Complex Analysis in Several Variables, MTH 3108
Home Assignement 2
Professor M.Shubin
Fall 1997


Textbook:

Introduction to Complex Analysis in Several Variables, by Lars Hörmander. 3rd edition, North Holland, 1990


1. Let $A:\hbox{{\bbb C}}^n\to\hbox{{\bbb C}}^n$ be a linear map (over $\hbox{{\bbb C}}$). Consider it as a map $A^{\hbox{{\bbb R}}}:\hbox{{\bbb R}}^{2n}\to\hbox{{\bbb R}}^{2n}$ using the canonical isomorphism $\hbox{{\bbb C}}^n\cong\hbox{{\bbb R}}^{2n}$.Prove that $\det A^{\hbox{{\bbb R}}}=\vert\det A\vert^2$.


2. Let $\Omega\subset \hbox{{\bbb C}}^n$ be open and connected, $u\in A(\Omega)$. Prove that $\Omega\setminus u^{-1}(0)$ is connected.


3. Let $u\in A(\Omega)$ where $\Omega\subset \hbox{{\bbb C}}^n$ is a neighbourhood of a closed bounded polydisc $\bar D$ with the center z0. Prove that

\begin{displaymath} \log \vert u(z_0)\vert\le {1\over \hbox{vol}\; D}\int_D \log \vert u(z)\vert d\lambda(z)\;,\end{displaymath}

where $d\lambda(z)$ is the Lebesgue measure, $\hbox{vol}\; D$ is the corresponding volume.


4. Let $\omega\in \Lambda^{p,q+1}(D)$ where D is an open polydisc in $\hbox{{\bbb C}}^n$, and $\bar{\partial}\omega=0$. Prove that there exists $\alpha\in\Lambda^{p,q}(D)$ such that $\bar{\partial} \alpha=\omega$ in D.


5. Let D be an open polydisc in $\hbox{{\bbb C}}^n$, $\{U_i\vert\;i\in I\}$ an open covering of D, and the functions $h_{ij}\in A(U_i\cap U_j)$ form an additive 1-cocycle i.e. satisfy the relations

\begin{displaymath} h_{ji}=-h_{ij}\ \hbox{on} \ U_i\cap U_j;\qquad h_{ij}+h_{jk}+h_{ki}=0\ \hbox{on}\ U_i\cap U_j\cap U_k\;. \end{displaymath}

Prove that there exist $h_i\in A(U_i)$ such that $h_{ij}=h_i-h_j\ \hbox{on}\ U_i\cap U_j$.


 
Sergey Bratus
11/8/1997