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\listok{6}{LCK manifolds 6: Orbifolds}

%\exercise
%Define a covering in the orbifold category without
%looking in the following definition.
%\ez
%
%\definition
%{\bf A covering of an orbifold}
%is a map from $(M, \{\phi_i:\; V_i \arrow V_i/G_i=U_i\})$
%to $(M', \{\phi_i':\; V_i' \arrow V_i'/G_i'=U_i'\})$
%induced by a continuous map $\Psi:\; M \arrow M'$ and inducing
%commutative squares \[ \begin{CD}
%V_i@>{\Psi_i}>>  V_i' \\
%@VVV  @VVV              \\
%U_i @>>>  U_i'
%\end{CD}
%\]
%with all $\Psi_i$ finite quotients by subgroups of $G_i$,
%such that for any $U_i'$ the set $\Psi^{-1}(U_i)$ is a union
%of several elements of the covering $\{U_i\}$ which do not intersect
%and map to $U_i'$ properly.
%An connected orbifold is {\bf orbifold simply-connected}
%if it has no non-trivial connected coverings. 
%\ed
%
%\exercise
%Prove that any orbifold admits a universal covering. Prove that
%it is unique up to an orbifold equivalence.
%\ez
%
%\definition
%An {\bf orbifold fundamental group} $\pi_1^\orb(M)$
%is a group of automorphism of the universal covering compatible
%with the projection to $M$. The {\bf topological fundamental group}
%$\pi_1^\top (M)$ is the fundamental group of $M$ as of the
%topological space.
%\ed
%

\definition
{\bf Orbipoints} are points of an orbispace with
$\Mor(x,x)$ non-trivial. {\bf An order} of an orbipoint
is $|\Mor(x,x)|$.  The group $\Mor(x,x)$
is called {\bf the monodromy group of an orbipoint}.
\ed

\exercise
Prove that a 1-dimensional
complex orbifold is uniquely defined by the 
following data: a smooth complex curve $M$, some orbipoints
$x_i$ marked on $M$, and order $p_i\in \Z^{>1}$ of monodromy at each $x_i$.
\ez


\exercise 
Let $S$ be a quasiregular Sasakian manifold, and $X:= S/\Reeb$
the corresponding orbifold. Prove that monodromy group of each
orbipoint is a cyclic group. 
\ez

\exercise
Let $\tilde M:\ \C^2 \backslash 0$, and $\C^*$ act on
$\tilde M$ as $h_t(x,y)=(tx, t^2 y)$. Find a Vaisman metric
on $\tilde M/h_\lambda$, where $\lambda>1$ is a fixed number.
Prove that $\tilde M/\C^*$ is $\C P^1$ with one orbipoint of order 2.
\ez

\exercise Let $X=\C P^1$ with two orbipoints of order 
$p$ and $q$. Find a quasiregular Sasakian
manifold $S$ with $S/\Reeb=X$. 
\ez

%\exercise
%Let $S$ be a Sasakian manifold with $\pi_1(S)$ infinite.
%Prove that $S$ is quasiregular.
%\ez

\exercise
Define a covering in the orbifold category, and prove
existence of a universal covering.
\ez

\definition
An {\bf orbifold fundamental group} $\pi_1^\orb(M)$
is a group of automorphisms of the universal covering compatible
with the projection to $M$. The {\bf topological fundamental group}
$\pi_1^\top (M)$ is the fundamental group of $M$ as of the
topological space.
\ed

\exercise
Construct a monomorphism $\pi_1^\top(M)\arrow \pi_1^\orb(M)$.
Find an orbifold $M$ such that $\pi_1^\top(M)$ is trivial, and
$\pi_1^\orb(M)$ is non-trivial.
\ez


\exercise
Let $M$ be a 1-dimensional complex orbifold with
a complete Hermitian metric of constant negative 
curvature, and $\pi_1^\orb(M)=0$. Prove that
$M$ is equivalent to a Poincare disk $\Delta$ and
has no orbipoints.
\ez

%\exercise
%Let $M$ be a 1-dimensional complex orbifold with
%a complete Hermitian metric of constant negative 
%curvature. Prove that $M$ is an orbifold quotient of 
%$\Delta$ by $\Gamma\subset PSL(2,\R)$, and
%$\Gamma = \pi_1^\orb(M)$. Prove, conversely,
%that any $\Gamma\subset PSL(2,\R)$ can be obtained
%as a fundamental group of such an orbifold.
%\ez

\exercise
Let $M$ be a complex curve of genus 1 with at least
one orbipoint. Prove that the universal covering of $M$
(in the orbifold category) is the Poincare disc $\Delta$.
\ez

\exercise
Let $M$ be $\C P^1$ with two orbipoints of order $p$ and $q$.
Find $\pi_1^\orb(M)$.
\ez




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