# Questions tagged [cv.complex-variables]

Complex analysis, holomorphic functions, automorphic group actions and forms, pseudoconvexity, complex geometry, analytic spaces, analytic sheaves.

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### Why are holomorphic $p$-forms parallel?

Let $X$ be a complex compact Kähler manifold, $\Omega_X$ its cotangent bundle and $D$ the Chern connection. It is, I believe, a standard fact that parallel $k$-forms $\alpha \in \mathcal{A}^k(X)$, i.e....
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### Complex Hölder space

I already posted this question on math.stackexchange, but got no response and was suggested to post it here. I came across a space in an ergodic theory paper, which I am calling here a (complex) ...
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### Construction of holomorphic function

I was trying to construct a holomorphic function $f$ on $\mathbb{C}$ such that $|f|^2(z)=e^{(|z|^2-\frac{1}{2})^2}$. I will be happy if someone can give me an idea how to do that. I would like also ...
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### Differential equation of Van Gorder type for zeta of global fields, or: Does the zeta function of a global field satisfy a differential equation?

Let \begin{equation*} \zeta(s):=\prod_{p\text{ prime}}\frac{1}{1-p^{-s}} \end{equation*} be the Riemann zeta function. Van Gorder has shown that $\zeta$ satisfies a differential equation \begin{...
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### Less strict holomorphy

Using the Grünwald-Letnikov definition of fractional derivative for complex-valued functions, can there be complex function $f(z)$ over all plane, that is not holomorphic but there is $r \in (0,1)$ s....
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### Component wise convergence of a sequence of complex harmonic functions

It is well known that a complex harmonic function $f$ on a simply connected domain $D$ has a canonical decomposition of the form $$f=g+\bar{h},$$ where $g$ and $h$ are analytic functions on $D.$ In ...