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)$.
2
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0answers
45 views
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
21 views
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?
11
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1answer
158 views
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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0answers
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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0answers
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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1answer
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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0answers
34 views
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
43 views
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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0answers
37 views
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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1answer
29 views
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}$$
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1answer
44 views
Questions About the Proof of Cauchy–Pompeiu Integral Formula.
I am studying function theory in several complex variables and the book I am using is "Tasty Bits of Several Complex Variables" by Jiří Lebl: https://www.jirka.org/scv/scv.pdf.
At the moment I am ...
2
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1answer
76 views
Show that $f(x,y,z)=x^2-y^2z$ is irreducible in $\mathbb{C}[x,y,z]$.
Let $p\in\mathbb{C}[x,y,z]$ be defined by $p(x,y,z)=x^2-y^2z$.
Goal: Prove that $p$ is irreducible.
Let $I\subset\mathbb{C}[x,y,z]$ be the ideal defined by
$$I:=(p).$$
My approach is to show that
...
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0answers
17 views
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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0answers
26 views
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
59 views
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
27 views
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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50 views
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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0answers
17 views
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
28 views
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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0answers
12 views
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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0answers
26 views
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
41 views
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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39 views
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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1answer
40 views
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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0answers
41 views
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
28 views
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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1answer
42 views
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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3answers
77 views
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 ...
2
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1answer
85 views
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
50 views
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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1answer
61 views
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
77 views
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
25 views
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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0answers
47 views
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
76 views
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 $z$.
...
3
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1answer
58 views
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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1answer
65 views
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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23 views
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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0answers
36 views
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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2answers
116 views
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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0answers
45 views
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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0answers
55 views
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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0answers
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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0answers
69 views
$M$ 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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0answers
35 views
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\...
