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# Questions tagged [cv.complex-variables]

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

2,034 questions
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### Dense Stein subset in complex manifold

Let $X$ be a smooth proper algebraic variety. Then $X$ has a dense affine open subset. In particular, any smooth proper algebraic variety has a dense Stein open subset as the complement of a divisor. ...
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### Holomorphic maps from upper half plane to itself (or equivalently Poincare disc to itself)

Suppose I parametrize complex plane by coordinates,$$z = x+i y,\ \bar z=x-i y$$ then the upper half plane, $\mathbb H_+$ is given by $y>0$. I am looking for chiral coordinate transformations, $f(z)$...
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### Does there exist a Runge Fatou-Bieberbach in each Fatou-Bieberbach domain?

A Fatou-Bieberbach domain $\Omega \subseteq \mathbb{C}^n$ is a domain that is a proper subset of $\mathbb{C}^n$ and is biholomorphic to $\mathbb{C}^n$. A domain is said to be Runge if for each ...
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### Ratio of exponentially weighted Selberg integrals

I'm interested in bounding the following ratio of integral: \frac{\int_{0<x_k<...<x_1<1}\prod_{i=1}^kx_i^{m-\frac{k+1}{2}}\prod_{i<j}(x_i-x_j)\exp(-\sum_{i=1}^kw_ix_i)}{\int_{0<x_k&...
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Let $K$ be a compact Hausdorff space and $A$ be a subset of $C(K)$. $A$ is said to be a Korovkin set if for every sequence $(T_n)$ of positive linear operators on $C(K)$, the condition $\|T_n(f)- f\|_\... 1answer 257 views ### Almost complex manifold of dimension 2… locally isomorphic to ?? I know that this is supposed to be standard, but I don't know how to search for it... hence the question: Let$J$be an almost complex structure on$M:=\mathbb R^2$, i.e., a$C^\infty$section of$\...
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### Criteria for a limit to be a proper function

This question is obviously broad; turning this broadness into something sharp is part of the problem. Given a sequence of functions defined on a Riemann Surface $R$, valued in $\Bbb C^2$, what ...
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### Positivity and zeros of Heun's function

I am interested in understanding where in the complex plane a Heun function might vanish, or where it (or its real part) is positive. Consider the case where the Frobenius indices at $0$ are \$(0,1- \...

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