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\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}

