\documentclass[12pt]{article} \usepackage{url} %version 1.0,\ \ 09.04.2018 \newcommand{\version}{version 1.0,\ \ 20.04.2018} \newcommand{\firstdate}{02.05.2018} %\addtolength{\topmargin}{-5mm} %\addtolength{\textheight}{10mm} %\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{14}{Hodge theory 14: Fubini-Study form} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% {\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{Plurisubharmonic functions} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \definition A real-valued function $f$ on a compact manifold is called {\bf strictly plurisubharmonic} if $dd^c f$ is a K\"ahler form. \ed \exercise Consider a function $l$ on $\C^n$, $l(z_1, ..., z_n)=\sum |z_i|^2$. Prove that $dd^c l$ is the standard K\"ahler form on $\C^n$. \ez \exercise Let $\omega$ be a K\"ahler form on a polydisk. Using the Poincar\'e-Dolbeault-Grothendieck lemma, prove that $\omega= dd^c(f)$ for some strictly plurisubharmonic function $f$. \ez \exercise[**] Let $f$ a plurisubharmonic funcction on $\C$. Prove that $f$ cannot be bounded. \ez \exercise[*] Let $f$ be a plurisubharmonic function on $M$. Prove that $f$ has does not have a maximum. \ez \exercise Prove that $dd^c f(\phi)= f' dd^c\phi + f''d\phi \wedge d^c \phi$. for any real-valued function $\phi$ on a complex manifold and any smooth $f:\; \R\arrow \R$. \ez \exercise[!] Let $f:\; \R\arrow \R$ be a convex smooth function with $f'>0$ everywhere. Prove that $f(\phi)$ is strictly plurisubharmonic whenever $\phi$ is strictly plurisubharmonic. \ez \exercise Let $f:\; \R^2 \arrow R$ be a convex function with $\frac {\6 f}{\6 x}$ and $\frac {\6 f}{\6 y}$ positive everywhere, and $\psi, \psi$ strictly plurisubharmonic functions. \enum \ite[!] Prove that the function $f(\phi, \psi)$ is also strictly plurisubharmonic. \ite[*] Prove that for any $\epsilon > 0$ there exists a strictly plurisubharmonic function $\xi$ such that $\xi= \max(\phi, \psi)$ whenever $|\phi-\psi| >\epsilon$ \ee \ez \exercise Prove that $dd^c \log \phi= \frac {dd^c \phi}{\phi} - \frac{d\phi \wedge d^c \phi}{\phi^2}$. for any real-valued function $\phi$ on a complex manifold. \ez \exercise For any real function $f$, the form $dd^c f= -\1 2 \6\bar\6 f$ is of type (1,1), hence the form $h_f :=dd^cf(I(\cdot), \cdot)$ is symmetric and pseudo-Hermitian. For any Hermitian form $s$ on $M$, prove that in a neighbourhood of each point there exists an orthonormal basis such that $h_f$ is diagonal. Let $\alpha_1, ..., \alpha_n$ be the eigenvalues of $h_fs^{-1}$. \enum \ite[*] Prove that the map $M \arrow \R^n/\Sigma_n$ mapping $x\in M$ to the un-ordered collection of all eigenvalues of $h_f$ is continuous. \ite[!] Prove that the sign of these eigenvalues at $x\in M$ is independent from the choice of $s$. \ee \ez \definition We say that {\bf $dd^cf$ has positive/negative/zero eigenvalues} when $h_fs^{-1}$ has positive (negative, zero) eigenvalues for some (hence, any) Hermitian forms $s$ on $M$. \ed %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Fubini-Study form} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \exercise Let $l(z_1, ..., z_n)=\sum |z_i|^2$ be the function on $\C^n$ defined above. Prove that $dd^c \log l= \frac {dd^c l}{l} - \frac{dl \wedge d^c l}{l^2}$. Prove that for $n >1$, the form $dd^c \log l$ has at least one positive eigenvalue. \ez \exercise \enum \ite Consider the function $|z|^2=z\bar z $ on $\C^*$, and let $\rho= z \frac d{dz}$, where $z$ is the complex coordinate on $\C$. Prove that $\Lie_\rho |z|^2=|z|^2$. \ite Prove that $\Lie_\rho (\log |z|)=\const$. \ee \ez \exercise Let $z_1, ..., z_{n+1}$ be the complex coordinates on $\C^{n+1}$. \enum \ite Prove that the vector fields $r:= \sum_{i=1}^{n+1} z_i \frac d{dz_i}$ and $\bar r:= \sum_{i=1}^{n+1} \bar z_i \frac d{d\bar z_i}$ are $\C^*$-invariant. \ite[!] Let $l(z_1, ..., z_n)=\sum |z_i|^2$. Prove that $\Lie_r(\log l)= 0$. \ite[!] Prove that $(d\log l)\cntrct r = (d\log l)\cntrct {\bar r}=0$. \ee \ez \hint Use the previous exercise. \eh \exercise Consider the tautological fibration $\C^{n+1}\backslash 0 \stackrel \pi \arrow \C P^n$. We consider $\pi$ as a quotient map, $\C P^n=(\C^{n+1}\backslash 0)/G$, where $G= \C ^*$, and take $r= \sum_{i=1}^{n+1} z_i \frac d{dz_i}$ as above. \enum \ite Prove that any $\C^*$-invariant form $\eta$ such that $\eta\cntrct r = \eta\cntrct \bar r=0$ is basic. \ite[!] Prove that $dd^c \log l$ is basic. \ee \ez \exercise Consider the tautological fibration $\C^{n+1}\backslash 0 \stackrel \pi \arrow \C P^n$. \enum \ite Prove that there exists a form $\omega\in \Lambda^{1,1}(\C P^1)$ such that $dd^c \log l=\pi^*(\omega)$. \ite Prove that this form is $U(n)$-invariant and has at least one positive eigenvalue. \ite[!] Prove that $\omega$ is a K\"ahler form. \ee \ez \remark This gives another definition of Fubini-Study form, clearly equivalent to the one we have seen in Handout 12. \er \exercise[!] Let $\C^n \subset \C P^n$ be an affine chart with affine coordinates $z_1, ..., z_n$. Prove that the Fubini-Study form on this chart is given by \[ \omega= \frac{\sum_{i=1}^n dz_i \wedge d\bar z_i}{1+ \sum_{i=1}^n|z_i|^2} - \frac{\sum_{i=1}^n\bar z_i dz_i}{1+ \sum_{i=1}^n|z_i|^2}\wedge \frac{\sum_{i=1}^nz_i d\bar z_i}{1+ \sum_{i=1}^n|z_i|^2} \] \ez \exercise[!] Let $f_1, ..., f_n$ be holomorphic functions without common zeroes on a complex manifold. Prove that $\log(\sum_i |f_i|)$ is a plurisubharmonic function. \ez \exercise Let $L$ be a Hermitian holomorphic bundle on a complex manifold $M$, and $\Tot L$ its total space. Denote by $V\subset \Tot L$ the set of non-zero vectors in $\Tot L$. Clearly, $V$ is equipped with a free action of $\C^*$, and $V/\C^* = M$. Denote by $\Sigma$ the corresponding foliation. Consider the function $l:\; V \arrow \R^{>0}$ mapping a vector $v\in L\restrict x$ to $|v|^2$. \enum \ite[!] Prove that $dd^c \log l$ is a $\Sigma$-basic form on $V$. \ite[**] Prove that the corresponding (1,1)-form on $M$ coincides with the curvature of the Chern connection on $L$. \ee \ez %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\subsection{Ehresmann connection} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % %\definition %Let $\pi:\;M \arrow Z$ be a smooth fibration, with %$T_\pi M$ {\bf the bundle of vertical tangent vectors} %(vectors tangent to the fibers of $\pi$). %{\bf Ehresmann connection} on $\pi$ %is a sub-bundle $T_\hor M\subset TM$ such that %$TM= T_\hor M\oplus T_\pi M$. %{\bf Parallel transport} along the path $\gamma:\; [0, a]\arrow M$ %associated with the Ehresmann fibration %is a diffeomorphism \\ $V_t:\; \pi^{-1}(0)\arrow \pi^{-1}(t)$ %smoothly depending on $t\in [0, a]$ and %satisfying $\frac {dV_t}{dt}\in T_\hor M$. %\ed % %\exercise[*] %Let $\pi:\;M \arrow Z$ be a smooth fibration with compact fibers. %Prove that the parallel trasport, associated with the Ehresmann %connection, always exists. %\ez % %\definition %Let $\pi_1:\;M_1 \arrow Z$, $\pi_2:\;M_2 \arrow Z$ %be smooth fibration. {\bf Fibered product} $M_1 \times_Z M_2$ %is %\[ %M_1 \times_Z M_2:= \{ (u, v)\in M_1 \times M_2 \ \ |\ \ \pi_1(u)=\pi_2(v)\}. %\] %\ed % %\exercise %Prove that the fibered product of two smooth fibrations %is again a smooth fibration. %\ez % %\exercise %Let $B_1, B_2$ be vector bundles on $M$, and %$\Tot(B_1)$ their total spaces. Prove that %$\Tot(B_1)\times_M \Tot(B_2)=\Tot(B_1 \oplus B_2)$. %\ez % %\exercise[*] %Suppose that $\pi_1, \pi_2$ are equipped with %the Ehresmann connections %Prove that $M_1 \times_Z M_2$ %is also equipped with an Ehresmann connection. %\ez % %\definition %Let $B$ be a vector bundle on $M$ and $\pi:\; \Tot B \arrow M$ %its total space. An Ehresmann connection %on $\pi$ is called {\bf linear} if it is preserved by %the homothety map $\Tot B \arrow \Tot B$ %mapping $v$ to $\lambda v$ and by the additon map %$\Tot B\oplus B = \Tot B\otimes_M \Tot B\arrow \Tot B$. %\ed % %\exercise[*] %Let $B$ be a vector bundle, $\pi:\; \Tot B \arrow M$ %its total space, and $T\Tot B= T_\hor \Tot B\oplus T_\pi \Tot B$ %a linear Ehresmann connection. Given a section %$f:\; M \arrow B$, consider the projection of %its differential $Df:\; TM \arrow T\Tot B$ to %$T_\pi \Tot B\restrict {\im f} = B$. %Prove that the corresponding map from sections of $B$ %to sections of $B \otimes \Lambda^1 M$ defines %a connection on $B$. Prove that all connections %on $B$ are obtained from the linear Ehresmann connections. %\ez % %\exercise %Let $B$ be a holomorphic Hermitian vector bundle, %$\pi:\; \Tot B \arrow M$ its total space, %on a complex manifold, $l:\; \Tot B \arrow \R$ %a map taking $v\in B\restrict x $ to $|v|$, and %$dd^c l$ the corresponding (1,1)-form on $\Tot B$. %\enum %\ite Prove that $dd^c l$ is non-degenerate %on the fibers of $\pi$. %\ite[!] Let $W\subset T\Tot B$ be the orthogonal %complement to $T_\pi \Tot B$ with respect to the %form $dd^c l$. Prove that $T_\pi \Tot B\oplus W = T\Tot B$, %that is, $W$ defines an Ehresmann connection on $\Tot B$. %\ite[*] Prove that this connection is linear. %\ite[**] Prove the the corresponding connection on $B$ %coincides with the Chern connection defined in the lectures. %\ee %\ez \end{document}