# 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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### 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 ...
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 $... 1answer 58 views ###$(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)$... 2answers 160 views ### Essential singularity of a holomorphic function of two variables I have a holomorphic function$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}$. I'm trying to ... 1answer 63 views ### About integration by parts of$\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$satisfies property of ... 0answers 33 views ### Calculating the Reeb vector field Given a smooth function$f: \mathbb{C}^2 \to \mathbb{R}$and a regular value$p$, then$f^{-1}(p)$is a 3-dimensional smooth submanifold, and we may see it as a CR manifold by defining the CR ... 1answer 51 views ### An equivalent characterization of the first chern class Let$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})$is the first Chern class of a ... 1answer 51 views ### Is the union of 2 complex analytic sets still a complex analytic set? Is the union of 2 complex analytic sets still a complex analytic set? The notion of the analytic set is the usual one in complex analysis and comeplex geometry. 0answers 38 views ### About the definition of logarithmic poles in the several complex variables In Demailly's book Analytic methods in algebraic geometry , he wrote : Let ϕ be a quasi-psh function. We say that$\phi$has logarithmic poles if ϕ is locally bounded outside an analytic set A and ... 0answers 16 views ### What's the difference between the formal adjoint and the Hilbert adjoint of$\bar\partial$operator? What's the difference between the formal adjoint and the Hilbert adjoint of$\bar\partial$operator? What's the difference between their definitions? Here we use the standard terminologies in the ... 1answer 53 views ### How are the$\sigma_k(x)$functions in the Weierstrass Polynomial derived from the Residue Theorem? On p. 70 in Griffith's "Introduction to Algebraic Curves", the author states in a proof of a Lemma regarding the construction of a polynomial as a Weierstrass Polynomial, the Newton Symmetric ... 1answer 73 views ### Is this simple condition enough to show$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$\begin{align} z_2f_3(z)-z_3f_2(z)\\ z_3f_1(z)-z_1f_3(z)\\ z_1f_2(z)-z_2f_1(z) ... 0answers 19 views ### Analytic continuation of a multivariable function composition Letf\colon \mathbb{R}^n\to\mathbb{R}$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 ... 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 ... 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$&...
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 $U$ ...