\documentclass[10pt]{article} \usepackage{url} %version 1.0,\ \ 20.10.2021 \newcommand{\version}{version 1.0,\ \ 20.10.20201} \newcommand{\firstdate}{20.10.2021} \addtolength{\topmargin}{-15mm} \addtolength{\textheight}{30mm} \addtolength{\oddsidemargin}{-5mm} \addtolength{\textwidth}{10mm} \input{defs-listki-en.tex} \setlength{\headheight}{15pt} \pagestyle{fancy} \lhead{\tiny Symplectic geometry, HSE} \lfoot{\tiny Issued \firstdate} \cfoot{-- \thepage \ -- } \rfoot{\tiny \sc\version} \rhead{{\tiny Misha Verbitsky}} \begin{document} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \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