\documentclass{slides} \usepackage{amssymb, amsmath, amscd, color, epsfig} %\usepackage[matrix,arrow]{xy} \newcommand{\green}{\color[rgb]{0,0.4,0}} \newcommand{\purple}{\color[rgb]{0.4,0,0.4}} \newcommand{\red}{\color[rgb]{0.7,0,0}} \newcommand{\blue}{\color{blue}} \def\eqref#1{(\ref{#1})} \newcommand{\goth}{\mathfrak} \newcommand{\g}{{\frak g}} \newcommand{\arrow}{{\:\longrightarrow\:}} \newcommand{\Z}{{\Bbb Z}} \newcommand{\C}{{\Bbb C}} \newcommand{\R}{{\Bbb R}} \newcommand{\Q}{{\Bbb Q}} \newcommand{\6}{\partial} \def\1{\sqrt{-1}\:} \newcommand{\restrict}[1]{{\left|_{{#1}}\right.}} \newcommand{\cntrct} % contraction with a vector field {\hspace{2pt}\raisebox{1pt}{\text{$\lrcorner$}}\hspace{2pt}} \def\Bbb#1{\mathbb #1} \newcommand{\calo}{{\cal O}} \newcommand{\cac}{{\cal C}} % Correcting TeX... %\let\oldtilde=\tilde %\renewcommand{\tilde}{\widetilde} \renewcommand{\bar}{\overline} \renewcommand{\phi}{\varphi} \renewcommand{\epsilon}{\varepsilon} \renewcommand{\geq}{\geqslant} \renewcommand{\leq}{\leqslant} % Operatornames \newcommand{\even}{{\rm even}} \newcommand{\ev}{{\rm even}} \newcommand{\odd}{{\rm odd}} \newcommand{\const}{{\sf const}} \newcommand{\fl}{{\rm fl}} \newcommand{\im}{\operatorname{im}} \newcommand{\End}{\operatorname{End}} \newcommand{\St}{\operatorname{St}} \newcommand{\Sym}{\operatorname{Sym}} \newcommand{\Hol}{\operatorname{{\cal H}ol}} \newcommand{\Tot}{\operatorname{Tot}} \newcommand{\Id}{\operatorname{Id}} \newcommand{\id}{\operatorname{\text{\sf id}}} \newcommand{\symb}{\operatorname{\text{\sf symb}}} \newcommand{\Vol}{\operatorname{Vol}} \newcommand{\Var}{\operatorname{Var}} \newcommand{\Hom}{\operatorname{Hom}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\Alt}{\operatorname{Alt}} \newcommand{\Iso}{\operatorname{Iso}} \newcommand{\Sec}{\operatorname{Sec}} \newcommand{\Can}{\operatorname{Can}} \newcommand{\Sing}{\operatorname{Sing}} \newcommand{\Spin}{\operatorname{Spin}} \newcommand{\Supp}{\operatorname{Supp}} \newcommand{\codim}{\operatorname{codim}} \newcommand{\dist}{\operatorname{\sf dist}} \newcommand{\coker}{\operatorname{coker}} \newcommand{\ind}{\operatorname{\sf ind}} \newcommand{\rk}{\operatorname{rk}} \newcommand{\Def}{\operatorname{Def}} \newcommand{\Lie}{\operatorname{Lie}} \newcommand{\Tw}{\operatorname{Tw}} \newcommand{\Tr}{\operatorname{Tr}} \newcommand{\Spec}{\operatorname{Spec}} \newcommand{\Diff}{\operatorname{Diff}} \renewcommand{\Re}{\operatorname{Re}} \renewcommand{\Im}{\operatorname{Im}} \newcommand{\inbfpare}[1]{{% \mbox{\tt (}\hspace{-5pt}\mbox{\tt (} #1 % \mbox{\tt )}\hspace{-5pt}\mbox{\tt )}% }} \newcommand{\comment}[1]{{}} \def\blacksquare{\hbox{\vrule width 10pt height 10pt depth 0pt}} \def\endproof{\blacksquare} \def\shortdash{\mbox{\vrule width 4.5pt height 0.55ex depth -0.5ex}} \makeatletter %\@ifundefined{Bbb} % {\newcommand{\Bbb}[1]{{\mathbb #1}}}% %{}% {\edef\Bbb#1{{\Bbb #1}}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Pagestyle % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newcommand{\ps@verbit}{% \renewcommand{\@oddhead}{% \scriptsize {\it \small Hodge theory, lecture 18 \hfil \tiny M. Verbitsky }} \renewcommand{\@evenhead}{\@oddhead} \renewcommand{\@oddfoot}{\hfil\thepage\hfil} \renewcommand{\@evenfoot}{\@oddfoot}} \pagestyle{verbit} \setlength\paperheight {10in}% \setlength\paperwidth {13.5in} \setlength{\textwidth}{0.8\paperwidth} \setlength{\textheight}{0.8\paperheight} \setlength{\pdfpageheight}{\paperheight} \setlength{\pdfpagewidth}{\paperwidth} \addtolength{\topmargin}{-20mm} \addtolength{\leftmargin}{-25mm} \addtolength{\rightmargin}{-25mm} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Lemma, sublemma, corollary, proposition, theorem, % % definition,example defined there: % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newcounter{section} \newcounter{Mycounter}[section] \newcounter{lemma}[section] \setcounter{lemma}{0} \renewcommand{\thelemma}{\noindent{Lemma \thesection.\arabic{lemma}}} \newcommand{\lemma}{% \setcounter{lemma}{\value{Mycounter}} \refstepcounter{lemma} \stepcounter{Mycounter} {\bf \green LEMMA:\ }} \newcounter{claim}[section] \setcounter{claim}{0} \renewcommand{\theclaim}{\noindent{Claim \thesection.\arabic{claim}}} \newcommand{\claim}{% \setcounter{claim}{\value{Mycounter}} \refstepcounter{claim} \stepcounter{Mycounter} {\bf \green CLAIM:\ }} \newcounter{corollary}[section] \setcounter{corollary}{0} \renewcommand{\thecorollary}{\noindent{Corollary \thesection.\arabic{corollary}}} \newcommand{\corollary}{% \setcounter{corollary}{\value{Mycounter}} \refstepcounter{corollary} \stepcounter{Mycounter} {\bf \green COROLLARY:\ }} \newcounter{theorem}[section] \setcounter{theorem}{0} \renewcommand{\thetheorem}{\noindent{Theorem \thesection.\arabic{theorem}}} \newcommand{\theorem}{% \setcounter{theorem}{\value{Mycounter}} \refstepcounter{theorem} \stepcounter{Mycounter} {\bf \green THEOREM:\ }} \newcounter{conjecture}[section] \setcounter{conjecture}{0} \renewcommand{\theconjecture}{\noindent{Conjecture \thesection.\arabic{conjecture}}} \newcommand{\conjecture}{% \setcounter{conjecture}{\value{Mycounter}} \refstepcounter{conjecture} \stepcounter{Mycounter} {\bf \green CONJECTURE:\ }} \newcounter{proposition}[section] \setcounter{proposition}{0} \renewcommand{\theproposition} {\noindent{Proposition \thesection.\arabic{proposition}}} \newcommand{\proposition}{% \setcounter{proposition}{\value{Mycounter}} \refstepcounter{proposition} \stepcounter{Mycounter} {\bf \green PROPOSITION:\ }} \newcounter{definition}[section] \setcounter{definition}{0} \renewcommand{\thedefinition} {\noindent{Definition~\thesection.\arabic{definition}}} \newcommand{\definition}{% \setcounter{definition}{\value{Mycounter}} \refstepcounter{definition} \stepcounter{Mycounter} {\bf \green DEFINITION:\ }} \newcounter{example}[section] \setcounter{example}{0} \renewcommand{\theexample}{\noindent{Example \thesection.\arabic{example}}} \newcommand{\example}{% \setcounter{example}{\value{Mycounter}} \refstepcounter{example} \stepcounter{Mycounter} {\bf \green EXAMPLE:\ }} \newcounter{remark}[section] \setcounter{remark}{0} \renewcommand{\theremark}{\noindent{Remark \thesection.\arabic{remark}}} \newcommand{\remark}{% \setcounter{remark}{\value{Mycounter}} \refstepcounter{remark} \stepcounter{Mycounter} {\bf \green REMARK:\ }} \newcounter{observation}[section] \setcounter{observation}{0} \renewcommand{\theobservation}{\noindent{Question \thesection.\arabic{observation}}} \newcommand{\observation}{% \setcounter{observation}{\value{Mycounter}} \refstepcounter{observation} \stepcounter{Mycounter} {\bf \green OBSERVATION:\ }} \newcounter{question}[section] \setcounter{question}{0} \renewcommand{\thequestion}{\noindent{Question \thesection.\arabic{question}}} \newcommand{\question}{% \setcounter{question}{\value{Mycounter}} \refstepcounter{question} \stepcounter{Mycounter} {\bf \green QUESTION:\ }} \newcommand{\exercise}{{\bf \green EXERCISE:\ }} \newcommand{\pstep}{{\bf \green Proof. Step 1:\ }} \newcommand{\proof}{{\bf \green Proof:\ }} \begin{document} \setcounter{page}{1} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \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 <... 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}