\documentclass{slides} \usepackage{amssymb, amsmath, amscd, color, epsfig, units} %\usepackage[matrix,arrow]{xy} \newcommand{\green}{\color[rgb]{0,0.4,0}} \newcommand{\purple}{\color[rgb]{0.4,0,0.4}} \newcommand{\red}{\color[rgb]{0.7,0,0}} \newcommand{\blue}{\color{blue}} \def\eqref#1{(\ref{#1})} \newcommand{\goth}{\mathfrak} \newcommand{\g}{{\frak g}} \newcommand{\arrow}{{\:\longrightarrow\:}} \newcommand{\Z}{{\Bbb Z}} \newcommand{\C}{{\Bbb C}} \newcommand{\R}{{\Bbb R}} \newcommand{\Q}{{\Bbb Q}} \renewcommand{\H}{{\Bbb H}} \newcommand{\6}{\partial} \def\1{\sqrt{-1}\:} \newcommand{\restrict}[1]{{\left|_{{#1}}\right.}} \newcommand{\cntrct} % contraction with a vector field {\hspace{2pt}\raisebox{1pt}{\text{$\lrcorner$}}\hspace{2pt}} \def\Bbb#1{\mathbb #1} \newcommand{\calo}{{\cal O}} \newcommand{\cac}{{\cal C}} % Correcting TeX... %\let\oldtilde=\tilde %\renewcommand{\tilde}{\widetilde} \renewcommand{\bar}{\overline} \renewcommand{\phi}{\varphi} \renewcommand{\epsilon}{\varepsilon} \renewcommand{\geq}{\geqslant} \renewcommand{\leq}{\leqslant} % Operatornames \newcommand{\even}{{\rm even}} \newcommand{\ev}{{\rm even}} \newcommand{\odd}{{\rm odd}} \newcommand{\const}{{\it const}} \newcommand{\capa}{{\sf capa}} \newcommand{\fl}{{\rm fl}} \newcommand{\im}{\operatorname{im}} \newcommand{\End}{\operatorname{End}} \newcommand{\Sym}{\operatorname{Sym}} \newcommand{\Hol}{\operatorname{{\cal H}ol}} \newcommand{\Tot}{\operatorname{Tot}} \newcommand{\Id}{\operatorname{Id}} \newcommand{\id}{\operatorname{\text{\sf id}}} \newcommand{\Vol}{\operatorname{Vol}} \newcommand{\Hom}{\operatorname{Hom}} \newcommand{\Ham}{\operatorname{Ham}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\Alt}{\operatorname{Alt}} \newcommand{\Ann}{\operatorname{Ann}} \newcommand{\Iso}{\operatorname{Iso}} \newcommand{\Sec}{\operatorname{Sec}} \newcommand{\Can}{\operatorname{Can}} \newcommand{\Sing}{\operatorname{Sing}} \newcommand{\Symp}{\operatorname{Symp}} \newcommand{\Spin}{\operatorname{Spin}} \newcommand{\codim}{\operatorname{codim}} \newcommand{\coim}{\operatorname{coim}} \newcommand{\coker}{\operatorname{coker}} \newcommand{\slope}{\operatorname{slope}} \newcommand{\rk}{\operatorname{rk}} \newcommand{\Def}{\operatorname{Def}} \newcommand{\Lie}{\operatorname{Lie}} \newcommand{\Tw}{\operatorname{Tw}} \newcommand{\Tr}{\operatorname{Tr}} \newcommand{\Spec}{\operatorname{Spec}} \newcommand{\Diff}{\operatorname{Diff}} \newcommand{\Teich}{\operatorname{Teich}} \newcommand{\Flux}{\operatorname{Flux}} \newcommand{\Hess}{\operatorname{Hess}} \newcommand{\Cyl}{\operatorname{Cyl}} \newcommand{\2}{{/\!\!/}} \renewcommand{\Re}{\operatorname{Re}} \renewcommand{\Im}{\operatorname{Im}} \newcommand{\inbfpare}[1]{{% \mbox{\tt (}\hspace{-5pt}\mbox{\tt (} #1 % \mbox{\tt )}\hspace{-5pt}\mbox{\tt )}% }} \newcommand{\comment}[1]{{}} \def\blacksquare{\hbox{\vrule width 10pt height 10pt depth 0pt}} \def\endproof{\blacksquare} \def\shortdash{\mbox{\vrule width 4.5pt height 0.55ex depth -0.5ex}} \makeatletter %\@ifundefined{Bbb} % {\newcommand{\Bbb}[1]{{\mathbb #1}}}% %{}% {\edef\Bbb#1{{\Bbb #1}}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Pagestyle % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newcommand{\ps@verbit}{% \renewcommand{\@oddhead}{% \scriptsize {\it \small Symplectic geometry, lecture 15 \hfil \tiny M. Verbitsky }} \renewcommand{\@evenhead}{\@oddhead} \renewcommand{\@oddfoot}{\hfil\thepage\hfil} \renewcommand{\@evenfoot}{\@oddfoot}} \pagestyle{verbit} \setlength\paperheight {10in}% \setlength\paperwidth {13.5in} \setlength{\textwidth}{0.8\paperwidth} \setlength{\textheight}{0.8\paperheight} \setlength{\pdfpageheight}{\paperheight} \setlength{\pdfpagewidth}{\paperwidth} \addtolength{\topmargin}{-20mm} \addtolength{\leftmargin}{-25mm} \addtolength{\rightmargin}{-25mm} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Lemma, sublemma, corollary, proposition, theorem, % % definition,example defined there: % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newcounter{section} \newcounter{Mycounter}[section] \newcounter{lemma}[section] \setcounter{lemma}{0} \renewcommand{\thelemma}{\noindent{Lemma \thesection.\arabic{lemma}}} \newcommand{\lemma}{% \setcounter{lemma}{\value{Mycounter}} \refstepcounter{lemma} \stepcounter{Mycounter} {\bf \green LEMMA:\ }} \newcounter{claim}[section] \setcounter{claim}{0} \renewcommand{\theclaim}{\noindent{Claim \thesection.\arabic{claim}}} \newcommand{\claim}{% \setcounter{claim}{\value{Mycounter}} \refstepcounter{claim} \stepcounter{Mycounter} {\bf \green CLAIM:\ }} \newcounter{corollary}[section] \setcounter{corollary}{0} \renewcommand{\thecorollary}{\noindent{Corollary \thesection.\arabic{corollary}}} \newcommand{\corollary}{% \setcounter{corollary}{\value{Mycounter}} \refstepcounter{corollary} \stepcounter{Mycounter} {\bf \green COROLLARY:\ }} \newcounter{theorem}[section] \setcounter{theorem}{0} \renewcommand{\thetheorem}{\noindent{Theorem \thesection.\arabic{theorem}}} \newcommand{\theorem}{% \setcounter{theorem}{\value{Mycounter}} \refstepcounter{theorem} \stepcounter{Mycounter} {\bf \green THEOREM:\ }} \newcounter{conjecture}[section] \setcounter{conjecture}{0} \renewcommand{\theconjecture}{\noindent{Conjecture \thesection.\arabic{conjecture}}} \newcommand{\conjecture}{% \setcounter{conjecture}{\value{Mycounter}} \refstepcounter{conjecture} \stepcounter{Mycounter} {\bf \green CONJECTURE:\ }} \newcounter{proposition}[section] \setcounter{proposition}{0} \renewcommand{\theproposition} {\noindent{Proposition \thesection.\arabic{proposition}}} \newcommand{\proposition}{% \setcounter{proposition}{\value{Mycounter}} \refstepcounter{proposition} \stepcounter{Mycounter} {\bf \green PROPOSITION:\ }} \newcounter{definition}[section] \setcounter{definition}{0} \renewcommand{\thedefinition} {\noindent{Definition~\thesection.\arabic{definition}}} \newcommand{\definition}{% \setcounter{definition}{\value{Mycounter}} \refstepcounter{definition} \stepcounter{Mycounter} {\bf \green DEFINITION:\ }} \newcounter{example}[section] \setcounter{example}{0} \renewcommand{\theexample}{\noindent{Example \thesection.\arabic{example}}} \newcommand{\example}{% \setcounter{example}{\value{Mycounter}} \refstepcounter{example} \stepcounter{Mycounter} {\bf \green EXAMPLE:\ }} \newcounter{remark}[section] \setcounter{remark}{0} \renewcommand{\theremark}{\noindent{Remark \thesection.\arabic{remark}}} \newcommand{\remark}{% \setcounter{remark}{\value{Mycounter}} \refstepcounter{remark} \stepcounter{Mycounter} {\bf \green REMARK:\ }} \newcounter{observation}[section] \setcounter{observation}{0} \renewcommand{\theobservation}{\noindent{Question \thesection.\arabic{observation}}} \newcommand{\observation}{% \setcounter{observation}{\value{Mycounter}} \refstepcounter{observation} \stepcounter{Mycounter} {\bf \green OBSERVATION:\ }} \newcounter{question}[section] \setcounter{question}{0} \renewcommand{\thequestion}{\noindent{Question \thesection.\arabic{question}}} \newcommand{\question}{% \setcounter{question}{\value{Mycounter}} \refstepcounter{question} \stepcounter{Mycounter} {\bf \green QUESTION:\ }} \newcommand{\exercise}{% {\bf \green EXERCISE:\ }} \newcommand{\proof}{% {\bf \green Proof:\ }} \newcommand{\pstep}{% {\bf \green Proof. Step 1:\ }} \begin{document} \setcounter{page}{1} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{center} {\Large\bf Symplectic geometry\\[15mm] \small lecture 15: Geometric invariant theory} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \\[14mm] {\small Misha Verbitsky } \\[20mm] {\tiny\bf HSE, room 306, 16:20, \\[2mm] October 23, 2021 } \end{center} \newpage {\bf \blue Moment maps (reminder)} \definition $(M,\omega)$ be a symplectic manifold, and $G$ a Lie group acting on $M$ by symplectomorphisms. {\bf \blue A moment map} $\mu$ of this action is a linear map ${\goth g}\arrow C^\infty M$ associating to each $g\in G$ its Hamiltonian. \remark It is more convenient to consider $\mu$ as an element of ${\goth g}^* \otimes_\R C^\infty M$, or (and this is most standard) {\bf \red as a function with values in ${\goth g}^*$}. \remark Moment map {\bf \purple always exists} if $M$ is simply connected. \definition A moment map $M \arrow {\goth g}^*$ is called {\bf \blue equivariant} if it is equivariant with respect to the coadjoint action of $G$ on ${\goth g}^*$. \remark $M\stackrel\mu \arrow {\goth g}^*$ is a moment map iff for all $g\in {\goth g}$, $\langle d\mu,g\rangle= i_{\rho_g}(\omega)$. Therefore, {\bf \purple a moment map is defined up to a constant ${\goth g}^*$-valued function.} An equivariant moment map is is defined up to {\bf \purple a constant ${\goth g}^*$-valued function which is $G$-invariant}, that is, up to addition of a central vector $c\in \g^*$. \claim {\bf \red An equivariant moment map exists whenever $H^1(G, {\goth g}^*)=0$.} In particular, when $G$ is reductive and $M$ is simply connected, an equivariant moment map exists. Further on, all moment maps will be tacitly considered equivariant. \newpage {\bf \blue Weinstein-Marsden theorem (reminder)} %\definition A $G$-invariant $c\in \goth g^*$ is called %{\bf\blue central}. \definition (Weinstein-Marsden) $(M,\omega)$ be a symplectic manifold, $G$ a compact Lie group freely acting on $M$ by symplectomorphisms, $M\stackrel\mu \arrow {\goth g}^*$ an equivariant moment map, and $c\in {\goth g}^*$ a central element. The quotient $\mu^{-1}(c)/G$ is called {\bf \blue symplectic reduction} of $M$, denoted by $M\2 G$. \claim {\bf \red The symplectic quotient $M\2 G$ is a symplectic manifold of dimension $\dim M - 2 \dim G$.}\\ \pstep $T_x (\mu^{-1}(c))= d\mu^{-1}(0)$. However, the space $\langle d\mu, g\rangle \subset \Lambda^1 M$ is $\omega$-dual to the space $\tau(\goth g)$ of vector fields tangent to the $G$-action, {\bf \purple hence $d\mu^{-1}(c)= \tau(\goth g)^\bot$.} {\bf \green Step 2:} Since $\mu$ is $G$-equivariant, $G$ preserves $\mu^{-1}(c)$, hence $\tau(\goth g)\subset d\mu^{-1}(0)$. This implies that {\bf \purple $\tau(\goth g)\subset TM$ is isotropic} (that is, $\omega\restrict{\tau(\goth g)}=0$). Its $\omega$-orthogonal complement in $T_x M$ is $T_x (\mu^{-1}(c))$ (Step 1). {\bf \green Step 3:} Consider the {\bf \blue characteristic foliation} ${\cal F}$ on $\mu^{-1}(c)$. It is a bundle because $\mu^{-1}(c)\subset M$ is coisotropic. From Step 2 {\bf \purple we obtain that ${\cal F}=\tau(\goth g)$.} {\bf \green Step 4:} Since $\omega\restrict {\mu^{-1}(c)}$ is closed, it satisfies $\Lie_v (\omega)=0$ for all $v\in {\cal F}$. This implies that it is {\em \green basic}, that is, lifted from the leaf space of characteristic foliation, identified with $M\2 G$. \endproof \newpage {\bf \blue Holomorphic vector fields} \definition {\bf\blue A holomorphic vector field} is a vector field $X\in TM$ satisfying $\Lie_X I=0$, that is, such that the corresponding diffeomorphism flow $e^{tX}$ is holomorphic. \remark Let $(M,I)$ be a compact complex manifold. Then the group of biholomorphisms of $M$ is a Lie group whose Lie algebra is the space of (real) holomorphic vector fields. \remark It is not hard to see that {\bf \purple $I(X)$ is holomorphic whenever $X$ is holomorphic} ({\bf \red prove this}). Then $X\arrow I(X)$ {\bf \purple defines the complex structure on the Lie algebra of holomorphic vector fields.} \newpage {\bf \blue Symplectic reduction and a K\"ahler potential (reminder)} \definition Let $d^c:= I dI^{-1}$. {\bf \blue K\"ahler potential} on a K\"ahler manifold $(M, \omega)$ is a function $\psi$ such that $dd^c \psi=\omega$. \proposition Let $G$ be a real Lie group acting on a K\"ahler manifold $M$ by holomorphic isometries, and $\psi$ a $G$-invariant K\"ahler potential. {\bf \red Then the moment map $\g \times M \stackrel {\mu_g} \arrow \R$ can be written as $g, m \arrow \Lie_{Iv}\psi$,} where $v=\tau(g)\in TM$ is the tangent vector field associated with $g\in \goth g$. \proof Since $\psi$ is $G$-invariant, and $I$ is $G$-invariant, we have $0 = \Lie_v d^c \psi = i_v (dd^c\psi) + d(i_v d^c \psi)$. Using $\omega=dd^c \psi$, we rewrite this equation as $i_v\omega = -d(\langle d^c \psi, v\rangle)$, giving an equation for the moment map $\mu_g= -\langle d^c \psi, v\rangle.$ Acting by $I$ on both sides, we obtain $\mu_g= \langle d \psi, Iv\rangle = \Lie_{Iv}\psi$. \endproof \corollary Let $V$ be a Hermitian representation of a compact Lie group $G$. {\bf \purple Then the corresponding moment map can be written as $\mu_g(v)= \Lie_{Ig}|v|^2= \frac 1 4 \langle v, Ig(v)\rangle.$} \endproof \newpage {\bf \blue Moment map and extrema on 1-parametric orbit} \remark Let $G\subset U(n)$ be a Lie group acting on a complex vector space $V=\C^n$, equipped with the standard Hermitian structure $h$, and $G_\C \subset GL(n, \C)$ its complexification. Denote by $\mu:\; V \arrow \g^*$ the moment map. Since $\mu_g(v)= \Lie_{Ig}|v|^2$, a vector $z\in V$ {\bf \purple belongs to $\mu^{-1}(0)$ if and only if the function $l:\; G_\C \cdot z\arrow \R$, $l(z)=|z|^2$ on the orbit $G_\C$ has extremum in $z$.} It turns own that $z\arrow l(z)$ {\bf \green is convex}, in the following sense. \claim 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. {\bf \red Then $\frac{d^2}{d{u}^2} |e^{(t + u) A}(z)|^2= 4 |A(e^{tA}(z))|^2$,} when $u\in \1 \R$. \proof Let $A$ be diagonalized in an orthonormal basis $x_1, ..., x_n\in \R$, such that $A(x_i) = \1 w_i x_i$, $w_i \in \R$, and $h$ the Hermitian form. The operator $e^{tA}(x_i) = e^{\1 tw_i} x_i$ is an isometry when $t$ is real, and is Hermitian self-adjoint when $t$ is imaginary. This gives $\frac{d}{d{u}} |e^{(t + u) A}(z)|^2= 2 h(Ae^{(t) A}(z), e^{(t) A}(z)).$ Taking the derivative in $u$ once again, we obtain $\frac{d^2}{d{u}^2} |e^{(t + u) A}(z)|^2=2 h(Ae^{(t) A}(z), A e^{(t) A}(z))$. \endproof \corollary The function $l$ {\bf \red is convex on $G_A$,} and {\bf \red strictly convex in imaginary direction,} unless $A=0$. \endproof \newpage {\bf \blue Moment map and extrema on 1-parametric orbit} \corollary Let $A$ be diagonalized in an orthonormal basis $x_1, ..., x_n\in \R$, such that $A(x_i) = \1 w_i x_i$, $w_i \in \R$, and $z= \sum \alpha_i x_i$, with all $\alpha_i\neq 0$. Then the function $l$ {\bf \red has a minimum on the line $e^{\1 \R A}(z)$ if and only if there are two basis vectors $x_l, x_k$ such that $w_l <0$ and $w_k >0$.} Moreover, {\bf \red $l$ has no other extrema on $e^{\1 \R A}(z)$, unless $A=0$.} \proof The function $u\arrow |e^{(t + u) A}(z)|^2$ is strictly convex along the imaginary axis, and constant along the real axis. Therefore, it has at most one extremum, and it is the minimum. It has minimum if and only if $\lim |e^{(t + u) A}(z)|^2=\infty$ as $t \arrow \pm \infty$, which happens if and only if $w_l <0$ and $w_k >0$ for some $k,l$. \endproof \newpage {\bf \blue Extrema of the length function on $G_\C$-orbits} \theorem Let $G\subset U(n)$ be a Lie group acting on a complex vector space $V=\C^n$, equipped with the standard Hermitian structure $h$, and $G_\C \subset GL(n, \C)$ its complexification. Let $\g_\C = \Lie(G_\C)= \g \otimes_\R \C$. Consider the function $\phi:\; \g_ \C \arrow \R$ taking $g$ to $l(e^{g}z)$. {\bf \red Then $\phi$ is constant in real direction, convex in imaginary direction, and satisfies $\frac{d^2}{d{u}^2}|\phi(u)(z)|^2= 4 h(u(z), u(z))$ where $u\in \im \g_\C$.} \proof Write $u\in I({\goth u}(n))$ in an appropriate orthonormal basis $x_1,...., x_n$ as $u(x_i) = w_i x_i$, $w_i\in \R$. Let $z= \sum \alpha_i x_i$. Then $\frac{d}{d{u}}|\phi(u)(z)|^2= 2\sum w_i |x_i|^2 = 2 h(u(z), z)$, and $\frac{d^2}{d{u}^2}|\phi(u)(z)|^2= 4 h(u(z), u(z))$. \endproof {\bf \green COROLLARY 1:} Either the function $l$ {\bf \red has no extremal points on the orbit $G_\C \cdot z$, or $l$ has a minimum.} In the second case, {\bf \red the set of points on $G_\C$ where $l$ has a minimum is an orbit of $G$.} \endproof \newpage {\bf \blue Stable and unstable orbits} \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\blue stable} if $l$ reaches minimum on $G_\C \cdot z$, {\bf\blue unstable} if $0$ belongs to the closure of $G_\C \cdot z$, and {\bf\blue (strictly) semistable} if if it is not stable and not unstable. \theorem Let $G\subset U(V)$ be a group acting on a complex Hermitian vector space $V= \C^n$, and $z\in V \backslash 0$. {\bf \red Then an orbit $G_\C\cdot z$ is stable if and only if it intersects the zero set of the moment map $\mu(v, z)= \Lie_{Iv}(l)(z)$. } Moreover, {\bf \red $G$ acts on $G_\C\cdot z\cap \mu^{-1}(0)$ transitively.} \proof Extrema of $l$ on $G_\C\cdot z$ are its minima because $l$ is convex. The extrema of $l$ are zero set of $\mu$ because $\mu(v, z)= \Lie_{Iv}(l)(z)$. The set of extrema is one $G$-orbit by Corollary 1. \endproof \newpage {\bf \blue Hilbert-Mumford criterion of stability} \exercise Let $\rho:\; U(1) \arrow \End(V)$, $V=\C^n$ be a complex Hermitian representation of $U(1)$. Then {\bf \red there exists an orthonormal basis $x_1, ..., x_n$ in $V$ such that $g(x_i)= \1 2\pi w_i x_i$,} where $w_i\in \Z$ are integer numbers called {\bf\blue the weights} of the action. \proof Since the action is unitary, $\rho(t)$ is diagonalizable. The numbers $w_i$ are integer because $\rho\restrict{\1 \R}$ factorizes through $U(1)$. \endproof \theorem Let $G=U(1)$ act on a complex Hermitian vector space $(V=\C^n, h)$, $G_\C= \C^*$ the corresponding complex Lie group, and $z\in V$ a non-zero vector, $z = \sum \alpha_{l_k} x_{l_k}$, where $\alpha_{l_k}$ are all non-zero. Consider the weight decomposition of the generator of this action: $A(x_i)= \1 2\pi w_i x_i$. Then $\bullet$ {\bf \red $G_\C\cdot z$ is unstable if and only if all $w_{l_k}$ are positive or negative,} $\bullet$ {\bf \red $G_\C\cdot z$ is stable if and only if some $w_{l_k}$ are positive while others are negative.} $\bullet$ {\bf \red $G_\C\cdot z$ is strictly semistable if some $w_{l_k}$ vanish and all others are positive or negative.} \proof If some $w_{l_k}$ are positive while others are negative, one has $\lim_{\im t \to \pm \infty} |e^{t\1 A}(z)| =\infty$, hence $l$ reaches the minimum somewhere on the imaginary axis. If all weights are positive or negative, one has $\lim_{\im t \to \infty} |e^{t\1 A}(z)| =\infty$ and $\lim_{\im t \to -\infty} e^{t\1 A}(z) =0$ or vice versa, and the orbit is unstable. If all weights are $\geq 0$ or $\leq 0$, with one of the weights equal to 0, one has $\lim_{\im t \to \infty} |e^{t\1 A}(z)| =\infty$ and $\lim_{\im t \to -\infty} e^{t\1 A}(z) =z$, and the orbit is strictly semistable. \endproof \newpage {\bf \blue Set of stable orbits} {\bf \purple For $G_\C=\C^*$, the following theorem immediately follows from the Hilbert-Mumford criterion.} \proposition Let $G\subset U(n)$ be a Lie group acting on a complex Hermitian vector space $(V=\C^n, h)$, and $G_\C \subset GL(n, \C)$ its complexification, and ${V_s}$ the set of all $z\in V$ such that the orbit $G_\C \cdot z$ is stable. {\bf \red Then ${V_s}\subset V$ is open.} \pstep Let $\bar B_R\subset V$ be a closed ball of radius $R$. The orbit $z\in {V_s}$ is stable if and only if for each $R\in \R^{>0}$, the intersection $\bar B_R\cap G_\C \cdot z$ is compact for all $R\in \R^{>0}$. Indeed, if it is compact, $l$ reaches minimum somewhere on $G_\C \cdot z$. On the other hand, of $l$ reaches its minimum, one has $\lim_{\im t \to \pm \infty} |e^{t\1 A}(z)| =\infty$ for all $\1 A \in \1 \g$, hence $\bar B_R\cap G_\C \cdot z$ is compact. {\bf \green Step 2:} If $\lim_{\im t \to \pm \infty} |e^{t\1 A}(z)| =\infty$ for all $\1 A \in \1 \g$, one has $\lim_{\im t \to \pm \infty} |e^{t\1 A}(z+u)| =\infty$ for $u$ sufficiently small, hence this condition is open in $z\in V$. \endproof \newpage {\bf \blue Geometric Invariant Theory} \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. \theorem Let $G\subset U(n)$ be a Lie group acting on a complex Hermitian vector space $(V=\C^n, h)$, and $G_\C \subset GL(n, \C)$ its complexification. Denote by $\mu:\; V \arrow \g^*$ the moment map, $\mu(g, z):= (\Lie_{Ig}l)(z)$. {\bf \red Then an orbit $G_\C \cdot z$ is stable if and only if $G_\C \cdot z\cap\mu^{-1}(0)\neq 0$.} Moreover, {\bf \red $G_\C \cdot z\cap\mu^{-1}(0)$ is precisely one $G$-orbit, and $\mu^{-1}(0)/G = V_s/G_\C$,} where $V_s\subset V$ is the union of all stable orbits. \proof $G_\C \cdot z$ is stable if and only if $G_\C \cdot z\cap\mu^{-1}(0)\neq 0$ because $\mu^{-1}(0)$ intersects the orbit in the points where the length $l$ is minimal. This intersection set is precisely one $G$orbit, which gives $\mu^{-1}(0)/G = V_s/G_\C$. \endproof \end{document}