Articles of several complex variables

Biholomorphic Equivalence in $\mathbb{C}^n$

I am stuck at the following problem : For $n \geq 2$, are the sets $\Delta_n$ and $H_n$ biholomorphically equivalent ? Here, $ \Delta_n = \{ ( z_1, z_2, \ldots z_n) : |z_i| < 1, i = 1, 2, \ldots, n\} $ and $ H_n := \{ (z_1, z_2, \ldots, z_n) : \Im (z_1) > […]

Calculus itegral on Riemann surfaces

Suppose that $\Omega$ is a compact region with a smooth boundary in a Riemann surface and that $\Phi$ is a real-valued function which is positive on $\Omega$ and vanishes on the boundary of $\Omega$. Show that $\int_{\partial\Omega} i \space\partial\Phi \ge0$. The problem makes a suggestion: Consider $\Omega ⊂ C$ and see that the integral is, […]

Sets of uniqueness in $\mathbb{C}^2$

A set $M$ is called a set of uniqueness for functions of a class $\mathcal{F}$ if any function $f \in \mathcal{F}$, equal to $0$ on $M$, is identically equal to $0$. Prove that the following sets are sets of uniqueness for functions holomorphic on $\mathbb{C}^2$: (a) a real hyperplane in $\mathbb{C}^2$; (b) the real two-dimensional […]

Fixed points of map $h$ in compact riemann surface

I have the following problem: Let $h : Σ → Σ$ be a conformal self-map, different from the identity, of a compact Riemann surface $Σ$ of genus $p$. Show that $h$ has at most $2p+2$ fixed points. The problem has a suggestion. Hint: Consider a meromorphic function $f : Σ → S^2$ with a single […]

Implicit function theorem for several complex variables.

This is the statement, in case you’re not familiar with it. Let $ f_j(w,z), \; j=1, \ldots, m $ be analytic functions of $ (w,z) = (w_1, \ldots, w_m,z_1,\ldots,z_n) $ in a neighborhood of $w^0,z^0$ in $\mathbb{C}^m \times \mathbb{C}^n $ and assume that $f_j(w^0,z^0)=0, \, j=1,\ldots,m $ and that $$ \det\left\{\frac{\partial f_j}{\partial w_k}\right\}^m_{j,k=1} \neq 0 […]

Holomorphic extension of a function to $\mathbb{C}^n$

I am stuck at the following question : Let $f$ be a holomorphic function on $\mathbb{C}^n \setminus \{(z_1, z_2, \ldots, z_n) | z_1=z_2=0\}$. Show that $f$ can be extended to a holomorphic function on $\mathbb{C}^n$. I think that I have to somehow use the Riemann extension theorem here. But I am not being able to […]

zero set of an analytic functio of several complex variables

In one variable complex theory, we have the result that zeroes of a non-zero analytic function are isolated. In several variable theory, this result does not hold. I read it somewhere that this fact can be proved using Hurwitz theorem. If anyone can help me with this.

Bergman-Shilov Boundary and Peak Points

Let $\Omega$ be a domain in $\mathbb{C}^n.$ Consider the Banach algebra $A(\Omega):=\mathcal{C}({\overline{\Omega}})\cap\mathcal{O}(\Omega).$ Denote the Bergman-Shilov boundary of $A(\Omega)$ by $\partial_S(\Omega)$ . From the very definiton of peak points we know that every peak point belongs to $\partial_S(\Omega).$ Now the examples that I know the set of peak points coincides with the $\partial_S(\Omega).$ My question here […]

Uniform convergence of the Bergman kernel's orthonormal basis representation on compact subsets

Consider the Bergman kernel $K_\Omega$ associated to a domain $\Omega \subseteq \mathbb C^n$. By the reproducing property, it is easy to show that $$K_\Omega(z,\zeta) = \sum_{n=1}^\infty \varphi_k(z) \overline{\varphi_k(\zeta)},\qquad(z,\zeta\in\Omega)$$ where $\{\varphi_k\}_{k=1}^\infty$ is any orthonormal basis of the Bergman space $A^2(\Omega)$ of Lebesgue square-integrable holomorphic functions on $\Omega$. This series representation converges at least pointwise, since the […]

Convergence domain: $\{(z,w):|z|^2+|w|^2 < 1\}$

I would like a HINT for this: Exhibit a two variable power series whose convergence domain is the unit ball $\{(z,w):|z|^2+|w|^2 < 1\}$. ($z$ and $w$ are complex numbers.) I think that it cannot be of the form $\sum P(z,w)^n$ where $P(z,w)$ is a polynomial. But I’m out of ideas. Thank you.