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\begin{document}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf Hodge theory \\[15mm]
\small lecture 18: Acyclic resolutions}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\\[14mm]
{\scriptsize NRU HSE, Moscow} \\[15mm]
{\small Misha Verbitsky, March 31, 2018 } 
\end{center}

\newpage

{\bf \blue Sheaves (reminder)}

\definition
{\bf\blue An open cover} of a topological space $X$ is a
family of open sets $\{U_i\}$ such that $\bigcup_iU_i=X$.

\definition
A {\bf\blue presheaf} on a
topological space $M$ is a collection of vector spaces
${\cal F}(U)$, for each open 
subset $U\subset M$,  together with {\bf \blue restriction maps}
$R_{UW}{\cal F}(U)\arrow {\cal F}(W)$ defined for each $W\subset U$,
such that for any three open sets $W\subset V\subset U$,
$\Psi_{UW}=\Psi_{UV}\circ \Psi_{VW}$. Elements of ${\cal F}(U)$
are called {\bf \blue sections of ${\cal F}$ over $U$},
and restriction map often denoted $f\restrict W$

\definition
A presheaf 
${\cal F}$ is called {\bf\blue a sheaf} 
if for any open set $U$ and any cover $U=\bigcup U_I$
the following two conditions are satisfied.\\
\phantom{xu} 1. Let $f\in {\cal F}(U)$ be a section of ${\cal F}$ on $U$
such that its restriction to each $U_i$ vanishes. {\bf \purple 
Then $f=0$.} \\
\phantom{xu}  2. Let $f_i\in{\cal F}(U_i)$ be a family of
sections 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 Direct limits}


\definition
{\bf\blue Commutative diagram} of vector spaces
is given by the following data. There is
a directed graph (graph with arrows). For each
vertex of this graph we have a vector space, 
and each arrow corresponds to a homomorphism of the
associated vector spaces. {\bf \purple These 
homomorphism are compatible, in the 
following way.} Whenever there exist two ways of going from
one vertex to another, the compositions of the
corresponding arrows are equal.


\definition
Let ${\cal C}$ be a commutative diagram of vector spaces,
 $A, B$ -- vector spaces, corresponding to two vertices
of a diagram, and  $a\in A, b\in B$ elements of these
vector spaces. Write  $a\sim b$ if $a$ and $b$ are mapped
to the same element $d\in D$ by a composition of 
arrows from  ${\cal C}$. Let $\sim$ be an
equivalence relation generated by such $a\sim b$.
A quotient $\bigoplus_i C_i/E$ is called
{\bf\blue a direct limit} of a diagram $\{C_i\}$.
The same notion is also called {\bf\blue colimit}
and {\bf\blue inductive limit}. Direct limit is denoted
$\lim\limits_\rightarrow$.

\definition
Let ${\cal F}$ be a sheaf on $M$, $x\in M$ a point,
and $\{U_i\}$ the set of all neighbourhoods of
$x$. Consider a diagram with the set of vertices 
indexed by $\{U_i\}$, and arrows from $U_i$ to $U_j$
corresponding to inclusions $U_j \hookrightarrow U_i$. 
The {\bf\blue space of germs} of ${\cal F}$ in
$x$ is a direct limit $\lim\limits_\rightarrow{\cal F}(U_i)$
over this diagram. The space of germs is also called
{\bf \blue a stalk} of a sheaf.

\newpage

{\bf \blue Sheaf morphisms (reminder)}

\definition
Let ${\cal B}, {\cal B}'$ be sheaves on
$M$. {\bf\blue A sheaf morphism} from ${\cal B}$ to ${\cal B}'$
is a collection of homomorphisms ${\cal B}(U)\arrow {\cal B}'(U)$,
defined for each open subset $U\subset M$,
and compatible with the restriction maps:
\[
\begin{CD}
{\cal B}(U) @>>> {\cal B}'(U)\\
@VVV@VVV\\
{\cal B}(U_1) @>>> {\cal B}'(U_1)
\end{CD}
\]

\remark
Morphisms of sheaves of modules are defined in the same
way, but in this case {\bf \purple the maps ${\cal B}(U)\arrow {\cal
  B}'(U)$ should be compatible with the module structure.}


\definition
A sheaf morphism is called {\bf\blue injective}
if it is injective on stalks
and {\bf\blue surjective}, if it is surjective on stalks.

\newpage

{\bf \blue \v Cech cohomology}

\definition
Let ${\cal F}$ be a sheaf on a topological space $M$
and $\{U_i\}$ an open cover of $M$ indexed by 
a linearly ordered set ${\cal I}$.
Define the space of {\bf \blue \v Cech chains}
\[ 
C_{k-1}:= \prod_{i_1< i_2 <... < i_k} {\cal F}
\left ( \bigcup_{j=1}^k U_{i_j}\right).
\]
Define {\bf \blue the \v Cech differential} $d:\; C_{k-1} \arrow C_{k}$ mapping
$f\in {\cal F}
\left ( \bigcap_{j=1}^k U_{i_j}\right)$
to 
\[ 
 \sum_{i\in {\cal I}\backslash \{i_1,..., i_k\}} (-1)^{\sigma} 
f\restrict{U_{i_1}\cap ...\cap U_{i_k}\cap U_i}
\]
where $\sigma-1$ is the number of $i$ in the sequence
$i_1< i_2 <... <i < ... < i_k$.

Consider the sequence
\[
... \stackrel d \arrow  C_i \stackrel d \arrow C_{i-1} \stackrel d \arrow ...
\]
Its cohomology are called {\bf \blue the \v Cech
  cohomology} of the sheaf ${\cal F}$, associated
with the cover $\{U_i\}$.
Elements of $\ker d$ are called {\bf \blue \v Cech
  cocycles} and elements of $\im d$ {\bf \blue the \v Cech
  coboundaries}. 

\newpage

{\bf \blue \v Cech cohomology and global sections}

\definition
A topological space $M$ is called {\bf\blue paracompact}
if any open cover of $M$ has a locally finite refinement.

\claim
Let $A$ be a sheaf 
on a paracompact topological space
such that its first \v Cech cohomology vanish for any
locally finite covering. Then {\bf \red for any exact sequence 
$0 \arrow A \arrow B \stackrel \psi \arrow C \arrow 0$
of sheaves, the sequence \\ $0 \arrow \Gamma(A) \arrow \Gamma(B) \arrow
\Gamma(C) \arrow 0$ is exact,} where $\Gamma$ denotes the space of global
sections.

\proof
Let $c$ be a global section of $C$.
Since $\psi$ is surjective, 
there exists a locally finite (by paracompactness)
covering $\{U_i\}$ and $b_i \in B(U_i)$
such that $\psi(b_i) = c\restrict {U_i}$.
Then $b_i -b_j \restrict{U_i\cap U_j}\in A(U_i\cap U_j)$ give a
\v Cech 1-cocycle. If it is a coboundary, this means
that $b_i - b_j = a_i -a_j$ for some collection of
sections $a_i \in A(U_i)$. Then $\tilde b_i:=b_i -a_i$ agree on
pairwise intersections; gluing all $\tilde b_i$ to a
global section $\tilde b$ of $B$, we obtain that $\psi(b)=c$.
\endproof


\newpage

{\bf \blue Fine sheaves}

\definition
Let $\{U_i\}$, 
be a locally finite open covering of a manifold
$M$, with the closure
of $U_i$ compact. Denote by $F^c\restrict U$ the group of sections
of a sheaf $F$ with compact support. {\bf \blue
Partition of unity} on a sheaf of rings is a set of sections
with compact support $\psi_i\in F^c(U_i)$,
such that  $\sum_i \psi_i=1$.
A sheaf of rings is called {\bf \blue fine} if it admits a 
partition of unity for any locally finite covering.

\remark {\bf \red The sheaf $C^\infty(M)$ is fine}.

\claim 
Let $F$ be a sheaf of modules over a fine sheaf of rings.
Then {\bf \red the \v Cech cohomology of $F$ vanish}
for any locally finite covering.

\proof
Let $\{U_i\}$ be a covering of $M$,
and $P= \prod_{i_1<i_2< ... < i_{k+1}} f_{i_1,..., i_{k+1}}\in F(U_{i_1}\cap ...\cap 
U_{i_{k+2}})$
a $k$-cocycle. Consider a partition of unity $\sum \psi_i =1$ associated with $\{U_i\}$.
Then for any $i$, the product
$\psi_{i}P$ is also a $k$-cocycle,
hence we may assume that $P$ is compactly supported
in some $U_i$, say, $U_{i_1}$. Put 
\[ g:= \prod_{i_2< ... < i_{k+1}} 
g_{i_2,..., i_{k+1}}\in \prod_{i_2< ... < i_{k+1}} F(U_{i_2}\cap ...\cap  U_{i_{k+2}})
\]
by taking $g_{i_2,..., i_{k+1}}=f_{i_1, i_2, ... i_{k+1}}$ 
and extending $f_{i_1, i_2, ... i_{k+1}}$ to 
$U_{i_2}\cap ...\cap  U_{i_{k+2}}$ using 
compactness of support of $f_{i_1, i_2, ... i_{k+1}}$ in $U_1$.
\endproof

\newpage

{\bf \blue Fine sheaves and flasque sheaves}

\definition
Let $F$ be a sheaf such that all restriction maps
$F(U)\arrow F(V)$ are surjective. Then $F$ is called {\bf \blue flasque},
or {\bf \blue flabby}.

\exercise
{\bf \purple
Prove that the \v Cech cohomology of flasque sheaves vanish.}

\corollary
Let $0 \arrow A \arrow B \arrow C \arrow 0$
be an exact sequence of sheaves, with $A$ fine or flasque.
{\bf \red Then the sequence of global sections
\[ 0 \arrow \Gamma(A) \arrow \Gamma(B) \arrow \Gamma(C)
  \arrow 0
\]
is also exact.}

\proof {\bf \purple This follows from vanishing of \v Cech cohomology,}
as shown above.
\endproof


\newpage

{\bf \blue Godement resolutions}

\definition
Let $F$ be a sheaf on $M$, and $F_x$ the stalk of $F$ in $x\in M$. 
It is clearly flasque.
Denote by $G(F)$ the sheaf $\prod_{x\in M} F_x$. We consider $F$
as a subsheaf of $G(F)$, and consider the following
flasque resolution of $F=F^0$
\[
0 \arrow F \stackrel d \arrow F^1 \stackrel d \arrow F^2 \arrow ... \ \ \  (***)
\]
with $F^{i+1}=G(F^{i}/d(F^{i-1}))$, and $d$ induced by the tautological map
\[ F^i \arrow F^{i}/d(F^{i-1})\hookrightarrow G(F^{i}/d(F^{i-1})).\]
The resolution (***) is called {\bf \blue Godement resolution}.

\remark
The same argument as used for fine sheaves above also
proves that {\bf \red the \v Cech cohomology of flasqye sheaves vanish.}
Therefore, for an exact sequence $0 \arrow A \arrow B \arrow C \arrow 0$
of sheaves with $A$ flasque,
{\bf \red the sequence of global sections
\[ 0 \arrow \Gamma(A) \arrow \Gamma(B) \arrow \Gamma(C)
  \arrow 0
\]
is also exact.}

\newpage

{\bf\blue Cohomology of a sheaf}

\definition
Let $F$ be a sheaf and 
$0 \arrow F \stackrel d \arrow F^1 \stackrel d \arrow F^2 \arrow ...$
is Godement resolution. Consider the complex of global sections\\
$0\arrow \Gamma(F^1) \arrow \Gamma(F^2) \arrow ...$.
Its cohomology are called {\bf\blue cohomology of the sheaf $F$},
denoted $H^i(F)$. {\bf \purple
The global sections $\Gamma(F)$ are identified with $H^0(F)$.}

\remark
Given an exact sequence of sheaves 
$0 \arrow A \arrow B  \arrow C \arrow 0$,
we obtain an exact sequence of their Godement resolutions
$0 \arrow A^* \arrow B^*  \arrow C^* \arrow 0$, 
{\bf \purple (prove that it is exact)}. The sequences of
sheaves $0 \arrow A^i \arrow B^i  \arrow C^i \arrow 0$
gives an exact sequence
\[0 \arrow \Gamma(A^{\geq 1}) \arrow \Gamma(B^{\geq 1})
\arrow \Gamma(C^{>1}) \arrow 0
\]
as shown above. Its cohomology are cohomology of $A, B, C$. {\bf
  \red This
gives an exact sequence of cohomology}
\[
0 \arrow H^0(A) \arrow H^0(B)\arrow  H^0(C) \arrow
H^1(A) \arrow H^1(B)\arrow  H^1(C) \arrow ...
\]

\newpage

{\bf \blue Roger Godement}


\begin{center}{\epsfig{file=Godement-004.jpg,width=0.43\linewidth}\\[10mm]
{\bf \small Roger Godement \\\tiny October 1, 1921 - July 21, 2016}}
\end{center}

\newpage

{\bf \blue Acyclic resolutions}

\definition
A sheaf $A$ on $M$ is called {\bf\blue acyclic} if $H^i(U, A)=0$
for any opens set $U\subset M$ and any $i>0$.
{\bf \blue An acyclic resolution} for $F=F^0$ is an exact
sequence
\[
0 \arrow F^0 \arrow F^1 \arrow F^2 \arrow ...
\]
where all $F^i$, $i>0$ are acyclic.

\example 
Let $x\in M$ and $A$ a vector space.
{\bf \blue A skyscraper sheaf} is a sheaf $F$
such that $F(U)=A$ for all $U\ni x$ and $F(U)=0$ for
$U\not\ni x$. 

\exercise Prove that
{\bf \purple product of skyscraper sheaves is acyclic.}
In particular, {\bf \purple 
the Godement sheaf $G(F)$ is acyclic for any sheaf $F$.}
Prove that {\bf \purple any fine sheaf is also acyclic.}

Further on, we shall prove the following resultat.

\theorem
Let $0 \arrow F^0 \arrow F^1 \arrow F^2 \arrow ...$
be an acyclic resolution for a sheaf $F=F^0$.
{\bf \red Then cohomology of the complex
$0 \arrow \Gamma(F^1) \arrow \Gamma(F^2)\arrow ...$
are equal to $H^*(F)$.}

\newpage

{\bf \blue Morphisms of complexes}

\definition
{\bf \blue A complex} is a sequence of objects of abelian category
(sheaves, groups, modules) 
$... \stackrel d\arrow C^{i-1} \stackrel d\arrow C^i \stackrel d\arrow C^{i+1} \stackrel d\arrow...$,
$i\in \Z$, with $d^2=0$. {\bf \blue Cohomology} of a complex
is $\ker d/\im d$. {\bf \blue A morphism} of complexes $(C^i, d) \arrow (C^i_1, d)$
is a sequence of maps $\psi_i:\; C^i \arrow C^i_1$ commuting with $d$.
{\bf \purple Category of complexes is also abelian.}

\exercise
Let $0 \arrow A^* \arrow B^* \arrow C^*\arrow 0$ be an exact sequence of 
complexes. {\bf \purple Prove that there exists a long exact sequence}
\[
... \arrow H^i(A) \arrow H^i(B)\arrow  H^i(C) \arrow
H^{i+1}(A) \arrow H^{i+1}(B)\arrow  H^{i+1}(C) \arrow ...
\]

\definition
A morphism of complexes is called {\bf \blue quasi-isomorphism}
if it induces an isomorphism on cohomology.

\newpage

{\bf \blue Cones of morphisms}

\definition
Let $(F^i, d_F) \stackrel {\psi_i}\arrow (G^i, d_G)$
be a morphism of complexes. {\bf \blue The cone} $C(\psi)$
is a complex $F^{i+1}\oplus G^{i}$, with differential given
by $d_F+d_G + (-1)^{i} \psi_{i+1}$.

\remark Denote by $F^*[1]$ the complex $(F^{i+1}, d)$, that is, $F^*$ shifted by 1.
Since the sequence of complexes $0\arrow G^* \arrow C(\psi) \arrow F^*[1]\arrow 0$
is exact, {\bf \purple we obtain an exact sequence}
\[
... \arrow H^i(G)\arrow  H^i(C(\psi)) \arrow
H^{i+1}(F) \arrow H^{i+1}(G)\arrow  H^{i+1}(C(\psi)) \arrow ...
\]

\corollary {\bf \red A morphism of complexes is a quasi-isomorphism
if and only if its cone has zero cohomology.}

{\bf \green Exercise 1:}
Let $0 \arrow C^1 \arrow C^2 \arrow ...$
be an exact sequence of acyclic sheaves.
{\bf \purple Prove that the sequence of global sections
$0\arrow \Gamma(C^1) \arrow \Gamma(C^2) \arrow ...$
is also exact.}


\newpage

{\bf \blue Cones and cohomology}

\claim
Let $0 \arrow A_1^0 \arrow A_1^1\arrow A_1^2 \arrow... $
be an acyclic resolution for a sheaf $A$, and
$0 \arrow A_2^0 \arrow A_2^1\arrow A_2^2 \arrow... $
another acyclic resolution. Suppose that there
exists a morphism $\phi$ of complexes inducing identity
on $A_2^0= A_1^0 =A$. {\bf \red Then the cohomology of
the complex $\Gamma(A^*_1)$ are equal to the cohomology of 
$\Gamma(A^*_2)$}.

\pstep
Consider the complex $X^*$, given by 
$0 \arrow A_1^1\arrow A_1^2 \arrow...$ and 
$Y^*$, given by $0 \arrow A_2^1\arrow A_2^2 \arrow...$ 
(we drop the first term $A_2^0= A_1^0 =A$).
Then the cohomology sheaves ${\cal H}^i(\cdot)$ of these complexes are
equal to $A$ in 0, and vanish in other terms.
The map $\phi$ induces a morphism of complexes 
$X^*\stackrel \phi \arrow Y^*$ which induces
identity on the cohomology sheaves $H^0(A_2^*)= H^0(A_1^*) =A$.
The long exact sequence\\
$ ... \arrow {\cal H}^{i}(A_1^*) \arrow {\cal H}^{i}(A_2^*)\arrow  
{\cal H}^{i}(C(\phi)) \arrow ...$
implies that {\bf \purple the cone $C(\phi)$ is an exact complex of 
acyclic sheaves.}

{\bf \green Step 2:}
Exercise 1 implies that the sequence\\
$... \stackrel d \arrow \Gamma(C^i(\phi)) \stackrel d \arrow \Gamma(C^{i+1}(\phi))
\stackrel d \arrow ...$
is exact. However, this sequence is a complex of vector spaces,
obtained as a cone of a morphism of complexes
$\Gamma(A_1^*) \arrow \Gamma(A_2^*)$, and {\bf \purple from the 
cone exact sequence we obtain that cohomology
of these complexes are equal.} \endproof



\newpage

{\bf \blue Bicomplexes}

\definition
{\bf \blue Bicomplex} is a collection $C^{i,j}$ of objects
in abelian category, enumerated by $i, j\in \Z^2$, and
equipped with two differentials $d^{1,0}:\; C^{i,j} \arrow C^{i+1,j}$
and $d^{0,1}:\; C^{i,j} \arrow C^{i,j+1}$, anti-commuting and
satisfying $(d^{0,1})^2=0$ and $(d^{1,0})^2=0$.

\definition
{\bf\blue Totalization} of a bicomplex $(C^{i,j}, d^{1,0}, d^{0,1})$
is a complex $\Tot^*(C^{i, j}, d)$ 
with $d=d^{1,0}+d^{0,1}$ and $\Tot^p(C^{i, j})= \bigoplus_{i+j=p} C^{i, j}$.

{\bf \green Exercise 2:}
Let $(C^{i,j}, d^{1,0}, d^{0,1})$ be a bicomplex, with $i, j \geq 0$.
Suppose that cohomology of $d^{1,0}$ are equal 0. {\bf \purple Prove that
cohomology of $\Tot^*(C^{i, j})$ vanish.}

\newpage

{\bf \blue Bicomplexes (2)}

{\bf \green Exercise 2:}
Let $(C^{i,j}, d^{1,0}, d^{0,1})$ be a bicomplex, with $i, j \geq 0$.
Suppose that cohomology of $d^{1,0}$ are equal 0. {\bf \purple Prove that
cohomology of $\Tot^*(C^{i, j})$ vanish.}

{\bf \green Claim 1:}
Let $(C^{i,j}, d^{1,0}, d^{0,1})$ be a bicomplex, with $i, j \geq 0$.
Suppose that cohomology of $(C^{i, *}, d^{0,1})$ vanish for all
$i>0$. {\bf \red Then the cohomology of $\Tot^*(C^{i, j})$
are equal to cohomology of $(C^{0, *}, d^{0,1})$.}

\proof
Consider the natural surjective morphism of complexes
$\Tot^*(C^{i, j}) \stackrel \Psi \arrow (C^{0, *}, d^{0,1})$.
Then $\ker \Psi= \Tot^*_{i>0}(C^{i, j})$, where
$\Tot^*_{i>0}(C^{i, j})$ is totalization of the subcomplex
$(C^{*+1, *},  d^{1,0}, d^{0,1})\subset (C^{*, *},  d^{1,0}, d^{0,1})$.
By Exercise 2, cohomology of $\Tot^*_{i>0}(C^{i, j})$ vanish.
Taking the long exact sequence associated with the
exact sequence of complexes
\[ 0 \arrow \Tot^*_{i>0}(C^{i, j})\arrow \Tot^*(C^{i, j})\arrow C^{0, *}\arrow 0
\]
we obtain that cohomology of $(C^{0, *}, d^{0,1})$
are equal to the cohomology of $(\Tot^*(C^{i, j}),d^{1,0}+d^{0,1})$.
\endproof



\newpage

{\bf \blue Godement bicomplex}

Let $A=A^0$ be a sheaf and
$0 \arrow A^0\arrow A^1 \arrow A^2 \arrow ...$
an acyclic resolution, and $G^n(A^i)$ the $n$-th term
of Godement resolution for $A^i$. This gives a bicomplex
$G^{*, *}$
 \[
\begin{CD}
&& 0 &&0 &&0 \\
&& @VVV @VVV @VVV\\
0 @>>> A^0 @>>> A^1 @>>> A^2 @>>>\\
&& @VVV @VVV @VVV\\
0 @>>> G^1(A^0) @>>> G^1(A^1) @>>> G^1(A^2) @>>>\\
&& @VVV @VVV @VVV\\
0 @>>> G^2(A^0) @>>> G^2(A^1) @>>> G^2(A^2) @>>>\\
&& @VVV @VVV @VVV
\end{CD}
\]
with all sheaves acyclic except $A^0$.


\newpage

{\bf \blue Acyclic sheaves}

\definition
A sequence $0 \arrow A \arrow B \arrow C \arrow 0$ 
of sheaves is called {\bf \blue an exact sequence}
if the corresponding sequences of stalks are exact.

\definition
A functor $\Phi$ from sheaves to vector spaces is called
{\bf \blue left exact} if any exact sequence of sheaves
$0 \arrow A \arrow B \arrow C \arrow 0$
is mapped to a left exact sequence
$0 \arrow \Phi(A) \arrow \Phi(B) \arrow \Phi(C)$.

\example {\bf \purple Functor of global sections
${\cal F} \arrow \Gamma_M({\cal F})$ 
is left exact.}

\definition
A sheaf is called {\bf \blue acyclic} if for any
open set $U\subset M$ and any exact sequence of sheaves
$0 \arrow A \arrow B \arrow C \arrow 0$, the sequence
\[ 0 \arrow \Gamma_U(A) \arrow \Gamma_U(B) \arrow
\Gamma_U(C) \arrow 0\]
is exact.

\remark
As shown above, {\bf \red a sheaf $A$ is acyclic if its \v Cech
cohomology $H^1(A)$ vanish} for any locally finite covering.
In particular, all sheaves of modules over $C^\infty M$ are acyclic.

\definition 
Let $0\arrow F\arrow F_1 \arrow F_2 \arrow ... $
be an exact sequence of sheaves. Assume that all
$F_i$ are acyclic. Then this sequence is called
{\bf \blue an acyclic resolution} for $F$.

\newpage

{\bf \blue Sheaf cohomology}
 
\definition
Let  $0 \arrow F\arrow F_1 \arrow F_2 \arrow ... $
be an acyclic resolution of $F$. {\bf \blue Cohomology
  group} $H^i(F)$ is defined as $i$-th group of cohomology
of the corresponding complex of global sections
\[ \Gamma_M(F)\arrow \Gamma_M(F_1) \arrow \Gamma_M(F_2) \arrow ... \]

\proposition {\bf \blue (Properties of cohomology sheaves):}\\
1. The groups $H^i(F)$ {\bf \purple don't depend on the choice of
acyclic resolution}.\\
2. {\bf \purple $H^i(F)=0$ for all $i>0$ if and only if
$F$ is acyclic.}\\
3. For any exact sequence of sheaves  $0 \arrow A \arrow B \arrow C \arrow 0$
there is {\bf \blue a long exact sequence}
\[
0 \arrow \Gamma(A) \arrow \Gamma(B)\arrow  \Gamma(C) \arrow
H^1(A) \arrow H^1(B)\arrow  H^1(C) \arrow ...
\]
{\bf \green Proof is later today.}


\newpage

{\bf \blue Independence of cohomology}

\remark The following theorem implies that 
the {\bf \purple cohomology of a sheaf are independent from the choice
of acyclic resolution.}

\theorem
Let $A=A^0$ and $0\arrow A^0 \arrow A^1 \arrow ...$ 
be an acyclic resolution of $A$. Then
{\bf \red the cohomology of $\Gamma(A^i)$ are equal to the cohomology
of the global sections of the Godement resolution
$\Gamma(G^i(A^0))$.}

\proof
Apply the functor $\Gamma(\cdot)$ to the bicomplex $G^{*,*}$.
Exercise 1 implies that the columns and rows of
$\Gamma(G^{*,*})$ are exact, with the possible exception of
$\Gamma(G^{0,*})$ and $\Gamma(G^{*,0})$. Then Claim 1 implies
that cohomology of totalization of $\Gamma(G^{*,*})$
are equal to the cohomology of $\Gamma(G^{0,*}), d^{0,1}$,
which is cohomology of $\Gamma(G^*(A^0))$
and to the cohomology of $\Gamma(G^{*,0}), d^{1,0}$
which is the same as cohomology of $\Gamma(A^*)$.
\endproof


\newpage

{\bf \blue Dolbeault resolution}

\remark
From Poincar\'e-Dolbeault-Grothendieck lemma
we obtain {\bf \purple an acyclic resolution
\[ 
0\arrow \Omega^p(M) \hookrightarrow \Lambda^{p,0}(M) \stackrel {\bar\6} \arrow 
\Lambda^{p,1}(M) \stackrel {\bar\6} \arrow 
\Lambda^{p,2}(M)  \stackrel {\bar\6} \arrow ...\ \ \  (****)
\]
of the sheaf of holomorphic $p$-forms.} Indeed,
the kernel of $\Lambda^{p,0}(M) \stackrel {\bar\6} \arrow 
\Lambda^{p,1}(M)$ is forms with holomorphic coefficients;
other terms of (**) are exact by
Poincar\'e-Dolbeault-Grothendieck lemma.
The sheaves $\Lambda^{p,0}(M)$ are all sheaves
of $C^\infty(M)$-modules, hence they are acyclic.

\corollary Let $M$ be a compact K\"ahler manifold. Then
{\bf \red the space $H^{p,q}(M)$ is identified with
the cohomology group $H^q(\Omega^p(M))$.}

\proof
Indeed, the cohomology of $\Gamma(\cdot)$ applied to (****)
is the kernel of the corresponding Laplacian $\Delta_{\bar
  \6}$,
which is the same as the kernel of $\Delta_d$ on $H^{p,*}(M)$.
\endproof


\end{document}




