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\begin{document}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf Complex geometry\\[15mm]
\small lecture 1: complex manifolds}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\\[14mm]

{\small Misha Verbitsky } 
\\[20mm]

{\tiny\bf HSE, room 306, 16:20,
\\[2mm] 
September 23, 2020
}
\end{center}

\newpage

{\bf \blue Complex structure on a vector space}

\definition
Let $V$ be a vector space over $\R$, and $I:\; V \arrow V$
an automorphism which satisfies $I^2 = - \Id_V$. 
Such an automorphism is called {\bf\blue a complex structure
operator} on $V$.

{\bf \blue We extend the
action of $I$ on the tensor spaces 
$V\otimes V \otimes ... \otimes V \otimes 
V^*\otimes V^* \otimes ... \otimes V^*$ by multiplicativity:}
$I(v_1 \otimes ... \otimes w_1 \otimes ... \otimes w_n)=
I(v_1) \otimes ... \otimes I(w_1)\otimes ... \otimes I(w_n)$.


{\bf \red Trivial observations:}\\
1. {\bf \purple The eigenvalues $\alpha_i$ of $I$ are $\pm \1$.}
Indeed, $\alpha_i^2=-1$.

2. {\bf \purple $V$ admits an $I$-invariant, positive definite
scalar product (``metric'') $g$.}
Take any metric $g_0$, and let $g:= g_0 + I(g_0)$.

3. {\bf \purple $I$ is orthogonal for such $g$.}\\
Indeed, $g(Ix,Iy)=g_0(x,y)+ g_0(Ix,Iy)=g(x,y)$.


4. {\bf \purple $I$ diagonalizable over $\C$.} 
Indeed, any orthogonal matrix is diagonalizable.

5. {\bf \purple There are as many $\1$-eigenvalues as there are
$-\1$-eigenvalues.} 

\newpage

{\bf \blue Complex structure operator in coordinates}

This implies that in an appropriate basis in $V \otimes_\R \C$, the 
complex structure operator is diagonal, as follows:

\[\left[
\begin{array}{c|c}
\begin{array}{cccc}
\1 &  & &\\
  &  \1 & & \\
  &    & \ddots &  \\
  &    &  &  \1 
\end{array}
 & 0 \\ \hline
0 & \begin{array}{cccc}
-\1 &  & &\\
  &  -\1 & & \\
  &    & \ddots &  \\
  &    &  &  -\1 
\end{array}
\end{array}\right]
\]
We also obtain its normal form in a real basis:
\[ \begin{bmatrix}
0 & -1 \\
1 &  0 \\
  &    & 0 & -1 \\
  &    & 1 &  0 \\
  &    &   &   & \ddots   \\
  &    &   &   & & \ddots \\
  &    &   &   & &       & 0 & -1 \\
  &    &   &   & &       & 1 &  0
\end{bmatrix}
\]

\newpage

{\bf \blue Hodge decomposition}

\definition
Let $(V,I)$ be a space equipped with a complex structure.
{\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.

\remark In the same way one defines the Hodge decomposition
on the dual space $V^*$.

{\bf \green Remark 1:} The space $V^{1,0} \subset V\otimes_\R \C$
{\bf \purple uniquely determines the operator $I$.} Indeed,
$I=\1$ on $V^{1,0}$ and $I=-\1$ on $V^{0,1}$.
This gives {\bf \red a bijection between the set of complex
structures on $V$ and the set of subspaces
$W \subset V\otimes_\R \C$ such that
$\dim_\C W = \frac 1 2 \dim_\R V$ and
$W \cap \bar W =0$.}



\newpage

{\bf \blue Hermitian structures}


\definition
Let $(V,I)$ be a space equipped with a complex structure.
{\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.

% Denote by $\nu$ the {\bf \blue real structure operator,}
%$\nu(\sum \lambda_i w_i)=\sum \bar \lambda_i w_i$, where $w_i\in V$
%is a basis. Then $\nu(I(z))=I(\nu(z))$, that is, {\bf \red $I$ is real}. 
%For any $\1$-eigenvector
%$w$, one has $I(\nu(w))=\nu(I(w))=\nu(\1 w)=-\1 w$,
%hence {\bf \purple 
%$\nu$ exchanges $\1$-eigenvectors and $-\1$-eigenvectors.}


\definition An $I$-invariant positive definite
scalar product on $(V,I)$ is called
{\bf \blue an Hermitian metric}, and $(V,I,g)$ -- 
an Hermitian space.

\remark
Let $I$ be a complex structure operator on a real 
vector space $V$, and $g$ -- a Hermitian metric. Then
{\bf \red
the bilinear form $\omega(x,y) := g(x, Iy)$ is skew-symmetric.}
Indeed, $\omega(x,y) = g(x, Iy) = g(Ix, I^2y) = -g(Ix, y) = -\omega(y, x)$.

\definition
A skew-symmetric form
 $\omega(x,y)$ is called {\bf \blue an Hermitian form on $(V,I)$}.

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

\newpage

{\bf \blue Holomorphic functions}

\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 $I(d/dx_i)=d/dy_i$, $I(d/dy_i)=-d/dx_i$.

\example 
In complex dimension 1, {\bf \purple almost complex structure
is the same as conformal structure with orientation}
{\bf \red (prove it).}

\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 
the unique $G_2$-invariant almost complex structure.


\newpage

{\bf \blue Holomorphic functions on $\C^n$}

\theorem
Let $f:\; M \arrow \C$ be a differentiable function on 
an open subset $M\subset \C^n$, with almost
complex structure as above. {\bf \red Then TFAE:}\\
\ \ (1) {\bf \purple $f$ is holomorphic}.\\
\ \ (2) The differential $df:\; TM \arrow \C$,
considered as a form on the vector space $T_x M=T_x\C^n=\C^n$
{\bf \purple is $\C$-linear.} \\
\ \ (3) For any complex affine line $L\in \C^n$,
the restriction $f\restrict L=\C$ is {\bf \purple holomorphic (complex analytic) 
as a function of one complex variable.}\\
\ \ (4) $f$ is expressed as a sum of Taylor series around any point $(z_1, ..., z_n)\in M$.
%\[
%f(z_1+t_1, z_2+ t_2, ..., z_n +t_n)= 
%\sum_{i_1, ..., i_n}a_{i_1, ..., i_n} t_1^{i_1}t_2^{i_2}...t_n^{i_n}.
%\]
%(here we assume that the complex numbers $t_i$ satisfy
%$|t_i|<\epsilon$, where $\epsilon$ depends on $f$ and $M$). 

\proof (1) and (2) are tautologically equivalent.
Equivalence of (1) and (3) is also clear, because a restriction of
 $\theta \in \Lambda^{1,0}(M)$ to a line is a $(1,0)$-form on a line,
and, conversely, if $df$ is of type (1,0) on each complex line, it is
of type (1,0) on $TM$, which is implied by the following linear-algebraic 
observation.

\lemma
Let $\eta\in V^*\otimes \C$ be a complex-valued linear form
on a vector space $(V,I)$ equipped with a complex structure.
{\bf \purple Then $\eta\in \Lambda^{1,0}(V)$ if and only if its restriction
to any $I$-invariant 2-dimensional subspace $L$ belongs to 
$\Lambda^{1,0}(L)$.}

\exercise {\bf \red Prove it.}

(4) clearly implies (2). 
(1) implies (4) by Cauchy formula.

%\question Who remembers the proof of Cauchy formula?

\newpage

{\bf \blue Sheaves}

\definition
A {\bf\blue presheaf of functions} on a
topological space $M$ is a collection of subrings 
${\cal F}(U)\subset C(U)$ in
the ring $C(U)$ of all functions on $U$, for each open 
subset $U\subset M$, such that the restriction of
every $\gamma\in{\cal F}(U)$ to an open subset $U_1\subset U$ belongs to
${\cal F}(U_1)$. 


\definition
A presheaf of functions
${\cal F}$ is called {\bf\blue a sheaf of functions} 
if these subrings
satisfy the following condition. Let $\{U_i\}$ be a cover of an open
subset $U\subset M$ (possibly infinite)
and $f_i\in{\cal F}(U_i)$ a family of
functions defined on the open sets of the cover and 
compatible on the pairwise intersections:
$$f_i|_{U_i\cap U_j}=f_j|_{U_i\cap U_j}$$
for every pair of members of the cover. {\bf \purple Then there exists
$f\in{\cal F}(U)$ such that $f_i$ is the restriction of $f$ to $U_i$ for
all $i$.}




\newpage

{\bf \blue Sheaves and exact sequences}


\remark
{\bf \purple A presheaf of functions} is a collection of subrings
of functions on open subsets, compatible with
restrictions. {\bf\purple A sheaf of fuctions is a presheaf
allowing ``gluing''} a function on a bigger open set
if its restrictions to smaller open sets are compatible.


\definition
A sequence $A_1 \arrow A_2 \arrow A_3 \arrow ...$
of homomorphisms of abelian groups or vector spaces
is called {\bf\blue exact} if the image of each map is
the kernel of the next one.


\claim
A presheaf
${\cal F}$ is a sheaf if and only if for every cover $\{U_i\}$ of
an open subset $U\subset M$, {\bf \red the sequence of restriction maps
$$0\to{\cal F}(U)\to\prod\limits_i{\cal F}(U_i)\to\prod\limits_{i\ne
j}{\cal F}(U_i\cap U_j)$$
is exact,} with $\eta\in {\cal F}(U_i)$ mapped to $\eta\restrict{U_i\cap U_j}$
and $-\eta\restrict{U_j\cap U_i}$. 


\newpage

{\bf \blue Sheaves and presheaves: examples}

{\bf \green Examples of sheaves:}

* Space of continuous functions

* Space of smooth functions, any differentiability class

* Space of real analytic functions


{\bf \green Examples of presheaves which are not sheaves:}

* Space of constant functions {\bf \purple (why?)}

* Space of bounded functions {\bf \purple (why?)}

\newpage

{\bf \blue Ringed spaces}

A {\bf\blue ringed space} $(M,{\cal F})$ is a
topological space equipped with a sheaf of
functions. A~{\bf \blue morphism}
$(M,{\cal F})\stackrel\Psi\longrightarrow(N,{\cal F}')$ of ringed spaces
is a continuous map $M\stackrel\Psi\longrightarrow N$ such that, for
every open subset $U\subset N$ and every function $f\in{\cal F}'(U)$, the
function $\psi^* f:=f\circ\Psi$ belongs to the ring
${\cal F}\big(\Psi^{-1}(U)\big)$. An {\bf\blue  isomorphism} of ringed spaces
is a homeomorphism $\Psi$ such that $\Psi$ and $\Psi^{-1}$ 
are morphisms of ringed spaces.

\example
Let $M$ be a manifold of class $C^i$ and let
$C^i(U)$ be the space of functions of this class. 
{\bf \purple Then $C^i$ is a sheaf of functions, and 
$(M, C^i)$ is a ringed space.}


\remark
Let $f:\; X \arrow Y$ be a smooth map of smooth manifolds.
Since a pullback $f^*\mu$ of a smooth function $\mu\in C^\infty(M)$
is smooth, {\bf \purple a smooth map of 
smooth manifolds defines a morphism of ringed spaces.}


{\bf \red Converse is also true:}


\newpage

{\bf \blue Ringed spaces and smooth maps}

\claim
Let $(M, C^i)$ and $(N,C^i)$ be manifolds of class $C^i$.
Then {\bf \red there is a bijection between smooth maps $f:\; M \arrow N$
and the morphisms of corresponding ringed spaces.}

{\bf \green Proof:}
Any smooth map induces a morphism of ringed spaces.
Indeed, {\bf \purple a composition of smooth functions is smooth,
hence a pullback is also smooth.}

Conversely, let $U_i \arrow V_i$ be a restriction
of $f$ to some charts; to show that $f$ is smooth,
it would suffice to show that $U_i \arrow V_i$ is smooth.
However, we know that a pullback of any smooth function
is smooth. {\bf \purple Therefore, Claim is implied by the following
lemma.}

\lemma
Let $M,N$ be open subsets in $\Bbb R^n$ and let
$f:\; M\to N$ map such that a pullback of any
function of class $C^i$ belongs to $C^i$. {\bf \red Then 
$f$ is of class $C^i$.}

{\bf \green Proof:} Apply $f$ to coordinate functions. \endproof


\newpage

{\bf \blue Smooth manifolds defined through  sheaves}

As we have just shown, this definition is equivalent to
the previous one.

\definition
Let $(M,{\cal F})$ be a topological manifold
equipped with a sheaf of functions. It is said to be a {\bf\blue smooth
manifold of class} $C^\infty$ or $C^i$ if every point in
$(M,{\cal F})$ has an open neighborhood isomorphic to the ringed space
$(\Bbb R^n,{\cal F}')$, where ${\cal F}'$ is a ring of 
functions on $\Bbb R^n$ of this class.

\definition
{\bf \blue A chart}, or 
{\bf\blue a coordinate system} on an open subset $U$
of a manifold $(M,{\cal F})$ is an isomorphism between $(U,{\cal F})$ and
an open subset in $(\Bbb R^n,{\cal F}')$, where ${\cal F}'$ are functions
of the same class on $\Bbb R^n$.

\definition
{\bf\blue Diffeomorphism} of smooth manifolds
is a homeomorphism $\phi$ which induces an isomorphim
of ringed spaces, that is, $\phi$ and $\phi^{-1}$ map 
(locally defined) smooth functions to smooth functions.


\newpage

{\bf \blue Complex manifolds}

\definition
{\bf \blue A holomorphic function} on $\C^n$ is a function
$f:\; \C^n \arrow \C$ such that $df$ is complex linear, that
is $df\in \Lambda^{1,0}(M)$.

\remark
Holomorphic functions form a sheaf.

\definition
{\bf \blue A complex manifold} $M$ is a ringed space which is locally isomorphic
to an open ball in $\C^n$ with a sheaf of holomorphic functions.

\remark 
In other words, {\bf \purple $M$ is covered with open balls embedded to $\C^n$}
and transition functions (being coordinate functions for one
ball considered in other coordinate system) {\bf \purple are holomorphic.}

\newpage

{\bf \blue Complex manifolds and almost complex manifolds}

\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.}


{\bf \green Another 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.

\exercise Prove that {\bf \purple this definition is equivalent to the one
with sheaves}.


\newpage

{\bf \blue Integrability of almost complex structures}


\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 \red  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 $\calo_M$, and $\calo_M$ is determined by $I$ as explained above.
Therefore, an integrable almost complex structure defines a complex structure.
Conversely, every complex structure gives a sub-bundle
in $\Lambda^{1,0}(M)=d\calo_M\subset \Lambda^1(M,\C)$, and
{\bf \purple  such a sub-bundle defines an almost complex structure 
by Remark 1.}
\endproof


\end{document}


