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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}

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\listok{13}{Hodge theory 13: Foliations, fiber bundles, and $dd^c$}

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{\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{Foliations}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\definition
{\bf Sheaf of submanifolds} on $M$ is a sheaf ${\cal F}$ of sets 
mapping each $U$ to a collection of its closed submanifolds, with
restriction maps ${\cal F}(U)\arrow {\cal F}(V)$
mapping each submanifold $Z\in {\cal F}(U)$ to $Z\cap V$.
{\bf A foliation} is a sheaf of submanifolds ${\cal F}$ on $M$ such that
each $x\in M$ has a neighbourhood $U$ which is decomposed onto
a product $U=A\times B$, with ${\cal F}(U)$ being all fibers
of the projection $U\arrow B$. {\bf A leaf} of the foliation
${\cal F}$ is a connected smooth manifold $Z$ immersed to $M$ in such a way
that any closed connected component of $Z\cap U$
is an element of ${\cal F}(U)$. {\bf Closed leaf}
is a leaf with closed image. 
\ed

\exercise
Let ${\cal F}$ be a foliation on $M$.
Prove that there exists a continuous map 
$\pi:\; M \arrow Z$ with all leaves of ${\cal F}$ obtained as
$\pi^{-1}(z)$ for some $z\in Z$ and $U\subset Z$ open 
if and only if $\pi^{-1}(U)$ is open.
\ez

\definition
In this case $Z$ is called {\bf the leaf space} of ${\cal F}$.
\ed

\exercise
\enum
\ite[!] Let ${\cal F}$ be a foliation on $M$ with all leaves compact.
Prove that in this case the leaf space of ${\cal F}$ is Hausdorff.
\ite[**] Is this true for all foliations with closed leaves?
\ee
\ez

\exercise
Find a foliation with all leaves dense.
\ez

\exercise[!]
Find a foliation with all leaves closed, but not all of them diffeomorphic.
\ez

\exercise[!]
Let ${\cal F}$ be a foliation with compact leaves
on a compact manifold $M$. Prove that its leaf space
is smooth, or find a counterexample.
\ez

\definition
A foliation on $M$ is called {\bf fiber bundle}
if all its leaves are closed and the
projection $M \arrow Z$ to its leaf space is 
locally trivial.
\ed

\exercise[!]
Let $(M, \omega)$ be a compact symplectic
manifold and $\pi:\; M \arrow Z$ a fiber bundle.
Assume that $\omega$ restricted to fibers of $\pi$
vanishes (in this case the fibers are called
{\bf Lagrangian submanifolds}, and $\pi$
{\bf a Lagrangian fibration}). Prove that
all fibers of $\pi$ have trivial tangent bundle.
\ez

\exercise
Let ${\cal F}$ be a foliation on $M$,
and $T{\cal F}$ the sheaf of all vector fields
tangent to leaves of ${\cal F}$.
\enum
\ite
Prove that $T{\cal F}\subset TM$ is a sub-bundle of $TM$.
\ite[!] Prove that the sub-bundle $T{\cal F}\subset TM$ uniquely determines the
foliation ${\cal F}$.
\ite[!] Prove that $[T{\cal F}, T{\cal F}]\subset T{\cal F}$.
\ite[!] (Frobenius theorem) Prove that any sub-bundle
$B\subset TM$ such that $[B,B]\subset B$ is tangent to a 
certain foliation determined by $B$.
\ee
\ez

\hint To prove the Frobenius theorem,
use the exercises from Handout 11.
\eh

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Basic forms}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\definition
Let ${\cal F}$ be a foliation and $B= T{\cal F}$
its tangent bundle. A differential form $\eta\in \Lambda^* M$
is called {\bf basic} with respect to $B$ if for all 
vector fields $X\in B$, one has $\Lie_X \eta =0$ and $\eta\cntrct X=0$.
\ed

\exercise
Prove that a closed form is basic if $\eta\cntrct X=0$
for all $X\in B$.
\ez

\exercise
Let $X_1, ... X_n \subset B$ be vector fields generating $B$
over $C^\infty M$, and $\eta$ a differential form such that 
$\Lie_{X_i} \eta =0$ and $\eta\cntrct {X_i}=0$ for all $i$.
Prove that $\eta$ is basic.
\ez

\remark This exercise is non-trivial, because
the Lie derivative $\Lie_X \eta$ is not $C^\infty$-linear in $X$.
\er

\exercise
Let $\pi:\; M \arrow Z$ be a differentiable
map of smooth manifolds with differential surjective
everywhere (further on, such maps will be called
{\bf smooth maps}). Prove that $\pi$ is {\bf open},
that is, the image $\pi(U)$ of an open set is always open.
\ez

\definition
Let $\pi:\; M \arrow Z$ be a smooth map and
$E$ a vector bundle on $Z$, considered as a 
locally free sheaf of $C^\infty Z$-modules.
Consider the sheaf-theoretic pullback
$\pi^\bullet E$, with sections of 
$\pi^\bullet E$ over an open subset $U\subset Z$
given by $\pi^\bullet E(U)= E(\pi(U))$.
{\bf The pullback} of the vector bundle
$E$ is $\pi^\bullet E\otimes_{\pi^\bullet C^\infty Z}C^\infty M$.
It is not hard to see that this is also a vector bundle,
of the same rank as $E$.
\ed

\exercise
Let $Z$ be the leaf space of the foliation ${\cal F}$;
we assume that the projection $\pi:\; M \arrow Z$
is smooth. 
\enum
\ite Let $\Lambda^*_\pi(M)$ be the bundle
of all forms $\eta\in \Lambda^* M$ such that $\eta\cntrct X=0$.
Prove that $\Lambda^*_\pi(M)=\pi^* \Lambda^*(M)$.
\ite For any bundle $E$ on $Z$, represent the sections
of $\pi^* E$ by linear combinations of $f\otimes \pi^\bullet e$,
where $f\in C^\infty M$, $e$ is a section of $E$, 
and $\pi^\bullet e$ the corresponding section of $\pi^\bullet E$.
For any $X\in B$, define $\Lie_X (f\otimes \pi^\bullet e):= 
\Lie_X (f) \otimes \pi^\bullet e$. Prove that this 
map satisfies the Leibitz rule. Prove that for all sections $e\in E$, one has
\[
e\in \pi^\bullet E \Leftrightarrow \Lie_X e=0\ \ \  \forall X\in B.
\]
\ite[!] Prove that a form $\eta$ is basic if and only if
$\eta$ lies in $\pi^\bullet\Lambda^*(Z)\subset \Lambda^*_\pi(M)
\subset \Lambda^*(M)$. 
\ite[!] Prove that the space of basic forms on $M$ is naturally
isomorphic to $\Lambda^* Z$.
\ee
\ez

\exercise[**]
For any given $0<i< n$ 
find a foliation ${\cal F}$ of codimension $n$
on a compact manifold $M$ such that all
basic $i$-forms vanish.
\ez

\exercise[*]
Denote basic forms by $\Lambda^*_B(M)$.
Prove that the de Rham differential of a basic form
is again basic. {\bf Basic cohomology}
is the quotient $\frac{\ker d\restrict {\Lambda^*_B(M)}}{d(\Lambda^*_B(M))}$.
Prove that the space of basic cohomology of a compact manifold
is finite-dimensional, for any foliation.
\ez


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{The twisted differential $d^c$}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\definition
Let $(M,I)$ be an almost complex manifold and
$T^{1,0}(M) \oplus T^{0,1}(M)$ the Hodge decomposition.
Consider the corresponding Frobenius form
$N \in \Hom(\Lambda^2(T^{1,0}(M)), T^{0,1}(M))$.
This map is called {\bf the Nijenhuis tensor}.
An almost complex structure is called
{\bf (formally) integrable} if its Nijenhuis tensor vanishes.
\ed

\remark
Please solve the exercises in this subsection
without the use of Newlander-Nirenberg theorem.
\er

\exercise
Let $M$ be an almost complex manifold, and
\[ d^{i,j}:\; \Lambda^{p, q}(M)\arrow \Lambda^{p+i, q+j}(M), \ \ \ i+j=1\]
be the Hodge component of de Rham differential. 
\enum
\ite[!]
Prove that
$d^{i, j}=0$ unless $(i, j)=(0,1), (1,0), (2, -1)$ or $(-1, 2)$.
\ite[!] Prove that the operators $d^{2,-1}$ and $d^{-1,2}$ are $C^\infty(M)$-linear.
\ite[!] Prove that $d^{2,-1}=d^{-1,2}=0$ when the almost complex structure on
$M$ is integrable.
\ite[*] Prove that the map $d^{2, -1}:\; \Lambda^{0,1}(M) \arrow \Lambda^{2,0}(M)$
is dual to the Nijenhuis tensor.
\ee
\ez

\definition
We extend $I$ to differential forms
multiplicatively, $I\restrict{\Lambda^{p,q}(M)}= (\1)^{p-q}$.
Let $d^c:= I d I^{-1}$. This operator is called {\bf the twisted differential}.
\ed

\exercise[!]
Prove that $I$ is integrable if and only if $dd^c = - d^c d$.
\ez

\exercise[!]
Prove that $I$ is integrable if and only if $d^{1,0}= \frac{d- \1 d^c}{2}$.
Prove that in this case $d^{0,1}= \frac{d+ \1 d^c}{2}$.
\ez

\definition
The operators $d^{1,0}$, $d^{0,1}$ on a complex manifold
are denoted $\6:\; \Lambda^{p, q}(M)\arrow\Lambda^{p+1, q}(M)$
and $\bar\6:\; \Lambda^{p, q}(M)\arrow\Lambda^{p, q+1}(M)$.
\ed

\exercise
Prove that $-2\1\6\bar\6 =  dd^c$.
\ez

\definition
The operator $dd^c$ is called {\bf the pluri-Laplacian},
and function $f$ with $dd^c(f)=0$ is called {\bf pluri-harmonic}.
\ed

\exercise
Let $B$ be an open subset in $\C$ and $\omega= dx \wedge dy$
the standard volume form. Prove that $dd^c(f)= \Delta(f) \omega$,
where $\Delta$ is the Laplacian.
\ez

\definition
Recall that a $\C$-valued function $f$ on a complex manifold is called
{\bf anti-holomorphic} if $\bar f$ is holomorpic.
\ed

\exercise
Let $f$ be a sum of a holomorphic and an anti-holomorphic
function is pluri-harmonic.
\ez

\exercise[!]
Let $f$ be a smooth real function on $\C^n$.
Suppose that $f$ is pluri-harmonic. Prove that 
restriction of $f$ to any complex curve $Z\subset \C^n$
is a real part of a holomorphic function.
\ez

\exercise
Let $f$ be a real pluri-harmonic function
on a poly-disc $U$ in $\C^n$. 
\enum \ite[*]
Using the Poincar\'e-Dolbeault-Grothendieck lemma,
prove that $f$ is a real part of a holomorphic function.
\ite[*] Deduce from the Poincar\'e-Dolbeault-Grothendieck lemma
 that a function is pluri-harmonic if and only
if it is represented locally as a sum of a holomorphic 
and antiholomorphic function.
\ee
\ez

\exercise[!]
Find a complex manifold $M$ and a real pluriharmonic
function $f$ which is not a real part of a holomorphic
function on $M$.
\ez





\end{document}

