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\begin{document}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf Hodge theory \\[15mm]
\small lecture 11: K\"ahler manifolds}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\\[14mm]
{\scriptsize NRU HSE, Moscow} \\[15mm]
{\small Misha Verbitsky, February 28, 2018 } 
\end{center}


\newpage

{\bf \blue Almost complex manifolds (reminder)}

\definition
Let $I:\; TM \arrow TM$ be an endomorphism of a tangent bundle
satisfying $I^2=-\Id$. Then $I$ is called {\bf \blue
almost complex structure operator}, and the pair $(M,I)$
{\bf \blue an almost complex manifold}.

\example
$M=\C^n$, with complex coordinates $z_i=x_i + \1 y_i$,
and $I(d/dx_i)=d/dy_i$, $I(d/dy_i)=-d/dx_i$.

\definition
Let $(V,I)$ be a space equipped with a complex structure $I:\; V \arrow V$,
$I^2 =-\Id$. {\bf \blue The Hodge decomposition} 
$V\otimes_\R \C:= V^{1,0}\oplus V^{0,1}$ is defined in such a way that
$V^{1,0}$ is a $\1$-eigenspace of $I$, and $V^{0,1}$ a $-\1$-eigenspace.


\definition
A function $f:\; M \arrow \C$ on an almost
complex manifold is called {\bf \blue holomorphic}
if $df \in \Lambda^{1,0}(M)$.

\remark
For some almost complex manifolds, {\bf \red there are no holomorphic
functions at all}, even locally. 

%Example: $S^6$ with a certain
%canonical ($G_2$-invariant) almost complex structure.

\newpage

{\bf \blue Complex manifolds and almost complex manifolds (reminder)}

\definition {\bf \blue Standard almost complex structure}
is $I(d/dx_i)=d/dy_i$, $I(d/dy_i)=-d/dx_i$ on $\C^n$ with
complex coordinates $z_i=x_i+\1 y_i$.

\definition
A map $\Psi:\; (M,I)\arrow (N,J)$ 
from an almost complex manifold to an almost complex
manifold is called {\bf \blue holomorphic} if 
$\Psi^*(\Lambda^{1,0}(N))\subset \Lambda^{1,0}(M)$.

\remark
This is the same as $d\Psi$ being complex linear; 
for standard almost complex structures, {\bf \purple this is the same
as the coordinate components of $\Psi$ being holomorphic functions.}


\definition
{\bf \blue A complex manifold} is a manifold equipped with an atlas
with charts identified with open subsets of $\C^n$ and transition
functions holomorphic.


\newpage

{\bf \blue Integrability of almost complex structures (reminder)}


\definition
An almost complex structure $I$ on a manifold is called {\bf\blue integrable}
if any point of $M$ has a neighbourhood $U$ diffeomorphic 
to an open subset of $\C^n$, in such a way that 
the almost complex structure $I$ is induced by the standard
one on $U\subset \C^n$.

\claim 
{\bf \purple Complex structure on a manifold $M$ uniquely determines
an integrable almost complex structure, and is determined by 
it.}

\proof
Complex structure on a manifold $M$ is determined by the sheaf of holomorphic
functions $\calo_M$, because $d\calo_M$ generates $\Lambda^{1,0}(M)$,
 and $\calo_M$ is determined by $I$, because $\calo_M=\{f\ \ |\ \ df\in \Lambda^{1,0}(M)\}$.
\endproof


\newpage

{\bf \blue Formal integrability (reminder)}


\definition
An almost complex structure $I$ on $(M,I)$
is called {\bf\blue formally integrable}
if $[T^{1,0}M, T^{1,0}]\subset T^{1,0}$, that is,
if $T^{1,0}M$ is involutive.

\definition The Frobenius form $\Psi\in \Lambda^{2,0}M\otimes TM$
is called {\bf \blue the Nijenhuis tensor}.

\claim 
{\bf \purple 
If a complex structure $I$ on $M$ is integrable, it is formally integrable.}

\proof Locally, the bundle $T^{1,0}(M)$ is generated by $d/dz_i$, where $z_i$ are
complex coordinates. These vector fields commute, hence satisfy
$[d/dz_i, d/dz_j]\in T^{1,0}(M)$. This means that the Frobenius
form vanishes. \endproof

\theorem
{\bf \blue (Newlander-Nirenberg)}\\
{\bf\red A complex structure $I$ on $M$ is integrable if and only if it is
formally integrable.}

\remark {\bf \purple In dimension 1, formal integrability is automatic.}
Indeed, $T^{1,0}M$ is 1-dimensional, hence all skew-symmetric
2-forms on $T^{1,0}M$ vanish.


\newpage

{\bf \blue Riemannian manifolds}

\definition
Let $h\in \Sym^2 T^* M$ be a symmetric 2-form on
a manifold which satisfies $h(x,x)>0$ for any
non-zero tangent vector $x$. Then $h$ is called
{\bf \blue Riemannian  metric}, of {\bf \blue Riemannian
structure}, and $(M,h)$
{\bf \blue Riemannian manifold}.

\definition For any $x, y\in M$, and any path 
$\gamma:\; [a, b] \arrow M$ connecting $x$ and $y$,
consider {\bf \blue the length} of $\gamma$
defined as $L(\gamma) = \int_\gamma |\frac{d\gamma}{dt}| dt$,
where $|\frac{d\gamma}{dt}|= h(\frac{d\gamma}{dt},\frac{d\gamma}{dt})^{1/2}$.
Define {\bf \blue the geodesic distance}
as $d(x, y) = \inf_\gamma L(\gamma)$,
where infimum is taken for all paths 
connecting $x$ and $y$.

\exercise Prove that the {\bf \purple geodesic distance 
satisfies triangle inequality and defines metric on $M$.}


\exercise
Prove that {\bf \purple 
this metric induces the standard topology on $M$.}

\example 
Let $M=\R^n$, $h= \sum_i dx_i^2$. {\bf \purple Prove that the
geodesic distance coincides with $d(x, y)= |x-y|$.}

\exercise 
Using partition of unity, {\bf \red prove that any manifold admits
a Riemannian structure.}


\newpage

{\bf \blue K\"ahler manifolds}

{\bf\green DEFINITION:} An Riemannian metric $g$ on
an almost complex manifiold $M$ is called 
{\bf \blue Hermitian} if $g(Ix, Iy)= g(x,y)$.
In this case, $g(x, Iy)= g(Ix, I^2y) = - g(y, Ix)$,
hence $\omega(x,y):= g(x, Iy)$ is skew-symmetric.


\remark
Given any Riemannian metric $g$ on an almost complex
manifold, {\bf \purple a Hermitian
metric $h$ can be obtained as $h= g + I(g)$,
where $I(g)(x,y)=g(I(x),I(y))$.}


{\bf\green DEFINITION:} The differential 
form $\omega\in \Lambda^{1,1}(M)$ is called
{\bf \blue the Hermitian form} of $(M,I,g)$.

\remark It is $U(1)$-invariant, hence {\bf \purple of Hodge type (1,1)}.

\remark 
In the triple $I, g, \omega$, {\bf \purple each element can 
recovered from the other two.}


{\bf\green DEFINITION:} A complex Hermitian manifold $(M,I,\omega)$
is called {\bf \blue K\"ahler} if $d\omega=0$. 
The cohomology class $[\omega]\in H^2(M)$ of a form $\omega$ 
is called {\bf \blue the K\"ahler class} of $M$, and
$\omega$ {\bf \blue the K\"ahler form}. 


\newpage

{\bf \blue Homogeneous spaces}

\definition
{\bf \blue A Lie group} is a smooth manifold equipped with a group
structure such that the group operations are smooth.
Lie group $G$ {\bf \blue acts on a manifold $M$}
if the group action is given by the smooth map
$G \times M \arrow M$. 


\definition
Let $G$ be a Lie group acting on a manifold $M$ transitively.
Then $M$ is called {\bf \blue a homogeneous space}.
For any $x\in M$ the subgroup $\St_x(G)=\{g\in G\ \ |\ \ g(x)=x\}$
is called {\bf \blue stabilizer of a point $x$}, or
{\bf \blue isotropy subgroup}.

\claim
For any homogeneous manifold $M$ with transitive
action of $G$, {\bf \purple one has $M=G/H$,} 
where $H=\St_x(G)$ is an isotropy subgroup.

\proof The natural surjective map $G\arrow M$ putting $g$ to $g(x)$
identifies $M$ with the space of conjugacy classes $G/H$.
\endproof

\remark Let $g(x)=y$. Then $\St_x(G)^g=\St_y(G)$:
{\bf \purple all the isotropy groups are conjugate.}

\newpage

{\bf \blue Isotropy representation}

\definition 
Let $M=G/H$ be a homogeneous space, $x\in M$ and
$\St_x(G)$ the corresponding stabilizer group.
The {\bf \blue isotropy representation}
is the natural action of $\St_x(G)$
on $T_x M$.

\definition
A tensor $\Phi$ on a homogeneous manifold $M=G/H$ 
is called {\bf \blue invariant} if it is mapped
to itself by all diffeomorphisms which come from
$g\in G$.

\remark
Let $\Phi_x$ be an isotropy invariant tensor
on $\St_x(G)$. For any $y\in M$ obtained as $y=g(x)$,
consider the tensor $\Phi_y$ on $T_y M$ obtained as
$\Phi_y:=g(\Phi)$. The choice of $g$ is not unique,
however, for another $g'\in G$ which satisfies $g'(x)=y$,
we have $g=g'h$ where $h\in \St_x(G)$. Since
$\Phi$ is $h$-invariant, {\bf \purple the tensor $\Phi_y$
is independent from the choice of $g$.}

We proved

{\bf \green Theorem 1:}
{\bf \red $G$-invariant tensors on $M=G/H$ are in bijective
correspondence with isotropy invariant tensors on
$T_x M$,} for any $x\in M$.
\endproof

\newpage

{\bf \blue Representations acting transitively on a sphere}

\theorem
Let $G$ be a group acting on a vector space $V$.
Suppose that $G$ acts transitively on a unit sphere
$\{ x\in V \ \ |\ \ g(x)=1\}$. {\bf \red Then a $G$-invariant
bilinear symmetric form is unique up to a constant multiplier.}

\pstep Since $G$ preserves the sphere, which is a level
set of the quadratic form $g$, $g$ is $G$-invariant.

{\bf \green Step 2:} For any $G$-invariant 
quadratic form $g'$, the function $x \arrow \frac {g'(x)}{g(x)}$
is constant on spheres and invariant under homothety, hence it is
constant.
\endproof

\exercise
Let $V$ be a representation of $G$, and suppose
$G$ acts transitively on a sphere. {\bf \purple Prove that
$V$ is an irreducible representation.}

\exercise
Prove the {\bf \blue Schur lemma:} 
let $V$ be an irreducible representation of $G$ over $\R$,
and $g$ a $G$-invariant positive definite bilinear symmetric
form. {\bf \red Then any $G$-invariant bilinear symmetric
form is proportional to $g$.}




\newpage

{\bf \blue Fubini-Study form}


\example
Consider the natural action of the unitary group
$U(n+1)$ on $\C P^n$. The stabilizer of a point
is $U(n)\times U(1)$.

\theorem
There exists an $U(n+1)$-invariant Riemann
form on $\C P^n$. Moreover, {\red \bf such a form is unique
up to a constant multiplier, and K\"ahler.} 

\remark This Riemannian structure is called {\bf \blue
the Fubini-Study metric}, and its Hermitian form 
{\bf \blue the Fubini-Study form}.

\pstep To construct a  $U(n+1)$-invariant Riemann
form on $\C P^n$, we take a $U(n)$-invariant form
on $T_x \C P^n$ and apply Theorem 1.
A $U(n)$-invariant form on $T_x \C P^n$ exists,
because it is a standard representation.

{\bf \green Step 2:} Uniqueness follows because 
the isotropy group acts transitively on a sphere.
\endproof

\claim {\bf \red The Fubini-Study form is closed,} and the corresponding
metric is K\"ahler.

\proof Let $\omega$ be a Fubini-Study form. Then $d\omega$
is an isotropy-invariant 3-form on $T_x \C P^n$. However,
the isotropy group contains $-\Id$, {\bf \red hence all isotropy-invariant
odd tensors vanish.} \endproof


\newpage

{\bf \blue Projective manifolds}

\definition
Let $M$ be a complex manifold, and $X\subset M$ a smooth
submanifold. It is called {\bf \blue a complex submanifold}
if $I(TX)\subset TX$, and the map $X \hookrightarrow M$
{\bf \blue a complex embedding}. A complex manifold which
admits a complex embedding to $\C P^n$ is called {\bf \blue
a projective manifold}.

\remark {\bf \purple
A complex submanifold of a K\"ahler manifold is K\"ahler.}
Indeed, restriction of a Hermitian metric is Hermitian, and
restriction of a closed form is closed. Therefore, 
{\bf \red all projective manifolds are K\"ahler}.

\definition
A subvariety of $\C P^n$ is called {\bf \blue complex algebraic} 
if can be obtained as common zeroes of a system of homogeneous
polynomial equations.

\theorem {\bf \blue (Chow theorem)} {\bf \red All complex submanifolds
in $\C P^n$ are complex algebraic.}

\newpage

{\bf \blue Kodaira embedding theorem}

\definition
{\bf \blue K\"ahler class} of a K\"ahler manifold
is the cohomology class $[\omega]\in H^2(M, \R)$ of its K\"ahler form.
We say that $M$ {\bf \blue has integer K\"ahler class}
if $[\omega]$ belongs to the image of $H^2(M, \Z)$ in 
$H^2(M, \R)$ 

\remark
$H^2(\C P^n, \R)=\R$. This implies that
{\bf \red the cohomology class of Fubini-Study form can be chosen integer.}
In particular, {\bf \red all projective manifolds admit
K\"ahler structures with integer K\"ahler classes.}

\theorem {\bf \blue (Kodaira embedding theorem)}
Let $M$ be a K\"ahler manifold with an integer K\"ahler
class. {\bf \red Then it is projective.}

{\bf \green This theorem will be proven later in these lectures.}



\newpage

{\bf \blue Classes of almost complex manifolds}



\vfill

\centerline{\epsfig{file=classes-mflds.eps,width=0.6\linewidth}}

\vfill



\end{document}
