# Questions tagged [several-complex-variables]

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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1answer
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### Problem book recommendations on complex manifolds

I came across the book on Cauchy Riemann manifolds, "CR manifolds and tangential Cauchy Riemann complexes". The book does not have a problem section. I would be grateful if anyone recommends ...
1answer
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### Kunneth formula for Čech cohomology

I want to inquire if there exists a Kunneth formula of underlined form: Let $X$ and $Y$ be two compact complex manifolds with two holomorphic vector bundles $E$ and $F$ on $X$ and $Y$ respectively, ...
1answer
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### Triangle inequality for Bergman metric

Please help to prove triangle inequality for d(z,w) Where
0answers
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### Trying to find whether a set is complex analytic

I haven't been able to decide whether the image $X$ of the holomorphic function $f: \mathbb{C} \rightarrow \mathbb{C}^2$ given by $f(z) = (z^2 - 1, z^3 - z)$ is analytic or not. Here is what I have ...
1answer
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0answers
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### Distinct ellipsoids are not biholomorphic in $\mathbb{C}^2$

I am stuck on the following problem: For each real $p\geq 1$ consider the the set $$D_p=\{(z,w)\in\mathbb{C}^2:|z|^{2p}+|w|^2<1\}.$$ Let $p\neq q$, then, there does not exist any biholomorphism ...
1answer
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1answer
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### Solving 2D Poisson equation using complex analysis

In 2D, the Poisson equation $$\nabla^2 \Phi(\vec{x}) = \delta^{(2)}(\vec{x})$$ admits the solution $$\Phi(\vec{x}) = \frac{1}{2\pi}\log(|\vec{x}|) + \text{constant}\,. \tag{1}$$ By using Fourier ...
0answers
35 views

### Contour integrals in complex manifold

We know that for contours $\gamma$ in $\mathbb{C}$, we can define winding numbers as $$\frac{1}{2\pi i} \oint_\gamma \frac{dz}{z}$$ Are there similar formulation for $\mathbb{C}^2$ or even for all ...
0answers
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0answers
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0answers
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