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).
562
questions
1
vote
0answers
14 views
Hartogs figure not holomorphically convex
Given $0 < a, b < 1$, consider the Hartogs figure $H$ given by
\begin{equation*}
H = \{ (z,w) \in \mathbb{D}\times \mathbb{D} \ \ | \ \ |z| > a \} \cup \{ (z,w) \in \mathbb{D} \times \mathbb{...
0
votes
0answers
42 views
Proving that holomorphic map $f : \mathbb{C}^n \to \mathbb{C}^m$ maps holomorphic tangent space at point $p$ to holomorphic tangent space at $f(p)$
I'm trying to to prove proposition 2.3.1 from Jiri Lebl's text on several complex variables here.
So far I have tried writing the jacobian in terms of $u_k$ and $v_k$ where $f_k = u_k + iv_k, k = 1,......
7
votes
1answer
124 views
What's nonsensical about this definition of the order of a meromorphic function?
According to this math.stackexchange.com answer, the following definition of Huybrechts in his book Complex Geometry is nonsensical:
Let $X$ be a complex manifold. Let $Y \subset X$ be a hypersurface ...
2
votes
0answers
24 views
A holomorphic sequence $u_{j}$ converges weakly to zero implies locally convergence to zero?
I have the following question:
Let $\Omega \subset \mathbb{C}$ be a bounded domain and let $\{u_{j}\}$ be a sequence of holomorphic functions in $\Omega$ such that $u_{j}\in L^{2}(\Omega)$, for each $...
0
votes
1answer
35 views
Irreducible analytic varieties with the same germ at some point
Let $V_1,V_2$ be irreducible analytic subvarieties of a complex manifold $M$. Suppose there exists $p\in V_1\cap V_2$ and a neighborhood of $p$ in $M$ such that $V_1\cap U=V_2\cap U$. Does it imply $...
0
votes
1answer
24 views
Dimension of irreducible analytic varieties
The dimension of an irreducible analytic variety is defined to be the dimension of its regular locus as a complex manifold. Suppose we have an irreducible analytic subvariety $V$ of $\mathbb{C}^n$ of ...
0
votes
2answers
42 views
What are the factors of (2+i) in Z[i]?
The complex number $2+i$ factors as $i\cdot (1-2i)$ and $(-i)\cdot (1-2i)$. But those factorizations seem trivial. Are there any other ways to factor 2+i within the Z[i]?
0
votes
1answer
20 views
Conformal mapping of the disk $| z | <R_1$ to disk $| w | <R_2$
could you help me with the following please:
Find the conformal mapping of the disk $| z | <R_1$ to disk $| w | <R_2$ such that $w (a) = b, Arg(wĀ“(a)) = \alpha $, $(| a | <R_1, | b | <R_2)$...
2
votes
0answers
17 views
Holomorphic dependence on initial data for complex ODEs with real time
Let $U \subset \mathbb{C}^n$ an open domain and $F \colon U \to \mathbb{C}^n$ a holomorphic map. Consider the autonomous differential equation
$$
\frac{dz}{dt} = F(z), \qquad (t,z) \in \mathbb{C} \...
0
votes
0answers
16 views
Order of a point in the amoeba complement is integer valued
I'm currently trying to understand the paper "Laurent Determinants and Arrangements of
Hyperplane Amoebas" by Forsberg, Passare and Tsikh, which can be found here https://core.ac.uk/download/...
1
vote
0answers
35 views
A complex analytic interpretation of multiplicity of projective hypersurface
Let $\{F_t\}$, $t\in \Delta$ be an algebraic family of projective hypersurfaces, where $\Delta=\{z\in\mathbb C||z|<1\}$ is the unit disk. Assume $F_0$ is reducible and $Z$ is one of the components ...
1
vote
0answers
33 views
Ramification index for a (fiberwise) generically finite morphism $M\to N$ over a disk
Assume $M,N$ are smooth varieties over $\mathbb C$. Let $f,g$ be projective and flat morphisms over $\Delta:=\{z\in\mathbb C||z|<1\}$ and $h:M\to N$ is a dominant map and is a generically finite ...
4
votes
2answers
170 views
Find maximum values of $p(z_1,z_2,z_3)=z_{1}^2 +z_{2}^{2} +z_{3}^2 ā2z_{1}z_{2} ā2z_{1}z_{3} ā2z_{2}z_{3}$
Let us consider the polynomial $p$ over a complex filed defiend by $$p(z_1,z_2,z_3):= z_{1}^2 +z_{2}^{2} +z_{3}^2 ā2z_{1}z_{2} ā2z_{1}z_{3} ā2z_{2}z_{3}~~~ \forall z_{1}, z_{2}, z_{3}\in \mathbb{C}.$$...
1
vote
0answers
35 views
Can positivity of currents implies positivity of forms?
Let $\alpha$ and $\beta$ be 2 continuous (or smooth) forms of $(1,1)$-type on a complex manifold $X$. Of course they can be considered as currents. Assume $\alpha\geq \beta$ in the sense of currents. ...
2
votes
2answers
69 views
prerequisites needed to Several complex variables
What are the necessary knowledge in order to learn Several complex variables? In the beginning I thought that complex analysis and multivariable calculus were the only thing, but then I realized that ...
0
votes
0answers
36 views
Fundamental matrix of ODE system with singularity
For the germ at $0$ of the following equation with irregular singularity, find explicitly a fundamental matrix of solutions
$$\dot z = \frac{A(t)}{t^2}z,\quad A(t)=
\begin{bmatrix}
1 & 0\\
0 &...
1
vote
0answers
12 views
Complexification of real linear map and Hermitian metric
Consider
$V$ a finite dimensional real vector space with Euclidean metric $\langle|\rangle$
$\langle|\rangle_{\mathbb{C}}$ the induced Hermitian metric on the complexification $V_{\mathbb{C}}=V\...
0
votes
1answer
51 views
The proof of that the Riemman sphere is not a complex submanifold of $\mathbb{C}^n$
As is well known , according to the maximal principle we can easily conclude that every compact connected complex manifold of $\mathbb{C}^n$ degenerate to a point . From the above point of view, we ...
1
vote
0answers
25 views
Vector field tangent to a singular holomorphic foliation
Is there an intuitive way to see that the vector field
$$\begin{equation}
v=\begin{cases}
\dot z=e^{\frac{1}{z}}\\
\dot w=e^{\frac{1}{w}}
\end{cases}
\end{equation}$$
is tangent to a singular ...
0
votes
2answers
35 views
Continuity of partial derivatives of harmonic function
If f is a real valued harmonic function in $C$ we know that $f_{xx} + f_{yy} = 0$. So, $f_{xx}$ and $f_{yy}$ exists, from this it is not correct to conclude that $f_x$ and $f_y$ are continuous right? ...
0
votes
1answer
32 views
Maximum principle in higher dimensions
Let $f$ be a function holomorphic on some connected open subset $D$ of the complex plane $\mathbb{C}$ and taking complex values. The maximum principle states that if $f$ is holomorphic within a ...
3
votes
1answer
47 views
zero extension of positive currents are always positive
In Demailly's Complex Analytic and Differential Geometry page 139:
He said the trivial (zero) extension of the positive current $T$ (on $X\setminus E$), which denoted by $\tilde T$ is always positive ...
1
vote
1answer
47 views
Soft Question: Texts for introduction into Complex/Analytic Geometry i.e. GAGA theorems, Several Complex variables.
I am currently an undergraduate mathematics student who is interested in studying more about complex/analytic geometry. However, most undergraduate modules (at least in my home university) do not ...
0
votes
0answers
11 views
Question about holomorphic image and distance
Say $X$ is a punctured polydisk of dimension $2$. More precisely, it is the set of pairs $(x,y)\in\mathbb{C}$ such that $0<|x|,|y|<1$. Say $f:X\to\mathbb{C}^2$ be a bounded holomorphic map, and ...
2
votes
2answers
59 views
Hartog's Theorem and Entire Functions
I'm interested in multivariable Complex Analysis, and I have two questions:
My first question is as follows: after reading about Hartog's Extension Theorem I started wondering about the following ...
0
votes
3answers
38 views
Is complex $sin ^{-1} z$ multivalued for real numbers bigger than one? [closed]
I am trying to understand the complex sin function. Let $a>1$ be a real number. How many solutions does
$\sin z= a$
has inside a circle of radius $R$ centered at the origin? Could it have ...
1
vote
0answers
16 views
Are the two holomorphic maps identical in the sense of unitary transformation?
If $D\in \mathbb{C}^m$, $f,g:D\rightarrow\mathbb{C}^n$ are holomorphic maps such that \begin{equation*} \langle f(z),f(z)\rangle=\langle g(z),g(z)\rangle ,\forall z\in D\end{equation*}then there ...
9
votes
2answers
197 views
Generators for the ideal of entire functions vanishing on $\mathbb Z\times\mathbb Z\subset \mathbb C\times\mathbb C$
For $n=1,2$, let $R_n$ be the ring of entire complex functions in $n$ complex variables. Let $I$ be the ideal $R_2$ of functions $f$ such that $f(\mathbb Z\times\mathbb Z)=\{0\}$. Is $I$ generated by ...
0
votes
0answers
39 views
Properties of conjugate function of an analytic function
I know that if $f(z)$ is an analytic function then $\ln|f(z)|$ is a harmonic function. I have tried to prove it but I have not found. Who can show me how to do?
2
votes
1answer
62 views
Showing that $\tau(t) = (t^2, t^3)$ is not a submanifold
Let $\tau : \mathbb{C} \to \mathbb{C}^2$ be the map $\tau(t) := (t^2, t^3)$. Show that $\tau$ defines an embedding map from $\mathbb{C}^*$ to $\mathbb{C}^2 \setminus{0}$. Is $\tau(\mathbb{C})$ a ...
0
votes
0answers
44 views
Understanding Hörmanders Theorem 1.3.3 and properties of hull
I am reading by myself Hƶrmander's book An Introduction to Complex Analysis and I have some doubts on the paragraphs after Runge's Theorem:
Let $\Omega$ an open subset of $\mathbb{C}$ and $K\subset\...
0
votes
0answers
28 views
Continuity and bijectivity of stereographic projection $\mathbb{C}\mathbb{P}^1\longrightarrow\mathcal{S}^2$
I am trying to prove that the stereographic projection from the sphere $\mathcal{S}^2$ to the complex projective line $\mathbb{C}\mathbb{P}^1$ is a homeomorphism. I have the function $f:\mathbb{C}\...
2
votes
1answer
36 views
Prove that $\mathbb{C}\mathbb{P}^n$ is connected
I need to show that the complex projective space $\mathbb{C}\mathbb{P}^n$ is connected. I know that $\mathbb{C}\mathbb{P}^1$ is connected, because there exists a homeomorphism (stereographic ...
1
vote
0answers
23 views
If $\iota(x,y,u,v):=(x+iy,u+iv)$, then $\iota(x,y,(x,y)A^T)$ is complex subspace of $\mathbb{C}^2$
Let $\iota\colon \mathbb{R}^4\longrightarrow\mathbb{C}^2$ be the function defined by $\iota(x,y,u,v):=(x+iy,u+iv)$. Let $A\in \mathrm{GL}(2,\mathbb R)$ be an invertible $2\times 2$-matrix with real ...
1
vote
1answer
38 views
Holomorphic function from $\mathbb{C}\mathbb{P}^n$ to $\mathbb{C}$ is constant
Let $f:\mathbb{C}\mathbb{P}^n\longrightarrow\mathbb{C}$ be some complex-differentiable function, where $\mathbb{C}\mathbb{P}^n$ is the complex projective space. I want to show that $f$ is constant. My ...
0
votes
0answers
22 views
An irreducible analytic sub-variety of a complex manifold is connected.
I am trying to see if the statement is correct. I think it should be.
An irreducible analytic sub-variety $V$ of a complex manifold $M$ is connected. The locus of smooth points of $V$ is denoted as $...
1
vote
1answer
58 views
$(p,q)$-forms and notation
I have a couple of questions about a notation I'm not very familiar with.
Let $D$ a bounded domain in $\mathbb{C}^n$, $n \geq 2$, not necessarily with a smooth boundary. Let $C_{(p, q)}^\infty(D)$ ...
3
votes
2answers
160 views
Essential singularity of a holomorphic function of two variables
I have a holomorphic function $G(z_1,z_2)$ in 2 variables, such that $G(z_1 + 1, z_2) = G(z_1,z_2)$, hence for a fixed $z_2$, $G(z_1,z_2)$ has a Laurent expansion in $e^{2\pi i z_1}$. I'm trying to ...
0
votes
1answer
63 views
About integration by parts of $\partial$ operator
As well known to us, operator $d$ satisfies property of integration by parts in the mathematical analysis.
My question is :
(1) Does the operator $\partial$ or $\bar\partial$ satisfies property of ...
0
votes
0answers
33 views
Calculating the Reeb vector field
Given a smooth function $f: \mathbb{C}^2 \to \mathbb{R}$ and a regular value $p$, then $f^{-1}(p)$ is a 3-dimensional smooth submanifold, and we may see it as a CR manifold by defining the CR ...
0
votes
1answer
51 views
An equivalent characterization of the first chern class
Let $X$ be a projective algebraic manifold. 92' Singular hermitian metrics on positive line bundles demailly wrote:
An integral cohomology class in $H^2(X,\mathbb{Z})$ is the first Chern class of a ...
0
votes
1answer
51 views
Is the union of 2 complex analytic sets still a complex analytic set?
Is the union of 2 complex analytic sets still a complex analytic set? The notion of the analytic set is the usual one in complex analysis and comeplex geometry.
0
votes
0answers
38 views
About the definition of logarithmic poles in the several complex variables
In Demailly's book Analytic methods in algebraic geometry , he wrote :
Let Ļ be a quasi-psh function. We say that $\phi$ has logarithmic poles if Ļ is locally
bounded outside an analytic set A and ...
0
votes
0answers
16 views
What's the difference between the formal adjoint and the Hilbert adjoint of $\bar\partial$ operator?
What's the difference between the formal adjoint and the Hilbert adjoint of $\bar\partial$ operator? What's the difference between their definitions? Here we use the standard terminologies in the ...
0
votes
1answer
53 views
How are the $\sigma_k(x)$ functions in the Weierstrass Polynomial derived from the Residue Theorem?
On p. 70 in Griffith's "Introduction to Algebraic Curves", the author states in a proof of a Lemma regarding the construction of a polynomial as a Weierstrass Polynomial, the Newton Symmetric ...
1
vote
1answer
73 views
Is this simple condition enough to show $f:\mathbb{C}^3\to\mathbb{C}^3$ is holomorphic?
Let
$$
f:\mathbb{C}^3\to\mathbb{C}^3,\quad f(z)=(f_1(z),f_2(z),f_3(z))
$$
be a smooth map such that the three functions
$$
\begin{align}
z_2f_3(z)-z_3f_2(z)\\
z_3f_1(z)-z_1f_3(z)\\
z_1f_2(z)-z_2f_1(z)
...
0
votes
0answers
19 views
Analytic continuation of a multivariable function composition
Let $f\colon \mathbb{R}^n\to\mathbb{R}$ and, for all $i=1,\dots,n$, let $g_i\colon\mathbb{R}\to\mathbb{R}$. Then $f(g_1(x),\dots,g_n(x))$ maps $\mathbb{R}\to\mathbb{R}$.
Suppose that each $g_i$ has ...
7
votes
1answer
151 views
Why is the sum of two algebraic functions algebraic?
Let $U\subset\mathbb{C}^n$ be a domain. A holomorphic function $f:U\to \mathbb{C}$ is called $\textbf{algebraic}$ if there exists a polynomial $p(x,y)$ in the variables of $U\times \mathbb{C}$ such ...
1
vote
1answer
74 views
Is the basin of attraction of a $p$ starshaped wrt to $p$?
Let $p\in\Bbb C^n$ (with $n\ge2$) be an attractive fixed point for $F\in\operatorname{Aut}\Bbb C^n$ (i.e. holomorphic bijection), that is $F(p)=p$ and all the eigenvalues of $F'(p)$ are in modulus $&...
1
vote
0answers
18 views
Is a bounded symmetric domain a Runge domain?
A domain in $\mathbb{C}^N$ is a bounded symmetric domain if there exists a norm $\|\cdot\|$ on $\mathbb{C}^N$ such that $U=\{z\in \mathbb{C}^N : \|z\|<1\}$ and if the automorphism group of $U$ ...
