\documentclass[11pt]{article} \usepackage{url} %version 1.0,\ \ 17.04.2018 %version 1.1,\ \ 19.04.2018, exercises 16(b) and 21 were the same %version 1.2,\ \ 02.05.2018, 12.2, 12.11 redone %version 1.3,\ \ 16.05.2018, many errors found by Ivan Frolov %version 1.4,\ \ 22.05.2018, more errors found by Ivan Frolov \newcommand{\version}{version 1.4,\ \ 22.05.2018} \newcommand{\firstdate}{18.04.2018} \addtolength{\topmargin}{-10mm} \addtolength{\textheight}{20mm} \addtolength{\oddsidemargin}{-5mm} \addtolength{\textwidth}{10mm} \input{defs-listki-en.tex} \setlength{\headheight}{15pt} \pagestyle{fancy} \lhead{\tiny Hogde theory, HSE} \lfoot{\tiny Issued \firstdate} \cfoot{-- \thepage \ -- } \rfoot{\tiny \sc\version} \rhead{{\tiny Misha Verbitsky}} \begin{document} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \listok{12}{Hodge theory 12: Complex and K\"ahler structures on symmetric spaces} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% {\scriptsize {\bf Rules:} You may choose to solve only ``hard'' exercises (marked with !, * and **) or ``ordinary'' ones (marked with ! or unmarked), or both, if you want to have extra stuff to work. To have a perfect score, a student must obtain (in average) a score of 10 points per week. If you have got credit for 2/3 of ordinary problems or 2/3 of ``hard'' problems, you receive $6t$ points, where $t$ is a number depending on the date when it is done. Passing all ``hard'' or all ``ordinary'' problems brings you $10t$ points. Solving of ``**'' (extra hard) problems is not obligatory, but each such problem gives you a credit for 2 ``*'' or ``!'' problems in the ``hard'' set. The first 3 weeks after giving a handout, $t=1.5$, between 21 and 35 days, $t=1$, and afterwards, $t=0.7$. The scores are not cumulative, only the best score for each handout counts. } %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Almost complex, Hermitian and K\"ahler structures} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \definition Let $M$ be a manifold. An endomorphism $I\in \End(TM)$, $I^2=-\Id_{TM}$ is called {\bf an almost complex structure}, and its $\1$-eigenbundle is denoted as $T^{1,0}M\subset TM\otimes \C$. An almost complex structure $I$ is called {\bf integrable} if $[T^{1,0}M,T^{1,0}M]\subset T^{1,0}M$. In this case $(M,I)$ is called {\bf a complex manifold}. A Riemannian metric on an almost complex manifold is called {\bf Hermitian} if it is $I$-invariant. \ed \exercise Let $U= V\oplus W$ be vector spaces. Prove that their Grassmann algebras are decomposed as follows: $\Lambda^n(U)= \bigoplus_{p+q=n}\Lambda^pV \otimes \Lambda^q W$.\footnote{This decomposition is not multiplicative.} \ez \definition Consider the eigenvalue decomposition $\Lambda^1(M, \C)= \Lambda^{1,0}(M)\oplus \Lambda^{0,1}(M)$ associated with the action of $I$, with $I\restrict{\Lambda^{1,0}(M)}=\1$ and $I\restrict{\Lambda^{0,1}(M)}=-\1$. It induces the decomposition on the de Rham algebra \[ \Lambda^k(M,\C)= \bigoplus_{p+q=k}\Lambda^p(\Lambda^{1,0}(M)) \otimes \Lambda^q (\Lambda^{0,1}(M)) \] as shown above. The bundles $\Lambda^p(\Lambda^{1,0}(M))$ and $\Lambda^q (\Lambda^{0,1}(M))$ are denoted $\Lambda^{p,0}(M)$ and $\Lambda^{0,q}(M)$, and the component $\Lambda^p(\Lambda^{1,0}(M)) \otimes \Lambda^q (\Lambda^{0,1}(M))$ is denoted $\Lambda^{p,q}(M)$. The decomposition $\Lambda^k(M,\C)=\bigoplus_{p+q=k}\Lambda^{p,q}(M)$ is called {\bf the Hodge decomposition}, the sections of $\Lambda^{p,q}(M)$ are called $(p,q)$-forms. \ed \exercise Let $(M,I)$ be an almost complex manifold, and $h$ an $I$-invariant Riemannian form. \enum \ite Prove that $\omega(x, y)=h(Ix, y)$ is a (1,1)-form. \ite[!] Prove that any Hermitian form $h$ is obtained from a (1,1)-form $\omega$ such that $\omega(x, Ix)> 0$ for all non-zero tangent vectors $x\in T_mM$. \ee \ez \exercise Prove that any almost complex manifold admits a Hermitian metric. \ez \exercise[*] Let $M$ be a manifold admitting a non-degenerate 2-form. Prone that $M$ admits an almost complex structure. \ez \definition Let $(M, I, h)$ be an almost complex Hermitian manifold The form $\omega(x, y)=h(Ix, y)$ is called {\bf the fundamental form} of $M$. The triple $(M, I, \omega)$ is called {\bf a K\"ahler triple} if $I$ is integrable and $\omega$ is closed. In this case $M$ is called {\bf the K\"ahler manifold}, $h$ {\bf the K\"ahler metric} and $\omega$ {\bf the K\"ahler form.} \ed \remark Recall that {\bf symplectic manifold} is a manifold equipped with a non-degenerate, closed 2-form. Clearly, the K\"ahler form is closed and non-degenerate. \er \exercise Find a complex, compact manifold not admitting a K\"ahler metric. \ez \exercise[**] Find a complex, compact manifold not admitting a K\"ahler metric, but admitting a symplectic structure. \ez %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Symmetric spaces} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \definition {\bf Homogeneous space} is a manifold with transitive action of a Lie group (often assumed connected). \ed \exercise Let $M$ be a connected manifold with transitive action of a Lie group $G$, and $H$ be a stabilizer of a point $x\in M$ (in this case, $H$ is called {\bf the isotropy group} of $x$). \enum \ite Prove that $M$ is identified with the space of orbits $G/H$. \ite[!] Let $x, y\in M$, and $H_x, H_y$ be the corresponding isotropy groups. Prove that $H_x$ and $H_y$ are conjugate by some element of $G$. \ee \ez \exercise Let $M=G/H$ be a homogeneous space with compact $H$. Assume that $M$ is connected and all non-unit $g\in G$ act non-trivially. \enum \ite[!] Prove that $M$ admits a $G$-invariant Riemannian structure. \ite[*] Prove that the natural map from the isotropy group of $x$ to $GL(T_x M)$ is injective. \ite[**] Is it always injective if $H$ is not necessarily compact? \ee \ez \definition {\bf A tensor} on a manifold $M$ is a section of the {\bf tensor bundle} $TM^{\otimes p}\otimes T^*M^{\otimes q}$. Whenever $G$ acts on $M$ by diffeomorphisms, it acts on the space of tensors, because tensors are functorial. \ed \exercise[!] Let $M=G/H$ be a homogeneous space, and $H_x$ the isotropy group of $x\in M$. Construct a bijective correspondence between $G$-invariant tensors on $M$ and $H_x$-invariant vectors in $T_xM^{\otimes p}\otimes T^*_xM^{\otimes q}$. \ez \definition A homogeneous space $M=G/H$ is called {\bf symmetric space} if $M$ admits a $G$-invariant Riemannian metric and $H_x$ contains an involution $\iota$ which acts as $-\Id$ on $T_x M$. \ed \definition $SO(n)$ denotes the {\bf special orthogonal} group (the group of all orthogonal matrices preserving the orientation). $U(n)$ is {\bf unitary group} (the group of all complex-linear matrices preserving a Hermitian form). $SU(n)$ is intersection of $U(n)$ and $SL(n,\C)$. \ed \exercise Consider the spaces $S^{2n}$, ${\Bbb C} P^n$, ${\Bbb H} P^n$ equipped with the natural action of $SO(2n+1)$, $U(n+1)$ and $Sp(n+1):= GL(n+1, {\Bbb H})\cap SO(4n+4)$. Prove that they are symmetric spaces. \ez \exercise Consider the Grassmannian $\Gr_\R(p,q):=\frac{SO(p+q)}{SO(p)\times SO(q)}$, \enum \ite[!] Prove that it is a symmetric space when $p$ or $q$ is even. \ite[*] Prove it for all $p, q$. \ee \ez \definition An {\bf odd tensor} on a symmetric space is a tensor $\Psi\in TM^{\otimes p}\otimes T^*M^{\otimes q}$ for $p+q$ odd. \ed \exercise Let $M=G/H$ be a symmetric space, and $\Psi$ a $G$-invariant odd tensor. Prove that $\Psi=0$. \ez \exercise Let $M=G/H$ be a symmetric space, and $\omega$ a $G$-invariant differential form. \enum \ite Prove that $d\omega=0$ and $\omega$ is even. \ite[!] Suppose that $M$ is equipped with a $G$-invariant Riemannian form. Prove that $\omega$ is harmonic. \ite[!] Assume in addition that $M$ is compact and $G$ is connected. Prove that any harmonic form is $G$-invariant. \ee \ez %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{K\"ahler structures on symmetric spaces} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \exercise[!] Let $M=G/H$ be a symmetric space, and $I$ a $G$-invariant almost complex structure. Prove that $I$ is integrable. \ez \exercise[!] $M=G/H$ be a symmetric space, $I$ a $G$-invariant almost complex structure, and $h$ a $G$-invariant Hermitian form. Prove that $(M, I, h)$ is K\"ahler. \ez \exercise Construct a structure of symmetric space and a $G$-invariant complex structure on the following spaces. \enum \ite $\C P^n$ (also prove that it is K\"ahler). \ite[!] $\Gr_\R(2,n):=\frac{SO(n+2)}{SO(n)\times SO(2)}$ \ee \ez \exercise[!] Let $M=G/H$ be a homogeneous space such that the isotropy group $H_x$ acts on the (real) projectivization ${\Bbb P} T_x M$ transitively. Prove that the $G$-invariant Riemannian metric on $M$ is unique up to a constant multiplier. \ez \exercise[*] Consider the Grassmannian space $\Gr_\R(p,q):=\frac{SO(p+q)}{SO(p)\times SO(q)}$. Prove that $\Gr_\R(p,q)$ admits a $SO(p+q)$-invariant metric. Prove that this metric is unique up to a constant multiplier, when $p>2$ or $q>2$. \ez \exercise Construct a $U(n+1)$-invariant Hermitian metric on $\C P^n$ (it is called {\bf Fubini-Study metric}). \enum \ite Prove that this metric is unique up to a constant. \ite Prove that it is K\"ahler. \ee \ez \definition $U(p, q)$ is the group of all complex-linear matrices preserving a pseudo-Hermitian metric $h$ of signature $(p, q)$, with $h(x_1, ..., x_{p+q})= \sum_{i=1}^p |x_i|^2- \sum_{i=q+1}^{p+q} |x_j|^2.$ \ed \exercise \enum \ite[!] Construct a $U(1,n)$-invariant metric and complex structure on $M:=\frac{U(1,n)}{U(1)\times U(n)}$. \ite[!] Prove that it is K\"ahler. \ite[*] Prove that $M$ is biholomorphic to an open ball in $\C^n$. \ite[**] Prove that all complex automorphisms of an open ball are isometries with respect to this metric. \ee \ez \remark This metric on an open ball is called {\bf Bergman metric}, or {\bf complex hyperbolic metric}. \er \exercise[*] Construct an $SO(n+2)$-invariant K\"ahler structure on the Grassmannian $\Gr_\R(2,n):=\frac{SO(n+2)}{SO(n)\times SO(2)}$. \ez \end{document}