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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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{...
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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,......
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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 ...
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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 $...
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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 $...
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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 ...
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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]?
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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(w´(a)) = \alpha $, $(| a | <R_1, | b | <R_2)$...
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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} \...
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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/...
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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 ...
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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 ...
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170 views

Find maximum values of $p(z_1,z_2,z_3)=z_{1}^2 +z_{2}^{2} +z_{3}^2 −2z_{1}z_{2} −2z_{1}z_{3} −2z_{2}z_{3}$

Let us consider the polynomial $p$ over a complex filed defiend by $$p(z_1,z_2,z_3):= z_{1}^2 +z_{2}^{2} +z_{3}^2 −2z_{1}z_{2} −2z_{1}z_{3} −2z_{2}z_{3}~~~ \forall z_{1}, z_{2}, z_{3}\in \mathbb{C}.$$...
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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. ...
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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 ...
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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 &...
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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\...
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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 ...
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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 ...
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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? ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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?
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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 ...
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44 views

Understanding Hörmanders Theorem 1.3.3 and properties of hull

I am reading by myself Hörmander'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\...
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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}\...
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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 ...
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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
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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 ...
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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 $...
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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 $C_{(p, q)}^\infty(D)$ ...
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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 ...
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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 ...
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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 ...
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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 ...
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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.
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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 ϕ be a quasi-psh function. We say that $\phi$ has logarithmic poles if ϕ is locally bounded outside an analytic set A and ...
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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 ...
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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 ...
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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) ...
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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 ...
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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 ...
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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 $&...
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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 $U$ ...

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