Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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).

2
votes
1answer
63 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 ...
-1
votes
1answer
42 views

This complex function has only real values, how can i find? [closed]

$$(1+i\tan\alpha)^{1+i\tan\beta}$$ is a complex function which possess only real values.And one of the real values is $(\sec\alpha)^{\sec\beta}$. How to start I am confused. With taking logarithm I ...
0
votes
0answers
16 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}\...
0
votes
0answers
25 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 ...
2
votes
1answer
54 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 ...
1
vote
1answer
23 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 ...
0
votes
0answers
43 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 ...
1
vote
0answers
12 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 ?
0
votes
0answers
20 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-...
0
votes
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 $(...
1
vote
0answers
35 views

The concavity of $\log\det\left(u_{j\overline{k}}\right)$

Let $F:\mathbb{C}^{n\times n}\rightarrow\mathbb{C}$ be the function $$ F\left(a_{1\overline{1}},a_{1\overline{2}},\ldots,a_{n\overline{n}}\right):=\log\det\left(\begin{array}{ccc} a_{1\overline{1}} &...
1
vote
0answers
20 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 \...
0
votes
0answers
20 views

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 ...
1
vote
1answer
38 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 $$\...
0
votes
0answers
24 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 ...
0
votes
1answer
39 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 $\varphi|_{U\backslash Z}$ ...
0
votes
0answers
35 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 `...
1
vote
1answer
26 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 ...
1
vote
1answer
41 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, ...
4
votes
3answers
68 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
votes
1answer
77 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 ...
0
votes
1answer
45 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 $\bar{p}\in D$, ...
0
votes
1answer
59 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: $$f(z) = \int^1_{-1} \frac{2z}{(1+ z^2) + t(1 - z^2)} \, d\mu(t),$$...
1
vote
1answer
48 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 ...
0
votes
1answer
24 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 $\mathbb C\setminus\{0\}\subset \mathbb C$. ...
2
votes
0answers
39 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 \...
2
votes
1answer
44 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 $f$ ...
0
votes
1answer
53 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 $N$ ...
2
votes
1answer
61 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 ...
0
votes
0answers
19 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 \...
0
votes
0answers
34 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}:\...
1
vote
2answers
90 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 ...
0
votes
0answers
39 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 ...
0
votes
0answers
53 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 $D_{\lambda}\subset L^2(\Bbb{R}^2)$. ...
1
vote
0answers
95 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\...
4
votes
0answers
65 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 ...
0
votes
0answers
34 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\...
0
votes
0answers
16 views

Ratio of functions of infinite vanishing order with finite-vanishing arguments

I'm attempting to understand a paper by Steven Krantz (proposition 1.3). If I assume the $\mathbb{C}^2$ case for simplicity's sake, on page five he seems to use the following fact: Let $\psi:\...
3
votes
0answers
38 views

How to get Riemann form on complex tori

Let $X$ be a closed compact Riemann surface of genus $g$. Then I can get $\operatorname{Jac}(X)$, Jacobian of the $X$ by Abel-Jacobi mapping. $\operatorname{Jac}(X)=C^g/\Lambda$ admits non-trivial $\...
2
votes
1answer
33 views

Log of multi-variable non-zero holomorphic function

Let $f$ be a non-vanishing holomorphic function in $z_1,\dots, z_n.$ Suppose $\operatorname{dom}(f)$ is open and connected. Then $\log(f)$ is a holomorphic function. $\textbf{Q:}$ I can see that $\...
1
vote
0answers
18 views

Reference request: local normal form of a holomorphic map

Let $U\subset\mathbb{C}^n$ and $V\subset\mathbb{C}^m$ be open subsets, and let $f : U\rightarrow V$ be a holomorphic map such that $df : TU\rightarrow TV$ has constant (complex) rank $k$. I'd like a ...
0
votes
1answer
25 views

Interprete $f(L_2)$ and $f(L_3)$.

Let $f(z)=z^2$ for $z \in \mathbb{C} $, $L_2$ = {$(0,y)$:y $\geqq0$}. Interprete $f(L_2)$ Let $f(z)=iz$ for $z \in \mathbb{C}$, $L_3$ = {$(x,y)$:$y=x+1$. Interprete $f(L_3)$. I don't know how to ...
1
vote
1answer
37 views

Given $u,v$ holomorphic functions not vanishing at $0\in C^n$, then we can suppose $u,v$ both non-vanishing along $z_2=\dots=z_n=0$ axis.

All questions below are over $C$,complex number or over $C^n$. $\textbf{Q1:}$ Given a $u$ holomorphic function not vanishing identically at $0\in C^n$, then one can assume $u$ is non-vanishing along $...
1
vote
1answer
59 views

Set of zeros of polynomial in several complex variable.

Question: the question is already asked here(link→If $p(z,w)=a_0(z)+a_1(z)w+\dots +a_k(z)w^k$ are non constant polynomial.) $p(z,w)=a_0(z)+a_1(z)w+\dots +a_k(z)w^k$ where $a_i(z)$ are non constant ...
0
votes
1answer
63 views

Contour deformation in several variables

Suppose that I'm dealing with a function of several complex variables, holomorphic in each variable separately. Should I expect that the contour deformation will work in essentially the same way as ...
0
votes
0answers
15 views

Injectivity of a morphism on a sheaf of finite type

Let $(X,\mathcal{A})$ be a ringed space and $\mathcal{S}$ be a sheaf of $\mathcal{A}$-modules on $X$. $\mathcal{S}$ is said to be a sheaf of finite type at $x\in X$ if there is a neighborhood $U$ of $...
3
votes
0answers
94 views

Dual of the space of all bounded holomorphic functions

Let $\mathbb{B}$ be the open unit ball in $\mathbb{C}^n, n\geq 1$ and let $H^\infty (\mathbb{B})$ be the space of all bounded holomorphic functions on $\mathbb{B}$. It is well known that $H^\infty (\...
3
votes
1answer
29 views

Is the collection of reduced points in a complex space is open?

A point $x$ in a complex (analytic) space $(X,\mathcal{O}_{X})$ is said to be reduced if $\mathcal{O}_{X.x}$ is a reduced ring i.e. it doesn't have any non-zero nilpotent elements. Defined the set $S=\...
2
votes
1answer
63 views

Irreducible point in a complex (analytic) space

I am reading the concept of complex spaces from "Several Complex Variables VII: Sheaf-Theoretical Methods in Complex Analysis" By Grauert, Peternell and Remmert. They defined a point $x$ in a complex ...
2
votes
0answers
36 views

On Shilov Boundary of Bounded Holomorphic Functions

Let $H^\infty (\mathbb{B}_n)$ be the space of all bounded holomorphic functions from the open unit ball $\mathbb{B}_n$ of $\mathbb{C}^n$ $(n\geq 1)$ to $\mathbb{C}$. It is a Banach algebra. Let $\...