# 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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### $L^1$ norm of Littlewood polynomials on the unit circle

A Littlewood polynomial is a polynomial with coefficients from $\{ 1, -1\}$ and the set of Littlewood polynomials with degree $n$ is denoted by $\cal{L}_n$. I'm interested in a "good" lower bound on ...
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### Reference request: normal form of k-differentials and flat surfaces at a puncture

Let $f(z)$ be a holomorphic function defined on a punctured neighborhood of $z=0$ with non-essential singularity of degree $d$ at $0$ (namely, $f(z)=z^dh(z)$, where $h(z)$ is a holomorphic function ...
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### Finitely connected planar domains and certain smooth maps with integrable Wirtinger derivative

Let $D$ be a finitely connected, bounded domain in $\mathbb C$. Suppose that the boundary of $D$ consists of finitely many, pairwise disjoint, piecewise $C^1$-Jordan curves. Does there exist an ...
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### Nekrasov Partition Function: $F^{inst}(\epsilon_1,\epsilon_2,\mathbf{a},\mathbf{q})$ analytic at $\epsilon_1 = \epsilon_2 = 0$?

Nakajima & Yoshioka [1] showed that F^{inst}(\epsilon_1,\epsilon_2,\mathbf{a},\mathbf{q}) = \sum_{n = 1}^\infty \mathbf{q}^nF^{inst}_n(\epsilon_1,\epsilon_2,\mathbf{a}) := \...
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### Padé multipoint approximants of the exponential function

One says that a pair of polynomials $(P_m,Q_n)$ over $\mathbb C[z]$, with $\text{deg }P_m=m$, $\text{deg }Q_n=n$, is a "multipoint Padé approximant of the exponential function" if $P_m(z)e^z-Q_n(z)$ ...
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### asymptotic behavior of a singular integral

What is the asymptotic behavior of the integral $$\int\int_{\mathbb C}\frac{f(z, \bar z)}{(z-a)(\bar z- \bar a_0)}|dz|^2$$ as $a\to a_0$? The function $f$ is from $C^\infty_0({\mathbb C})$ with ...
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### An inner product on the vector space $\mathbb{R}[x_1,\cdots,x_n]_m$

For any given integers $m,n\geq1$, let $\mathbb{R}[x_1,\cdots,x_n]_m$ be the vector space of homogeneous polynomials of degree $m$ in $x_1,\cdots,x_n$ over the field of real numbers $\mathbb{R}$. ...
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### An equality of inner products of holomorphic curves

The following is the main result in the paper by Vinnikov, Putinar, Alpay: A Hilbert space approach to bounded analytic extension in the ball, 2003, Communications on Pure and Applied Analysis. The ...
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### Hunting for holomorphic functions part 2: this time with non-local boundary condition

This is a follow-up to Looking for holomorphic function on a sector with specified boundary behavior. There I was looking for a holomorphic function on a sector with real boundary condition on one ...
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### Super global dimension

Let $R$ be a ring of finite global dimension. Define the small super global dimension as $sgl(R):= \sup \{ pd(X)+id(X) | X \in mod-R$ and indecomposable $\}$. Here $id(X)$ stands for the injective ...
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### Bound of the measure of the support of a set of divisors in a fixed linear system

Let $(X,L)$ be a compact polarized complex manifold of dimension $n$. Let $\varphi$ be a smooth positive metric on $L$. Define $\omega=dd^c\varphi$. We shall use $MA(\varphi)=\omega^n$ as the measure ...
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### Is there Cauchy-Goursat for $1$-cycles without invoking winding numbers?

Depending on one's approach to Complex Analysis in One Variable, Cauchy's Integral Theorem is one of the first interesting results about holomorphic functions in any course. There are several related ...
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### Trees and Shabat polynomials

Recently, I read the relation between Shabat polynomials and trees. The book [0] says that if $p: \mathbb{C} \to \mathbb{C}$ is a degree-$n$ polynomial such that the segment $[-1,1]$ contains no ...
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### Asymptotics of an integral by two methods

This was asked in MSE, here, but the answer was not satisfactory. I want to compute the asymptotic behavior of the integral $$f(K,a)=\int_0^1 (1-x)^Ke^{iKa\frac{x}{1-x}}x^2dx$$ when $K$ is large ...
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### Differentials on tori realised as double of annuli

In this question it was described how to realise a torus as the double of an annulus Explicit construction of mirror surface and complex double for an annulus. In short, the torus is realised ...
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### $\det(I-K(z)+\varepsilon(z,x))$ versus $\det(I-K(z))$

First let me ask the general question that might interest others dealing with determinantal formulas. We are trying to compare the following two quantities C_{\varepsilon} := \oint \det(I-K(z)+\...
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### Euler characteristic of an exhaustion of compacts of a surface

Let $X$ be an open (connected) Riemann surface of finite Euler characteristic. And $K_1 \subset \cdots K_n \subset$ be an sequence of closures of bounded open subsets with smooth boundary of $X.$ ...
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What is the state of the Krzyż Conjecture? It states for that for all $f:\mathbb{D}\to \bar{\mathbb{D}}$ holomorphic and non-vanishing, the coefficients $a_n$ in the power series of $f$ are at most $2/... 0answers 117 views ### Path components of complex analytic spaces [closed] Let$(X,\mathcal{O}_X)$be a complex analytic space. Are the path components of$X$the same as the connected components? 1answer 137 views ### An estimate on deviation of two smooth tangent$J$-holomorphic curves Take$\mathbb C^2$with coordinates$(z,w)$. Suppose that$J$is a$C^{\infty}$almost complex structure on$\mathbb C^2$such that the line$w=0$is$J$-holomorphic and$J(0,0)$is given by$(z,w)\to ...
Trying to find the pdf of the real part x of the product $z_1z_2$ of two uncorrelated complex random Gaussian variates . The pdf of the modulus $r \equiv |z_1z_2|$ is known $f_r(r)=rK_0(r)$ from ...
It is well-known that any convergence in $L^p$ for $p \in [1,\infty]$ implies convergence in $L^1_{\text{loc}}$ by Hölder's inequality. It is also known that for $p>1$ it holds that \$L^p(\mathbb R)...