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\begin{document}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf Complex geometry\\[15mm]
\small lecture 5: Real analytic manifolds}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\\[14mm]

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

{\tiny\bf HSE, room 306, 16:20,
\\[2mm] 
October 7, 2020
}
\end{center}

\newpage

{\bf \blue Real analytic manifolds}

\definition
{\bf \blue A real analytic function} on an open set $U\subset \R^n$
is a function which admits a Taylor expansion near each point
$x\in U$:
\[
f(z_1+t_1, z_2+ t_2, ..., z_n +t_n)= 
\sum_{i_1, ..., i_n}a_{i_1, ..., i_n} t_1^{i_1}t_2^{i_2}...t_n^{i_n}.
\]
(here we assume that the real numbers $t_i$ satisfy
$|t_i|<\epsilon$, where $\epsilon$ depends on $f$ and $M$). 

\remark
Clearly, {\bf \purple real analytic functions constitute a sheaf.}

\definition
A {\bf\blue real analytic manifold} is a ringed
space which is locally isomorphic to an open ball $B\subset \R^n$
with the sheaf of of real analytic functions.


\newpage


{\bf \blue Real structures on complex vector spaces}

\definition
{\bf \blue An involution} is a map $\iota:\; M \arrow M$
such that $\iota^2=\Id_M$.
{\bf \blue A real structure}
on a complex vector space $V=\C^n$
is an $\R$-linear involution $\iota:\; V \arrow V$ such
that $\iota(\lambda x) = \bar\lambda \iota(x)$ for any
$\lambda\in \C$.

\claim
Let $\iota$ be a real structure on a complex vector space $V$,
and $V^\iota\subset V$ the space of $V$-invariant vectors.
{\bf \red Then $\dim_\R V^\iota=\dim_\C V$, and 
the natural map $V^\iota\otimes_\R \C\arrow V$
is an isomorphism.}

\pstep
Let $x_1, ..., x_n$ be a basis in $V^\iota$,
and  $\sum_i \alpha_i x_i=0$ a linear relation in $V$, with 
$\alpha_i \in \C$. Then $0=\iota\left(\sum_i\alpha_i x_i\right)=
\sum_i\bar\alpha_i x_i$. Summing up these two relations, we obtain
$\sum_i \Re\alpha_i x_i=0$. Since $x_i$ are linearly independent
over $\R$, this implies $\Re\alpha_i=0$ for all $i$.
Applying the same argument to $\sum_i \1\alpha_i x_i=0$,
we obtain that $\Im\alpha_i=0$ for all $i$.
Then {\bf \purple
the natural map $V^\iota\otimes_\R \C\arrow V$ is injective.}

{\bf \green Step 2:} This map is also surjective. Indeed,
for any $v\in V$, one has $(v+\iota(v))\in V^\iota$
and $(v-\iota(v))\in V^\iota$, hence $v$ can b
expressed as a linear combination of vectors from $V^\iota$
with complex coefficients.
\endproof


\newpage


{\bf \blue Real structures on complex manifolds}

\definition 
A smooth map $\Psi:\; M \arrow N$ on an almost complex manifold 
$(M,I)$ is called {\bf \blue antiholomorphic} if 
$d\Psi(I)=-I$.  A function $f$ is called {\bf\blue antiholomorphic}
if $\bar f$ is holomorphic.

\exercise Prove that {\bf \purple an antiholomorphic function on $M$ defines
an antiholomorphic map from $M$ to $\C$.}

\exercise Prove that a map $\Psi:\; M \arrow N$ of almost complex
manifolds is antiholomorphic
{\bf \purple if and only if $\Psi^*(\Lambda^{0,1}(N))\subset \Lambda^{1,0}(M)$.}


\exercise
Let $\iota$ be a smooth map from a
complex manifold $M$ to itself. Prove that {\bf \purple $\iota$ is
antiholomorphic if and only if $\iota^*(f)$ is
antiholomorphic for any holomorphic function $f$
on $U\subset M$.}

\definition
{\bf \blue A real structure} on a complex manifold $M$
is an antiholomorphic involution $\tau:\; M \arrow M$.


\example {\bf \purple Complex conjugation defines a real structure on
$\C^n$.}

\newpage


{\bf \blue Fixed points of real structures on manifolds}

\proposition
Let $M$ be a complex manifold and
$\iota:\; M\arrow M$ a real structure.
Denote by $M^\iota$ the fixed point set of $\iota$.
Then, {\bf \red for each $x\in M^\iota$ there exists
a $\iota$-invariant coordinate neighbourhood
with holomorphic coordinates $z_1, ..., z_n$,
such that $\iota^*(z_i)=\bar z_i$.}

\pstep
For each basis of 1-forms
$\nu_1, ..., \nu_n \in \Lambda^{1,0}_x(M)$,
there exists a set of holomorphic coordinate functions
$u_1, ..., u_n$ such that $du_i \restrict x=\nu_i$.
To obtain such a coordinate system, {\bf \purple we chose any
coordinate system $v_1, ..., v_n$ and apply a linear transform
mapping $dv_i\restrict x$ to $\nu_i$.}

{\bf \green Step 2:} The differential $d\iota$
acts on $T_x M$ as a real structure. Using the structure
theorem about real structures, we obtain that any real basis
$\zeta_1, .., \zeta_n$ of $T_x^* M^\iota$ is a complex basis in the complex vector
space $T_x^* M$. Then $\nu_i:=\zeta_i+ \1 I(\zeta_i)\}$ is a
basis in $\Lambda^{1,0}_x(M)$. Choose the coordinate
system $u_1, ..., u_n$ such that $du_i \restrict
x=\nu_i$ (Step 1). {\bf \purple Replacing $u_i$
by $z_i:=u_i+\iota^*(\bar u_i)$, we obtain a holomorphic
coordinate system $z_i$ on $M$} (compare with Theorem 1 in
Lecture 4)  {\bf \purple which satisfies $\iota^*(z_i)=\bar z_i$.}
\endproof


\definition
Let $\{U_i\}$ be an complex atlas on $M$. Assume that
any $U_i$ intersecting $M^\iota$ satisfies the conclusion of
this proposition. Then $\{U_i\}$
is called {\bf \blue compatible 
with the real structure}.

\newpage


{\bf \blue Real analytic manifolds and real structures}


\proposition
Let $M^\iota\subset M$ be a fixed point set of an
antiholomorphic involution $\iota$ on a complex manifold $M$, 
$\{U_i\}$ a complex analytic atlas, and $\Psi_{ij}:\; U_{ij}\arrow
U_{ij}$ the  gluing functions. Assume that the atlas $U_{i}$
is compatible with the real structure, in the sense of the 
previous proposition. {\bf \red Then all $\Psi_{ij}$ are
real analytic on $M^\iota$, and define a real analytic
atlas on the manifold $M^\iota$.}

\proof  All gluing functions from one
coordinate system compatible with the real structure
to another {\bf \purple commute with $\iota$, acting on coordinate
functions as the complex conjugation.} This gives
$\Psi_{ij}(\bar z_i)= \overline {\Psi_{ij}(z_i)}$.
Therefore, $\Psi_{ij}$ preserve $M^\iota$, and
are expressed by real-valued functions on $M^\iota$.
\endproof

\newpage


{\bf \blue Real analytic manifolds and real structures (2)}


\proposition {\bf \red Any real analytic manifold can be obtained
from this construction.}

\pstep
Let $\{U_i\}$ be a locally finite
atlas of a real analytic manifold $M$,
and $\Psi_{ij} :\; U_{ij}\arrow U_{ij}$
the gluing maps. We realize $U_i$ as an open ball
with compact closure in $\Re(\C^n)=\R^n$.
By local finiteness, there are only 
finitely many such $\Psi_{ij}$ for any given $U_i$. 
Denote by $B_\epsilon$ an open ball of radius $\epsilon$
in the $n$-dimensional real space $\im(\C^n)$.

{\bf \green Step 2:}
Let $\epsilon>0$ be a sufficiently small real number
such that all $\Psi_{ij}$ can be extended to
gluing functions $\tilde \Psi_{ij}$ on the open sets
$\tilde U_i:=U_i\times B_\epsilon\subset \C^n$. 
{\bf \purple Then $(\tilde U_i, \Psi_{ij})$ is an atlas for
a complex manifold $M_\C$.} Since all $\Psi_{ij}$ are
real, they are preserved by the natural involution
acting on $B_\epsilon$ as $-1$ and on $U_i$ as identity.
This involution defines a real structure on $M_\C$.
Clearly, $M$ is the set of its fixed points.
\endproof

\newpage

{\bf \blue Complexification}

\definition
Let $M_\R$ be a real analytic manifold, and
$M_\C$ a complex analytic manifold equipped with
an antiholomorphic involution, such that $M_\R$
is the set of its fixed points. Then $M_\C$ is called
{\bf \blue complexification} of $M_\R$.

\definition
A tensor on a real analytic manifold is called 
{\bf\blue real analytic} if it is expressed locally
by a sum of coordinate monomials with real
analytic coefficients.

\claim
Let $M_\R$ be a real analytic manifold, $(M_\C, \iota)$ its complexification,
and $\Phi$ a tensor on $M_\R$. {\bf \red Then $\Phi$ is real analytic
if and only if $\Phi$ can be extended to a holomorpic tensor
$\Phi_\C$ in some neighbourhood of $M_\R$ inside $M_\C$.}
Moreover, {\bf \red $\Phi$ is real on $M_\R$ if $\iota^*\Phi_\C =\bar\Phi_\C$.}

\proof
The ``if'' part is clear, because every complex analytic 
tensor on $M_\C$ is by definition real analytic on $M_\R$.

Conversely, suppose that $\Phi$ 
is expressed in coordinates by a sum of
tensorial monomials with real analytic coefficients $f_i$.
Let $\{U_i\}$ be a cover of $M$, and 
$\tilde U_i:=U_i\times B_\epsilon$ the corresponding cover
of a neighbourhood of $M_\R$ in $M_\C$ constructed above.
Chosing $\epsilon$ sufficiently small, we can assume that
the Taylor series giving coefficients of $\Phi$ 
converges on each $\tilde U_i$. {\bf \purple We define $\Phi_\C$
as the sum of these series.}
\endproof

\newpage


{\bf \blue Categories}

\newcommand{\Ob}{\operatorname{\mathcal {Ob}}}
\newcommand{\Mor}{\operatorname{\mathcal{Mor}}}

\definition 
{\bf \blue A category} $\cac$ is a collection of data
called ``objects'' and ``morphisms between objects''
which satisfies the axioms below.

{\bf \green DATA.}\\
\phantom{XX} {\bf \red Objects:} A class
$\Ob(\cac)$ of {\bf \blue objects}
of $\cac$. \\
\phantom{XX} {\bf \red Morphisms:} For each $X, Y\in \Ob(\cac)$,
one has a set $\Mor(X,Y)$ of {\bf \blue morphisms from $X$ to $Y$}.\\
\phantom{XX} {\bf \red Composition of morphisms:} For each
$\phi\in \Mor(X,Y), \psi \in \Mor(Y,Z)$ there exists
{\bf \blue the composition} $\phi\circ \psi \in \Mor(X, Z)$\\
\phantom{XX} {\bf \red Identity morphism:} For each 
$A\in \Ob(\cac)$ there exists a morphism $\Id_A \in \Mor(A,A)$.


{\bf \green AXIOMS.}\\
\phantom{XX} {\bf \red Associativity of composition:}
$\phi_1\circ(\phi_2\circ\phi_3)=(\phi_1\circ\phi_2)\circ\phi_3$.\\
\phantom{XX} {\bf \red Properties of identity morphism:}
For each $\phi\in \Mor(X,Y)$, one has
$\Id_x\circ \phi = \phi = \phi\circ \Id_Y$\\

\newpage

{\bf \blue Categories (2)}

\definition
Let $X, Y\in \Ob(\cac)$ -- objects of $\cac$.
A morphism $\phi\in \Mor(X,Y)$ is called {\bf \blue an isomorphism}
if there exists $\psi\in \Mor(Y,X)$ such that
$\phi \circ \psi = \Id_X$ and $\psi\circ\phi = \Id_Y$.
In this case, the objects $X$ and $Y$ are called
{\bf \blue isomorphic}.


{\bf \green Examples of categories:}\\
{\bf \purple Category of sets:} its morphisms are arbitrary maps.\\
{\bf \purple Category of vector spaces:} its morphisms are linear maps.\\
{\bf \purple Categories of rings, groups, fields:} morphisms are 
homomorphisms.\\
{\bf \purple Category of topological spaces:} morphisms are continuous maps.\\
{\bf \purple Category of smooth manifolds:} morphisms are smooth maps.

\newpage

{\bf \blue Functors}

\definition
Let $\cac_1, \cac_2$ be two categories.
A {\bf \blue covariant functor} from $\cac_1$ to
$\cac_2$ is the following set of data.

1. {\bf \purple A map $F:\; \Ob(\cac_1)\arrow \Ob(\cac_2)$}.

2. {\bf \purple A map $F:\; \Mor(X,Y) \arrow \Mor(F(X), F(Y))$
defined for any pair of objects $X, Y\in \Ob(\cac_1)$.}

These data define a functor if they are {\bf \purple compatible with
compositions,} that is, satisfy $F(\phi) \circ F(\psi) = F(\phi\circ\psi)$
for any $\phi\in \Mor(X,Y)$ and $\psi\in \Mor(Y,Z)$, and
{\bf \purple map identity morphism to identity} morphism.

\newpage

{\bf \blue Small categories}

\remark This way, one could speak of {\bf \blue category of
all categories}, with categories as objects and functors as
morphisms.

{\bf \red A caution}
To avoid set-theoretic complications, Grothendieck 
added another axiom to set theory, ``universum axiom'',
postulating existence of ``universum'', a very big set, and
worked with ``small categories'' -- categories where the set of all
objects and sets of morphisms belong to the universum.
In this sense, ``category of all categories'' is not
a ``small category'', because the set of its object
(being comparable to the set of all subsets of the
universum) is too big to fit in the universum.

In practice, mathematicians say ``category''
when they mean ``small category'', tacitly assuming that any
given category is ``small''. This is why not many people call
``category of all categories'' a category: nobody wants
to deal with set-theoretic complications. 

\newpage

{\bf \blue Example of functors}

{\bf \purple A ``natural operation'' on mathematical objects is usually
a functor.} Examples:

1. A map $X \arrow 2^X$ from the set $X$ to the set of all subsets
of $X$ is a functor from the category $\Sets$ of sets to itself.

2. A map $M \arrow M^2$ mapping a topological space
to its product with itself is a functor on topological spaces.

3. A map $V \arrow V \oplus V$ is a functor on vector spaces;
same for a map $V \arrow V\otimes V$ or $V \arrow (V \oplus V)\otimes V$.

4. {\bf \blue Identity functor} from any category to itself.

5. A map from topological spaces to $\Sets$,
putting a topological space to the set of its connected components.

\exercise {\bf \purple Prove that it is a functor.}

\newpage

{\bf \blue Equivalence of functors}

\definition
Let $X, Y\in \Ob(\cac)$ be objects of a category $\cac$.
A mprphism  $\phi\in \Mor(X,Y)$ is called {\bf \blue an isomorphism}
if there exists $\psi\in \Mor(Y,X)$ such that
$\phi \circ \psi = \Id_X$ and $\psi\circ\phi = \Id_Y$.
In this case $X$ and $Y$ are called {\bf \blue isomorphic}.

\definition
Two functors  $F, G:\;\cac_1\arrow \cac_2$ 
are called {\bf\blue equivalent} if for any $X \in \Ob(\cac_1)$
we are given an isomorphism $\Psi_X:\; F(X) \arrow
G(X)$, in  such a way that for any  $\phi\in \Mor(X,Y)$,
one has $ F(\phi) \circ \Psi_Y= \Psi_X\circ G(\phi)$.

\remark
Such commutation relations are usually expressed by 
{\bf \blue commutative diagrams}. For example, the condition
$F(\phi) \circ \Psi_Y= \Psi_X\circ G(\phi)$ is expressed
by a commutative diagram
\begin{equation*}
\begin{CD}
F(X) @>{F(\phi)}>> F(Y)\\
@V{\Psi_X}VV @VV{\Psi_Y}V\\
G(X) @>{G(\phi)}>> G(Y)
\end{CD}
\end{equation*}


\newpage

{\bf \blue Equivalence of categories}

\definition
A functor $F:\; \cac_1 \arrow \cac_2$
is called {\bf \blue equivalence of categories}
if there exists a functor  $G:\;\cac_2 \arrow \cac_1$
such that the compositions $G\circ F$ and $G\circ F$
are equivaleent to the identity functors
$\Id_{\cac_1}$, $\Id_{\cac_2}$.

\remark
It is possible to show that this is equivalent to the following
conditions: {\bf \purple $F$ defines a bijection on the set of isomorphism
classes of objects of $\cac_1$ and $\cac_2$, and a bijection
\[ \Mor(X,Y) \arrow \Mor(F(X), F(Y)).\]
for each $X, Y \in \Ob(\cac_1)$}.

\remark
From the point of view of category theory, {\bf \red equivalent categories
are two instances of the same category} (even if the cardinality
of corresponding sets of objects is different).


\newpage

{\bf \blue Germ of a complex manifold}

\definition
Let $K\subset M$ be a closed subset of a complex manifold,
homeomorphic to $K_1\subset M_1$, where $M_1$ is also
a complex manifold. Fixing the homeomorphism $K\cong K_1$,
we may identify these sets and consider $K$ as a subset $M_1$.
We say that $M$ and $M_1$ {\bf\blue have the
same germ in $K$} if there exist biholomorphic open subsets
$U_1 \subset M_1$ and $U\subset M$ containing $K$,
with the biholomorphism $\phi:\; U \arrow U_1$ identity on $K$.

\definition
{\bf \blue Germ of a manifold $M$ in $K\subset M$}
is an equivalence class of open subsets $U \subset M$
containing $K$, with this equivalence relation.

\definition
Consider category $\cac_\iota$, with objects
complex manifolds $(M, \iota)$ equipped with a
real structure, and morphisms holomorphic maps
commuting with $\iota$.



\theorem {\bf \blue (Grauert)} {\bf \red Category of
real analytic manifolds is equivalent to the category
of germs of $M\in \cac_\iota$ in $M^\iota\subset M$.}

\exercise {\bf \purple Prove this theorem.}

\newpage

{\bf \blue Hans Grauert}


\begin{center}
\epsfig{file=Grauert.png,width=0.40\linewidth}\\[10mm]
{ \it \small\green Hans Grauert in Bonn, 2000\\
(8.02.1930 - 4.09.2011)} 
\end{center}


\newpage

{\bf \blue Extension of tensors to a complexification}

{\bf \green Lemma 1:}
Let $X$ be an open ball in $\C^n$ equipped with the
standard anticomplex involution, $X_\R=X \cap \R^n$ its fixed point set, 
and $\alpha$ a holomorphic
tensor on $X$ vanishing in $X_\R$.
{\bf \red Then $\alpha=0$.}

\proof 
{\bf \purple Any holomorphic function which vanishes on $\R^n$ has all
its derivatives vanishing.}  Therefore its Taylor series vanish.
Such a function vanishes on $\C^n$ by
analytic continuation principle. This argument
can be applied to all coefficients of $\alpha$.
\endproof

\definition An almost complex structure $I$ on a real analytic
manifold is {\bf \blue real analytic} if $I$ is a real analytic tensor.

\corollary
Let $(M,I)$ be a real analytic almost complex manifold,
$M_\C$ its complexification, and $I_\C:\; TM_\C \arrow TM_\C$ the 
holomorphic extension of $I$ to $M_\C$. 
{\bf \red Then $I_\C^2 = -\Id$.}

{\bf \green Proof:} {\bf \purple The tensor $I_\C^2 +\Id$ is holomorphic 
and vanishes on $M_\R$,} hence the previous lemma can be applied.
\endproof


\newpage

{\bf \blue Underlying real analytic manifold}

\remark {\bf \red A complex analytic map
$\Phi:\; \C^n \arrow \C^n$ is real analytic as a map
$\R^{2n} \arrow \R^{2n}$.} Indeed, the coefficients of $\Phi$
are real and imaginary parts of holomorphic functions, and
real and imaginary parts of holomorphic functions
can be expressed as Taylor series of the real variables.

\definition
Let $M$ be a complex manifold. 
The {\bf \blue underlying real analytic manifold} $M_\R$
is the same manifold, with the same gluing functions,
considered as real analytic maps.

\remark
The sheaf of real analytic functions on $M_\R$
can be defined as {\bf \purple the sheaf of converging power series
generated by holomorphic and antiholomorphic functions.}
Indeed, such functions are real analytic in any of the
real analytic map; conversely, {\bf \purple any real analytic
function on $M_\R$ is a converging power serie on $\Re z_i, \Im z_i$, where
$z_i$ are holomorphic coordinates on $M$.}



\newpage

{\bf \blue Complexification of the underlying real analytic manifold}

\definition
Let $M$ be a complex manifold. The {\bf \blue complex conjugate manifold}
is the same manifold with almost complex structure $-I$ and
antiholomorphic functions on $M$ holomorphic on $\bar M$.

\claim
Let $M$ be an integrable almost complex manifold.
Denote by $M_\R$ its underlying real analytic manifold.
{\bf \red 
Then a complexification of $M_\R$ can be given as $M_\C:=M \times \bar M$,}
with the anticomplex involution $\tau(x,y)=(y,x)$.

\proof
Clearly, the fixed point set of $\tau$ is the diagonal, identified with
$M_\R =M$ as usual. Both holomorphic and antiholomorphic functions 
on $M_\R$ are obtained as restrictions of holomorphic functions from
$M_\C$, hence the sheaf of real analytic functions on $M_\R$
is a subsheaf of $\calo_{M_\C}$ of holomorphic functions
on $M_\C$ restricted to $M_\R$.
\endproof


\end{document}
