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).
1
vote
2answers
73 views
Prove that: $\sup_{z \in \overline{D}} |f(z)|=\sup_{z \in \Gamma} |f(z)|$
Suppose $D=\Delta^n(a,r)=\Delta(a_1,r_1)\times \ldots \times \Delta(a_n,r_n) \subset \mathbb{C}^n$
and
$\Gamma =\partial \circ \Delta^n(a,r)=\left \{ z=(z_1, \ldots , z_n)\in ...
5
votes
0answers
37 views
Branch locus along a smooth curve
Let $f : S \to X$ be a dominant morphism of smooth complex surfaces. Let $C \subset S$ be a smooth curve such that $df$ is of rank $1$ along $C$ and that in a neighborhood of $C$, $df_x$ is an ...
3
votes
0answers
135 views
What's the difference between analytic singularity and algebraic singularity?
Let $f$ be a holomorphic function defined in $U\in \mathbb{C}^n$, and $f=0$ gives an analytic variety. Suppose $f$ is irreducible as a germ of holomorphic function in $\mathcal{A}_z$, here $z$ is a ...
2
votes
0answers
55 views
Holomorphic function vanishing in a real hyperplane
A real hyperplane of $\mathbb{C}^{n}$ is given by a equation $\textrm{Re}( \ell(Z)) = c$, where $c \in \mathbb{R}$ and $\ell(Z) = \sum_{j=1}^{n} a_{j}z_{j}, a_{j} \in \mathbb{C}$. How to prove that if ...
2
votes
0answers
65 views
Eigenvalue of a form
I came across the following matrix while reading an article..Can you please help me to understand the following.
We are defining following form:
...
2
votes
0answers
83 views
Uniform Convergence of Bergman Kernel
In my complex analysis class we've shown that if $\Omega \subset \Bbb C^n$ is compact and if $H^2(\Omega)$ is the set of square integrable holomorphic functions on $\Omega$, then the Bergman kernel is ...
1
vote
0answers
25 views
Rational Singularities in dimenson 2 or highter and square integrability
I was wondering if there are any results available on square integrability of quotients of polynomials $P(z_1,z_2,\ldots,z_n)$ and $Q(z_1,z_2,\ldots,z_n)$ like
$$\int_{\mathbb{T}^n} ...
1
vote
0answers
30 views
Analytic variety
Let $\Omega$ be a convex domain in $\mathbb{C^{n}}$ , $\Delta$ be unit disk in $\mathbb{C}$ and f: $\Delta \rightarrow b\Omega $ be a non-constant holomorphic map.Then the convex hull of f($\Delta$) ...
0
votes
0answers
44 views
How does the complex convex set look like?
The usual convex set is the real linear convex set, if we change the real linear map into complex linear map, we can get the complex convex set. A system way to do this is in the several complex ...
0
votes
0answers
108 views
Harmonic function?
Let $\varphi : U \rightarrow \mathbb{C}$ be a holomorphic function, where $U \subset \mathbb{C}^{n}$ containing $0$. Is the function $u(x_{1}, ..., x_{n}, y_{1}, ..., y_{n}) := Im( \int_{0}^{(x_{1} + ...
