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\listok{7}{LCK manifolds 7: Hopf manifolds}

\definition
{\bf Contraction} of a manifold $M$ to a point
$x\subset M$ is an automorphism $\phi$ such that for any 
open subset $U\ni x$ with compact
closure there exists $N>0$ such that for all
$n>N$, the map $\phi^n$ maps $U$ to a compact subset $K\subset U$.
\ed

\exercise
Let $\phi:\; M \arrow M$ be a contraction to $x$.
Prove that $\lim_n \phi^n(m)=x$, for each $m\in M$.
\ez


\exercise
Let $\phi:\; \C^n \arrow \C^n$ be a polynomial map
preserving $0$, and with $D\phi\restrict 0$ invertible
with all eigenvalues $|\alpha_i|<1$.
\enum
\item Prove that $\phi$ defines a contraction on an open
ball $B_\epsilon(0)$. 
\item Consider the equivalence
relation on $B_\epsilon(0)\backslash 0$ generated by $x\sim\phi(x)$.
Prove that $(B_\epsilon(0)\backslash 0)/\sim$ is a complex manifold
diffeomorphic to $S^1\times S^{2n-1}$.
\ee
\ez

\definition
The manifold $B_\epsilon(0)/\sim$ defined above is called
{\bf Hopf manifold}. When $\phi$ is linear, it is called
{\bf a linear Hopf manifold}. In dimension 2, Hopf manifold 
is also called {\bf Hopf surface}, or {\bf primary Hopf surface}. 
A {\bf secondary Hopf surface} is a quotient of a Hopf surface
by a finite group freely acting on it.
\ed

\exercise Find secondary Hopf surfaces with $\pi_1(H)=\Z
\oplus \Z/n\Z$, for each $n\in \Z^{>0}$.
\ez

\exercise Find a secondary Hopf surface with $\pi_1(H)=\Z
\oplus S_3$, where $S_3$ is a symmetric group.
\ez

%\exercise
%Let $A\in GL(n,\C)$ be a diagonal matrix with integer eigenvalues
%$|\alpha_i|>0$. Prove that $\C^n\backslash 0/\langle A\rangle$
%is Vaisman.
%\ez

\exercise
Let $H_1, H_2$ be Hopf manifolds associated with
holomorphic contractions $P_1, P_2$ of an open ball
$B\subset \C^n$. Assume that $H_1$ is biholomorphic
to $H_2$. Prove that there exists a biholomorphism
$F:\; B\arrow B$ giving $P_1=FP_2 F^{-1}$.
\ez

\definition
Let $A\in GL(n,\C)$ be a linear map, written in a basis
$x_i\in \C^n$ diagonally as $A(x_i)=\alpha_i x_i$.
A vector $\lambda=(\lambda_1, ..., \lambda_n)\in {\Bbb N}^n$, 
$|\lambda|>1$ is called {\bf resonant} for $X$ if
$\alpha_s = \prod_{i=1}^n \alpha_i^{\lambda_i}$.

Consider a polynomial map $P=(P^1, ..., P^n):\; \C^n \arrow \C^n$,
with each $P^i\in \C[x_1, ..., x_n]$; assume that the 
linear part of $P$ is equal to $A$. We write each
$P^s$ as a sum of monomials 
$P^s_{h_1h_2...h_n}=\alpha_{h_1h_2...h_n}x_1^{h_1}x_2^{h_2}...x_n^{h_n}$.
A monomial $P^s_{h_1h_2...h_n}$ is {\bf resonant} if
$(h_1,h_2,...,h_n)$ is a resonant vector.
\ed

\exercise
Let $P=(P^1, ..., P^n):\; \C^n \arrow \C^n$
be a polynomial map, $A\in GL(n,\C)$ its linear part,
and $F:\; \C^n \arrow \C^n$ a polynomial automorphism
preserving 0 and satisfying $dF\restrict 0=\Id$.
Assume that all monomials in $P-A$ are of degree $\geq d$,
and let $P_d$, $F_d$ be the degree $d$ term of $P$, $F$.
\enum
\item Consider the polynomial map $FPF^{-1}$.
Show that the degree $d$ term of $FPF^{-1}$
is equal to $P_d+AF_d-F_d A$.
\item 
Given $P$ as above, prove that there following are 
equivalent: (i) there exists $F$ 
such that the degree $d$ terms of $FPF^{-1}$ vanish
(ii) all resonant monomials of degree $d$
in $P$ vanish.
\ee
\ez

\exercise
Let $P:\; \C^2 \arrow \C^2$ be a polynomial mapping
$(x,y)$ to $(\frac 1 2 x, \frac 1 4 y+x^2)$,
and $H$ the corresponding Hopf surface. Prove that
$H$ is not biholomorphic to a linear Hopf surface.
\ez 


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