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\lhead{\tiny Symplectic geometry, HSE} 
\lfoot{\tiny Issued \firstdate} 
\cfoot{-- \thepage \ -- } \rfoot{\tiny  \sc\version}
\rhead{{\tiny  Misha Verbitsky}}


\begin{document}

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\listok{7}{Symplectic handout 7: Geometric invariant theory}

We freely use the definitions given in assignment 3 and 6
(``Symplectic reduction'', ``K\"ahler reduction''). 


\definition
Let $M$ be a complex manifold.
Define $d^c:\; \Lambda^i(M) \arrow \Lambda^{i+1}(M)$
as $d^c:= I d I^{-1}$, where $I$ acts on $k$-forms
multiplicatively. {\bf K\"ahler potential}
is a function $\phi:\; M \arrow \R$ such that
$dd^c \phi$ is a K\"ahler form.
\ed

\definition
{\bf A holomorphic vector field} is a vector field
satisfying $\Lie_X I=0$, that is, such that the
corresponding diffeomorphism flow $e^{tX}$ is holomorphic.
\ed 

\exercise
Let $G$ be a compact Lie group acting
on a complex manifold holomorphically and
preserving a K\"ahler potential $\phi$. 
Denote by $\omega:=dd^c \phi$ the corresponding K\"ahler form. 
\enum
\ite Prove that $G$ acts on $(M, \omega)$ by 
isometries.
\ite
Let $X\in \Lie(G)$ be a vector field on $TM$ 
tangent to the action of $G$.
Prove that $i_X \omega =i_X (d d^c \phi) = -d (i_X(d^c \phi))$.
\ite
Prove that the function $-\langle X, d^c \phi\rangle$
is a Hamiltonian for $X$.

\ite Prove that the moment map for the action of $G$
can be written as \\ $(m, X) \arrow \Lie_{IX}(\phi)(x)$.
\ee
\ez


\exercise
Let $G\subset U(n)$ be a Lie group acting 
on a complex vector space $V=\C^n$, equipped
with the standard Hermitian structure. Prove that
its action admits an equivariant moment
map $\mu$, given by $\langle \mu(v), g\rangle =
\Lie_{Ig} l$, where $v\in V$, $g\in \Lie(G)$, and
$l\in C^\infty V$ the function $l(v)=\frac 1 4|v|^2$.
\ez

\hint Use the previous exercise.
\eh


\exercise\label{_mome_map_extre_Exercise_}
Let $G\subset U(n)$ be a Lie group acting 
on a complex vector space $V=\C^n$, equipped
with the standard Hermitian structure, and 
$G_\C \subset GL(n, \C)$ its complexification.
Denote by $\mu:\; V \arrow \g^*$ the moment map.
Prove that a vector $z\in V$ belongs to $\mu^{-1}(0)$
if and only if the function $l:\; G_\C \cdot z\arrow \R$
on the orbit $G_\C$ has extremum in $z$.
\ez

\exercise
Let $V=\C^n$, $A\in {\goth u}(V)$ be an anti-Hermitian endomorphism,
and $G_A:=e^{t A}\subset GL(V)$, $t\in \C$ the corresponding 1-parametric
subgroup. 
\enum
\ite Consider $l(e^{t A}z):=|e^{t A}z|^2$ as a function on 
$G_A\cdot z$.
Prove that $l(e^{(t+u)A}(z))=l(e^{tA}(z))$ for all $u\in \R$.
Prove that $\frac{d^2}{du^2} |e^{(t +\1 u) A}(z)|^2= |A(e^{tA}(z))|^2$.

\ite  Assume that $A(z)\neq 0$.
Prove that $l$ is convex on the complex line
$G_A:=e^{\C A}$ and has at most one minimum
on the real line $e^{\1 \R A}(z)$.

\ite 
Let $A$ be diagonalized in an orthonormal basis 
$x_1, ..., x_n\in \R$, such that $A(x_i) = \1 w_i x_i$, 
and $z= \sum \alpha_i x_i$. Prove that the function
$l$ has a minimum on the line $e^{\1 \R A}(z)$
if and only if there are two basis vectors $x_l, x_k$  with
$\alpha_l, \alpha_k \neq 0$, such that $w_l <0$ and
$w_k >0$.
\ee
\ez

\exercise\label{_minima_on_orbit_Exercise_}
Let $G\subset U(n)$ be a Lie group acting 
on a complex vector space $V=\C^n$, equipped
with the standard Hermitian structure, and 
$G_\C \subset GL(n, \C)$ its complexification.
\enum
\ite
 Consider $l(z)=|z|^2$ as a function on $G_\C \cdot z$.
We parametrize $G_\C \cdot z$ by $\g_\C = \g\otimes_\R \C$
using the map $\g_\C\xlongrightarrow{g \mapsto e^{g}z} G_\C \cdot z$.
Prove that $\phi(g):=l(e^{g}z)$ is a convex function which satisfies
$\frac{d^2}{dg^2}(\phi)(g)=|\im(g)(z)|^2$.

\ite
Prove that  either $l$ has no extremal points on 
$G_\C \cdot z$, or $l$ takes minimum somewhere on 
$G_\C \cdot z$. Prove that $G$ acts transitively on the set
of minima of $l$ on $G_\C \cdot z$.
\ee
\ez

\definition
Let $G\subset U(n)$ be a Lie group acting 
on a complex vector space $V=\C^n$, equipped
with the standard Hermitian structure, and 
$G_\C \subset GL(n, \C)$ its complexification.
An orbit $G_\C \cdot z$, $z\neq 0$ is called {\bf stable}
if $l$ reaches minimum on $G_\C \cdot z$,
{\bf unstable} if $0$ belongs to the closure
of $G_\C \cdot z$, and {\bf (strictly) semistable} 
if if it is not stable and not unstable.
\ed

\exercise
Let $G_\C \cdot z\subset V$ be a stable orbit.
\enum
\ite Prove that for any non-zero $g\in \1 \g$,
one has $\lim_{t\to\infty} |e^{tg}(z)|=\infty$.
\ite
Let $\bar B_R\subset V$ a closed ball of radius $R$.
 Prove that $\bar B_R\cap G_\C \cdot z$
is compact for all $R\in \R^{>0}$.
\ite
Prove that there is a neighbourhood $U\ni z$
such that for all $z_i \in U$, the orbit
$G_\C \cdot z_1$ is stable.
\ee
\ez

\remark
The following theorem is identifies 
``the GIT reduction'' (taking a $G_\C$-quotient
of the union of all stable orbits) and the 
symplectic reduction.
\er

\exercise
Let $G\subset U(n)$ be a Lie group acting 
on a complex vector space $V=\C^n$, equipped
with the standard Hermitian structure, and 
$G_\C \subset GL(n, \C)$ its complexification.
Denote by $\mu:\; V \arrow \g^*$ the moment map,
$\mu(g, z):= (\Lie_{Ig}l)(z)$.
\enum
\ite Prove that
an orbit $G_\C \cdot z$ is stable if and only if
$G_\C \cdot z\cap\mu^{-1}(0)\neq 0$.
\ite Prove that  $G_\C \cdot z\cap\mu^{-1}(0)$ is
precisely one $G$-orbit.
\ite Prove that $\mu^{-1}(0)/G = V_s/G_\C$,
where $V_s\subset V$ is the union of all stable orbits.
\ee
\ez

\hint Use Exercises \ref{_mome_map_extre_Exercise_} and
\ref{_minima_on_orbit_Exercise_}.
\eh

\exercise (``Hilbert-Mumford criterion of stability'')\\
Let $G=U(1)$ act on a complex vector space $V=\C^n$, equipped
with the standard Hermitian structure, $G_\C= \C^*$
the corresponding complex Lie group, and $z\in V$ 
a non-zero vector.
\enum
\ite
Prove that there exists an orthonormal basis $x_1, ..., x_n$
in $V$ such that $g(x_i)= \1 w_i x_i$,
where $w_i\in \Z$ are integer numbers called
{\bf the weights} of the action.

\ite Let $z = \sum \alpha_{l_k} x_{l_k}$, where $\alpha_{l_k}$
are all non-zero. Prove that $G_\C\cdot z$ is unstable if and only if all
$w_{l_k}$ are positive or negative.

\ite Prove that $G_\C\cdot z$ is stable if and only if some
$w_{l_k}$ are positive while others are negative.
Prove that it is strictly semistable if some
$w_{l_k}$ vanish and all others are  positive or negative.
\ee
\ez

\exercise
Let $G=U(1)$ act on $V=\C^2$ as $\rho_t(z_1, z_2) = (tz_1, t^{-1}z_2)$.
Find all stable and non-stable orbits, and
prove that $V\2 G= \C^*$.
\ez




\end{document}



\exercise
Let $G_1 \times G_2$ be a product of Lie
groups acting on $(M, \omega)$ by Hamiltonian symplectomorphisms,
and $\mu_1, \mu_2$ equivariant moment maps for $G_1, G_2$.
Prove that $\mu_1 + \mu_2$ is an equivariant moment map
for $G_1 + G_2$.
\ez

\exercise
Let $M_1\subset M_2$ be a closed symplectic submanifold,
and $G$ a Lie group acting on $M_2$ by Hamiltonian
symplectomorphisms and preserving $M_1$.
Prove that the moment map on $M_1$ is obtained by
restriction from $M_2$, and the symplectic
quotient $M_1 \2 G$ admits a natural symplectic
embedding to $M_2\2 G$.
\ez


\definition
A {\bf projective manifold} is a submanifold of $\C P^n$
obtained as the set of common zeros of a system of
homogeneous polynomial equations.
\ed


\exercise
Let $G\subset U(n)$ be a Lie group acting 
on a complex vector space $V=\C^n$, equipped
with the standard Hermitian structure,
and let $X\subset {\C P}^{n-1}$ be
a $G$-invariant projective manifold.
\enum
\ite Prove that the zero set of the
moment map $\mu_V$ for the action of $G$ on $V$ 
is invariant under the natural
action of $\C^*$ on $V$. 
Prove that this gives an
action of $\C^*$ by symplectic 
homotheties on the symplectic
quotient $V\2 G$.

\ite Let $C(X)\subset V\backslash 0$ be the 
{\bf algebraic cone of $X$}, that is,
the set of lines projecting to $X$.
Prove that the symplectic
quotient $X\2 G$ is equal to 
the space of $\C^*$-orbits in 
$C(X)\2 G$.
\ee
\ez
