexity restrict the topology?
A domain of holomorphy in $\mathbb{C}^n$ has vanishing de rham cohomology in real dimensions greater than $n$ - half of it's cohomology is missing. Are there any other restrictions? If I give you a ...
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Complex manifolds with corner?
I was reading Dominic Joycee article on Manifold with corner. He talk about manifold with corner modeled over $[0,\infty)^k\times \mathbb R^{n-k}$ for some $k\leq n$. From here I moved to Melrose ...
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Reference for the converse of Cartan's Theorem B
Cartan's theorem B states that every coherent analytic sheaf on a Stein manifold is acyclic. Hartshorne claims that the converse to this theorem also holds, but doesn't supply a reference. Does ...
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Snazzy applications of Several Complex Variables techniques
I am starting to dive into a study of Several Complex Variables. I would like to have a few guiding examples of "big payoffs". These should be very natural sounding theorems which depend on a lot of ...
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Functions of several complex variables: book recommendations?
Can anyone recommend a good comprehensive introduction to functions of several complex variables that a) is fairly up to date, b) isn't a geometry or an algebra book only, but takes multiple ...