\documentclass[11pt]{article}
\usepackage{url}

%version 1.0,\ \   12.11.2020

\newcommand{\version}{version 1.0,\ \   12.11.2020}
\newcommand{\firstdate}{14.11.2020}

%\addtolength{\topmargin}{-5mm}
%\addtolength{\textheight}{10mm}
%\addtolength{\oddsidemargin}{-5mm}
%\addtolength{\textwidth}{10mm}

\input{defs-listki-en.tex}


\setlength{\headheight}{15pt}
\pagestyle{fancy} 
\lhead{\tiny Complex geometry, HSE} 
\lfoot{\tiny Issued \firstdate} 
\cfoot{-- \thepage \ -- } \rfoot{\tiny  \sc\version}
\rhead{{\tiny  Misha Verbitsky}}


\begin{document}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\listok{8}{Complex geometry handout 8: Plurisubharmonic functions}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\exercise
Let $f$ be a smooth real function on $\C^n=\R^{2n}$ with coordinates $z_i$,
and $v_{2i-1}=\Re z_i$, $v_{2i}=\Im z_i$ corresponding real coordinates.
Let $h:= \frac{d^2f}{dv_i dv_j}$ be the Hessian of $f$,
considered as a bilinear symmetric form, and $g:= \frac{h+ I(h)}{2}$.
Prove that $dd^cf(x,y)= g(Ix, y)$.
\ez
 
\definition
Let $f$ be a smooth real function on a complex manifold,
such that the (1,1)-form $dd^cf$ is Hermitian (hence, K\"ahler).
Then $f$ is called {\bf strictly plurisubharmonic.}
In that case $f$ is called {\bf the K\"ahler potential}
of the K\"ahler form $dd^c f$.
If all eigenvalues of $dd^cf$ are non-negative,
$f$ is called {\bf plurisubharmonic}. 
\ed

\exercise
Prove that any smooth convex function
on $\C^n$ is plurisubharmonic.
\ez

\exercise
Let $f$ be a smooth function on $\R^n$,
and $\phi\in C^\infty(\C^n)$ map
$(z_1, ..., z_n)$ to $f(\Re z_1, ..., \Re z_n)$.
Prove that $\phi$ is plurisubharmonic
if and only if $f$ is convex.
\ez

\exercise
Let $\omega$ be a K\"ahler form.
Using the Poincar\'e-Dolbeault-Grothendieck lemma,
prove that $\omega$ locally admits a K\"ahler potential.
\ez

\exercise
Let $f$ be plurisubharmonic. Prove that $e^f$ 
is also plurisubharmonic.
\ez


\exercise
Let $f$ be a plurisubharmonic function
satisfying $f<0$. Prove that $\log(-f)$ 
is also plurisubharmonic.
\ez


\exercise
Let $f$ be a strictly plurisubharmonic function.
Prove that $f$ cannot have a maximum.
\ez

\exercise
Let $f$ be a plurisubharmonic function.
Prove that $f^2$ is also plurisubharmonic
or find a counterexample.
\ez


\end{document}