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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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Holomorphic Morse functions

For a holomorphic $f:\mathbb{C}^n\rightarrow \mathbb{C}$ and $a=(a_1, \dots, a_n) \in \mathbb{C}^n$, let $f_a:\mathbb{C}^n\rightarrow \mathbb{C}$ be the function $$(z_1, \dots, z_n) \mapsto a_1z_1 + ...
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How can i incorporate that z(t)=e^it. I understand the process but that part is throwing me off. Also how can I show that it lies above the real axis? [on hold]

Let $f(z) = P.V. z^i$ does not equal $0$. $\exp(i Log z)$, $z$ does not equal $0$. Compute the integral of $C$ for $f(z) dz$ when $C$ is the semicircle $z(t) = e^{it}$ $(0 \leq t \leq \pi)$.
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Is there a by-hand prove that $\Gamma(\mathbb{C}P^n,E)$ is finite dimensional for a holomorphic vector bundle $E$?

Let $E$ be a finite holomorphic vector bundle (or more generally a coherent analytic sheaf) on a compact complex manifold $X$. By the Cartan-Serre finiteness theorem, the cohomology $H^q(X,E)$ is a ...
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1answer
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Example of a complex meromorphic functions which is not locally Lipschitz [on hold]

Let $U\subset \mathbb{C}^n$ be an open subset and let $\varphi$ be a meromorphic function on $U$ i.e. locally given by the ratio of two holomorphic functions. Can $\varphi$ be not locally Lipschitz?
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Does every non-compact Riemann surface embed holomorphically into $\mathbb{C}^2$?

Question: Can every non-compact Riemann surface be holomorphically embedded into $\mathbb{C}^2$? If not, what are some (all?) of the obstructions to such an embedding? This question is partially ...
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108 views

How to visualize such regions in complex plane?

I was wondering whether the open sets in complex plane which are bounded by rectifiable simple closed curves are in fact simply connected? In particular, what is the significance of considering the ...
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1answer
35 views

Differential of logarithm on complex plane

I will be grateful if someone could explain the following statement from p. 26 of the book "Analytic functions of several complex variables" by Gunning and Rossi: As a function of $\zeta$ for $z$ ...
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58 views

Show the zero set of a holomorphic section can be written as the union of two disjoint closed set

Let $X$ be a complex manifold and $E$ be a holomorphic vector bundle over $X$. Denote $\tilde{E}$ as the pull-back bundle of $E$ over $X\times X$ via the map $\pi: X\times X\to X$ sending $(x,y)\to x$....
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66 views

Show the set is closed

Let $X$ be a Hausdorff locally compact topological space, $s:X\times X\to \mathbb{R}$ be a continuous map such that $s(x,x)=0$ for all $x\in X$. Set $\triangle=\{(x,y): x=y, x,y\in X\}$ and $A=\{(x,y):...
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Analytic proof that infinite level sets of complex polynomials aren't compact

I've heard it stated that if you have a family $p_{1}, \ldots, p_{\ell} \in \mathbb{C}[x_1 , \ldots , x_n]$ of $n$-variate complex polynomials, then the set $F = \bigcap_{j = 1}^{\ell} p_{j}^{-1} (\{0\...
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1answer
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series convergence lang page 26 exercise 7

Show that the series $\sum \frac{z^{n-1}}{(1-z^n)(1-z^{n+1})}$ where $z$ is a complex number. converges to $\frac{1}{(1-z)^2}$ for $|z|<1$ and to $\frac{1}{z(1-z)^2}$ for $|z|>1$... I try ...
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How to minimize $\sum_i |a_{i1}|x_1|^2 + a_{i2}x_1\overline{x_2} + a_{i3}\overline{x_1}x_2 + a_{i4}|x_2|^2 - b_i|^2$ over $x_1, x_2 \in \mathbb C$?

Consider the following nonlinear minimization problem \begin{align} \tag{P1} \min_{x_1, x_2 \in \mathbb C} \sum_{i=1}^m \big|a_{i1}|x_1|^2 + a_{i2}x_1\overline{x_2} + a_{i3}\overline{x_1}x_2 + a_{i4}|...
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roots of a several variable for the equation $y-3x^2-y^3=0$

How can I get the roots of the next equation? $$y-3x^2-y^3=0$$ I just dont get the same answer than my teacher: $$x = \frac {- \sqrt2}{3(3^{1/4})}, y = \frac{-2}{\sqrt3}$$
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1answer
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Questions About the Proof of Cauchy–Pompeiu Integral Formula.

I am studying function theory in several complex variables and the book I am using is "Tasty Bits of Several Complex Variables" by Jiří Lebl: https://www.jirka.org/scv/scv.pdf. At the moment I am ...
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Show that $f(x,y,z)=x^2-y^2z$ is irreducible in $\mathbb{C}[x,y,z]$.

Let $p\in\mathbb{C}[x,y,z]$ be defined by $p(x,y,z)=x^2-y^2z$. Goal: Prove that $p$ is irreducible. Let $I\subset\mathbb{C}[x,y,z]$ be the ideal defined by $$I:=(p).$$ My approach is to show that ...
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Simplify a symmetric complex function .

Lets consider the following symmetrical complex relation $f(z) = \frac{B(z)B(\frac{1}{z})}{B(z)B(\frac{1}{z}) + \alpha A(z)A(\frac{1}{z})}$ where $\alpha$ is a real value, $B(z)=\sum_{i=1}^{N}\...
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Factorising vector spaces

Suppose I have some complex (unit) vector $V$ on a vector space $\mathcal{H}$ of dimension $N$, where $N$ has prime factorisation $N=\prod^d_i p_i$ (multiplicity is allowed- the $p_i$ need not be ...
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1answer
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Calculate the integral of exp(z)/sin(z) over |z|=4 using the residue theorem.

Click to view the integral in correct format. Calculate the integral of exp(z)/sin(z) (as in the image above) over the positively oriented circle defined by |z|=4 using the residue theorem. This is ...
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1answer
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Upper envelope of plurisubharmonic functions

Suppose that $\{u_\alpha\}_{\alpha \in A}$ is a family of plurisubharmonic functions (psh function) on $\Omega \subset\subset \mathbb{C}^n$. Then, let $u(z) = \sup_{\alpha \in A} u_{\alpha}(z)$ be the ...
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Maximal Natural Domain of An Analytic Function on Complex Plane

I want to ask a question of the maximal natural domain of an analytic function. We know that on $\mathbb{C}^n$, a domain $U$ is a domain of holomorphy if and only if $U$ is a pseudoconvex domain, and ...
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Finding Levi Matrix [closed]

Who represent Levi Matrix $n\times n$ in several complex variables? I read many documents and books but I can't find a Levi Matrix explicitly. Can you help me ?
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1answer
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A real-analytic radial function on closed unit ball which peaks at zero is strictly decreasing

Let $\mathbb{B}$ be the closed unit ball in $\mathbb{C}^n$ and let $g:\mathbb{B}\rightarrow \mathbb{C}$ be a real-analytic radial function such that $g(0)=1$ and $|g(z)|<1\, \forall\, \text{non-...
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Some problem of complex multi-function

Assume that $f:U\rightarrow \mathbb{C}$ is a holomorphic function on a connected open subset of $\mathbb{C}^n$. Then can we prove that for every point $x\in U$. There would be a local coordinates $(...
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Extension of holomorphic function to exceptional divisors

Let $(X,0) \subset \mathbb{C}^n$ be complex analytic set with an isolated singularity. Let $f:(X,0) \to (\mathbb{C},0)$ be a germ of a reduced holomorphic function germ defined on it. Let $\pi:E \to \...
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On constancy of a complex multivariable polynomial

consider,a nonconstant polynomial $f(x,y,z,w)$ in $\mathbb C$ $[x,y,z,w]$ as a continuous map from $\mathbb C^4$ to $\mathbb C$. Now, if we have all four partial derivatives of $f$ vanishes at all ...
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1answer
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Several variables contour integral

Suppose I have an entire function of $n$ complex variables $f \colon \mathbb C^n \to \mathbb C$ and $f$ is rapidly decreasing (or enough conditions to make the following work). I want to show that $$\...
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Nonlinear optimization with complex residual and jacobian

I am trying to minimize the following function $\chi^2=(f(t_i;\vec{p})-y_i)^{H}\text{Cov}^{-1}_{ij}(f(t_j;\vec{p})-y_j)$ where $A^H$ is the hermitian conjugate of A, $f(t;\vec{p})-y$ is a complex ...
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Can we throw away points to make a Holomorphic injection into a homeomorphism?

Let $U\subset\mathbb{C}$ and let $\varphi:U\to\mathbb{C}^n$ be a holomorphic injection. Is it true that there is a discrete (or better yet finite) set $Z\subset U$ such that $\varphi|_{U\backslash Z}$ ...
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Spherical derivative in several variables

This is more of a reference request, really. The spherical derivative of a holomorphic function (in one variable) $f$ is defined by $ f^\# := \frac{|f^{'}|}{1 + |f|^2}. $ Is there a corresponding `...
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1answer
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real holomorphic field definition equivalence problem.

I'm reading the book of Andrei Moroianu, "Lectures on kahler geometry" and at the page 67 is this lemma: And when it comes time for the proof he sais that $(2)$ and $(3)$ are "tautological". I'm ...
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If $U$ is a connected open subset of $\mathbb{C}^n,n>2$ and $V$ is a closed subvariety of $U$, show that $U-V$ is connected

The question is taken from the notes by Joseph L.Taylor http://people.math.sfu.ca/~kya17/seminars/Taylor_Notes_On_Several_Complex_Variables.pdf problem 5.2. Since $D$ is already given to be closed, ...
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A function holomorphic on all of $\mathbb{C}^n$ cannot have a nonempty bounded set as it set of zeroes

I took this question from the book 'Several Complex Variables with connections to algebraic geometry and lie groups' chapter 2. The statement to be proven is that for $n>1$, a function that ...
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1answer
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How to show $f$ is identically equal to 0 in several complex variables

This is a homework question in a SCV course, and I'm struggling to approach the problem: Suppose $f:\mathbb{C}^2 \rightarrow \mathbb{C}$ is an entire holomorphic funtion whose zero set contains the ...
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1answer
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How to show a holomorphic function in several variables that vanishes at some points vanishes identically?

Let $D\subset \mathbb{C}^n$ be connected and suppose $f:D\times D\to\mathbb{C}$ is holomorphic in the $2n$ complex variables $(z,w)\in D\times D$. If there is a point $p\in D$ with $\bar{p}\in D$, ...
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Proving that a family of functions is compact

Suppose that I'm dealing with a family of complex functions analytic in the right-half plane and that each $f$ has a representation: $$f(z) = \int^1_{-1} \frac{2z}{(1+ z^2) + t(1 - z^2)} \, d\mu(t),$$...
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1answer
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How to properly deduce the Holomorphic Implicit Function Theorem from the Smooth Real Implicit Function Theorem?

I have seen at several places, incl. some notes and books, the following inference of the Holomorphic Implicit Function Theorem from the Smooth Real Function Theorem, but I believe this proof to be ...
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1answer
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Generic CR submanifolds

So the circle $S^1$ is a hypersurface of $\mathbb C$ and hence a generic CR submanifold. But $\mathbb C\setminus\{0\}$ is the complexification of $S^1$ and $\mathbb C\setminus\{0\}\subset \mathbb C$. ...
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Germ at any point determines analytic function

If I remember by Riemann surfaces course correctly, then the following should be true: Let $X$ be a Riemann surface, $U\subset X$ be open, and $x\in U$. Then the map $\Gamma(U,\mathcal{O}_X)\to \...
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1answer
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Analytic continuation of several complex variables

Let $f(w_1,\ldots,w_n;z)$ be a holomorphic function of $n+1$ variables. For every fixed $w_1\ldots w_n$, let $g(w_1,\ldots,w_n;z)$ be an analytic continuation of $f$ as a holomorphic function of $z$. ...
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Vector-Valued Holomorphic Functions of Constant Norm

Suppose throughout that $E$ is a complex normed vector space. Question: For which $E$ does it hold that if $D\subset\Bbb C$ is a domain, $f:D\to E$ is holomorphic and $||f(z)||$ is constant then $f$ ...
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1answer
56 views

How to complexify a harmonic function with an isolated singularity?

I would really appreciate if you could direct me to a reference for the following fact. Given a harmonic function $h$ defined in $R^N\backslash\{0\}$ we can find a holomorphic function $g$ of $N$ ...
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Why do holomorphic functions separate points?

I have just started looking into complex manifolds and in particular Stein manifolds. Stein manifolds are defined to be both holomorphically convex and holomorphically separable. It is claimed in ...
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Convergence of the multidimensional stationary phase asymptotic expansion

One section of the Wikipedia article on the method of steepest descent appears to be internally inconsistent. The beginning of the section assumes that $\Re(S(z))$ has a single maximum: $\max_{z \...
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function $\Gamma$ and analytic continuation

Fix $\mu\in \Bbb{C}$. Denote $\Omega=\{z\in \Bbb{C}, Re(z)> 0 \}$. Consider the function $\Big[\Bbb{R}^*_+ \setminus\cup_{m\in\Bbb{N}}\big\{\frac{\mu}{2m+1}\big\}\Big]\longrightarrow \Bbb{C}:\...
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on several complex variables on a ball

I want to solve the following problem: Let $B=B(0,R)$ be a ball in $\mathbb{C}^n, n\geq 2$. Let $f$ be holomorphic on $B$ and continuous on $\overline {B}$. if $f(a)=0$ for some $a\in B$, show that ...
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What is the Jacobian determinant of a complex transformation?

For two real $N\,x\,1$ vectors ${\underline {\bf{x}} _R}$ and ${\underline {\bf{y}} _R}$, and real full rank $N\,x\,N$ matrix ${\underline {\overline {\bf{A}} } _R}$, it is easy to show that, for the ...
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analytic continuation and spectrum

Let $\Omega$ denote the open set $\{z\in \Bbb{C}: \Re(z)>0\}$. Let $(L_\lambda)_{\lambda\in\Omega }$ be a family of closed unbounded operators, with domain $D_{\lambda}\subset L^2(\Bbb{R}^2)$. ...
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104 views

On Several Complex variables

On R. Michael range, Holomorphic Functions and Integral Representations in several Complex Variables there is a problem saying that: Let $D\subset \mathbb{C}^n$ be connected and suppose $f:D\times D\...
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$M$ is a complex manifold $\Leftrightarrow \overline{\partial}^2=0$

One way of stating Newlander-Niremberg's theorem is by saying an almost complex manifold $(M,J)$ is complex $\Leftrightarrow \overline{\partial}^2=0$. I'm confused by this, because I can't see why ...
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Domain of analyticity

Denote $\Omega=\{z\in\Bbb{C}: \Re(z)>0 \}$; $Y_m=\{\big(\lambda, (2m+1)\lambda\big);\lambda\in\Omega\}$ and $Y=\cup_{m\in\Bbb{N}} Y_m$. Define the function $(\Omega\times\Bbb{C}\setminus Y)\to\...