# Questions tagged [cv.complex-variables]

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

1,979 questions
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### Why only $\bar\partial$ but not $\partial$ in Dolbeault cohomology

While I learn about $\partial$ and $\bar{\partial}$ operators, I had some questions about the reason why people prefer $\bar\partial$ over $\partial$. Specifically, When defining Dolbeault ...
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### How to prove that weighted Bergman space is separable.

Let $D$ be a bounded domain in $\mathbb{C}^n$ and $\varphi$ be a non-positive plurisubharmonic function on $D$. The weighted Bergman space $A^2(D,e^{-\varphi})$ is the space of holomorphic functions ...
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### An upper bound for minimum

For any polynomials of degree $n$ having all its zeros in $|z|\leq K,K\geq 1,$ is it true $\max_{|z|=1}|nP(z)+(a-z)P'(z)|\geq n\min_{|z|=K}|P(z)|$ where $a$ is any complex number with $|a|\geq K?$
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### Bounding the derivative of a holomorphic function on a disk by its absolute value

Let $f(z)$ be a holomorphic function defined on the disk $|z|\le 2$. Suppose $|f(z)|<1$ for $|z|\le 2$. It looks like there is a constant $c>0$ such that $|f(z)'|<c$ on the disk $|z|\le 1$ (...
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### Does exist a Kahler-Einstein metric on the blow-up of $\mathbb{P}^3$ along a smooth plane cubic?

This might be well known for the experts but I am not able to find a reference. I was wondering if there exists a Kahler-Einstein metric on the Fano threefold given by blow-up of $\mathbb{P}^3$ along ...
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### What are good ways to 'relax' a uniform approximation into independent saddle-point expressions once the uniform approach is no longer needed?

I am doing long-running project that involves asymptotic saddle-point estimation of integrals (for flavour, it's this sort of stuff) and I would like to ask if there are established ways in the ...
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### How do analysts think about functions with poles at all roots of unity?

In branches of algebra impinging on the enumeration of partitions, one often encounters formulas like $$\prod_i \left( \frac{1}{1-q^i} \right)^{n_i}$$ for some integers $n_i$. E.g., with $n_i = 1$, ...
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### Defining “addition” on the Riemann surface of log(z)

The title of this question is a bit awkward, as adding two points together on a manifold is usually not considered possible, but in this case there appears to be a nice little hack. Consider the ...
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### Roots of a polynomial inside the unit circle

Let $k$ be a even positive integer. Now, consider the polynomial $$p(x)=x^k-px^{k-1}-qx^{k-2}-x^{k-3}-\cdots -x-1,$$ with $p$ and $q$ integers satisfying $q-1>p\geq 1$. How to prove that this ...
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### Zeros of MacLaurin polynomials for the exponential function

Asked but never answered at MSE. Let $\exp_n(z)$ denote the nth degree Taylor polynomial of $e^z$ : $\exp_n(z) = 1 + z + z^2/2! + ... + z^n/n! \;$ . The zeros of $\exp_n(z)$ were studied by ...
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### Countable dense subset of functions of exponential type 1 that decay along the positive real axis

I am interested in the space of all holomorphic function of exponential type one, that decay exponentially along the positive real axis. I tried to define it as follows. Let \|f\|_n = \sup_{z\in\...