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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{\proof}{% {\bf \green Proof:\ }} \newcommand{\pstep}{% {\bf \green Proof. Step 1:\ }} \begin{document} \setcounter{page}{1} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{center} {\Large\bf Complex geometry\\[15mm] \small lecture 16: Riemann-Hilbert correspondence} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \\[14mm] {\small Misha Verbitsky } \\[20mm] {\tiny\bf HSE, room 306, 16:20, \\[2mm] November 14, 2020 } \end{center} \newpage {\bf \blue Sheaves} \definition {\bf\blue An open cover} of a topological space $X$ is a family of open sets $\{U_i\}$ such that $\bigcup_iU_i=X$. \remark The definition of a sheaf below {\bf \purple is a more abstract version of the notion of ``sheaf of functions''} defined previously. \definition A {\bf\blue presheaf} on a topological space $M$ is a collection of vector spaces (or abelian groups) ${\cal F}(U)$, 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 $W\subset V\subset U$, $R_{UW}=R_{UV}\circ R_{VW}$. Elements of ${\cal F}(U)$ are called {\bf \blue sections of ${\cal F}$ over $U$}, and the 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 Morphisms of sheaves} \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: \[ \begin{CD} {\cal B}(U) @>>> {\cal B}'(U)\\ @VVV@VVV\\ {\cal B}(U_1) @>>> {\cal B}'(U_1) \end{CD} \] \definition {\bf\blue A sheaf isomorphism} is a homomorphism $\Psi:\; {\cal F}_1 \arrow {\cal F}_2$, for which there exists an homomorphism $\Phi:\; {\cal F}_2 \arrow {\cal F}_1$, such that $\Phi\circ \Psi =\Id$ and $\Psi\circ \Phi =\Id$. \newpage {\bf \blue Sheaves of modules} \remark Let $A:\; \phi \arrow B$ be a ring homomorphism, and $V$ a $B$-module. {\bf \purple Then $V$ is equipped with a natural $A$-module structure: $a v:= \phi(a) v$.} \definition Let ${\cal F}$ be a sheaf of rings on a topological space $M$, and ${\cal B}$ another sheaf. It is called {\bf\blue a sheaf of ${\cal F}$-modules} if for all $U\subset M$ the space of sections ${\cal B}(U)$ is equipped with a structure of ${\cal F}(U)$-module, and for all $U'\subset U$, the restriction map ${\cal B}(U) \stackrel{\phi_{U,U'}}\arrow {\cal B}(U')$ is a homomorphism of ${\cal F}(U)$-modules (use the remark above to obtain a structure of ${\cal F}(U)$-module on ${\cal B}(U')$). \definition A {\bf \blue free sheaf of modules} ${\cal F}^n$ over a ring sheaf ${\cal F}$ maps an open set $U$ to the space ${\cal F}(U)^n$. \definition {\bf\blue Locally free sheaf of modules} over a sheaf of rings ${\cal F}$ is a sheaf of modules ${\cal B}$ satisfying the following condition. For each $x\in M$ there exists a neighbourhood $U\ni x$ such that the restriction ${\cal B}\restrict U$ is free. \definition {\bf\blue A vector bundle} on a smooth manifold $M$ is a locally free sheaf of $C^\infty M$-modules. \example Clearly, {\bf \purple tangent bundle is a vector bundle}. \newpage {\bf \blue Locally constant sheaves} \definition Let ${\cal F}$ be a sheaf on $M$ which takes a connected non-empty open subset $U\subset M$ to a vector space or abelian group ${\Bbb V}$. Extend ${\cal F}$ to all open sets using the gluing axiom. Then ${\cal F}$ is called {\bf \blue the constant sheaf}, denoted ${\Bbb V}_M$. \exercise Prove that {\bf \purple the constant sheaf ${\Bbb V}_M$ exists, and is unique up to isomorphism.} \exercise Let $W$ be an open set in $M$, and $S_W$ its set of connected components. Prove that {\bf \purple ${\Bbb V}_M(W)={\Bbb V}^{|S_W|}$}. \definition A {\bf \blue locally constant sheaf} is a sheaf which is locally isomorphic to a constant sheaf. \example Let $\pi:\; M' \arrow M$ be a covering. Given $U\subset M$, let $S_U$ be the set of connected components of $\pi^{-1}(U)$, and set ${\cal F}(U)={\Bbb V}^{|S_W|}$. We are going to define the restriction map $r$ as follows. For an open subset $W\subset U$, consider the map $S_W\arrow S_U$ induced by the natural embedding $\pi^{-1}(W)\stackrel j \hookrightarrow \pi^{-1}(U)$. For each direct sum component ${\Bbb V}_u\subset {\Bbb V}^{|S_U|}$ corresponding to $u \in \im j$, let $r_u:\; {\Bbb V}_u\arrow {\Bbb V}_{j(u)}$ be identity. For a component ${\Bbb V}_u\subset {\Bbb V}^{|S_U|}$ corresponding to $u \notin \im j$, we set $r_u=0$. Then $r:=\bigoplus_{u\in S_U} r_u:\; \bigoplus_{u\in S_U}{\Bbb V}\arrow \bigoplus_{w\in S_W}{\Bbb V}$. {\bf \purple This defines a locally constant sheaf on $M$} {\bf \red (prove it).} \newpage {\bf \blue \'Etal\'e space of a sheaf} \definition Let ${\cal F}$ be a sheaf on $M$, and $U, V\supset x$ be two open set containing $x\in M$. Two sections $f\in {\cal F}(U)$, $g\in {\cal F}(V)$ are called {\bf \blue equivalent in $x$} if there exists an open set $W\ni x $ such that $W\subset U\cap V$ and $f\restrict W= g\restrict W$. %{\bf \blue A germ of ${\cal F}$ %in $x$} is the set of all sections on all $U\ni x$ up to %this equivalence. %When $K=x$ is a point, %we say that {\bf \blue $f$ is equivalent to $g$ in a point $x$.} {\bf \blue A germ of a sheaf ${\cal F}$ in $x$} is a class of equivalence of sections of ${\cal F}$ in all open sets $U \ni x$ under this equivalence relation. {\bf \blue The stalk} of a sheaf ${\cal F}$ in $x$ is the space ${\cal F}_x$ of all germs in $x$. \definition Let $E({\cal F})$ be the set of all stalks of a sheaf ${\cal F}$ in all points $x\in M$. A germ $f \in {\cal F}_m$ is called {\bf \blue a limit of a sequence of germs $f_i \in {\cal F}_{m_i}$} if $\lim_i m_i =m$ and there exists a section $\tilde f$ of ${\cal F}$ over $U\ni x$ such that almost all $f_i$ are germs of $\tilde f$. The {\bf \blue \'etal\'e topology} on $E({\cal F})$ is defined as follows: a subset $K\subset E({\cal F})$ is {\bf \blue closed in \'etal\'e topology} if it contains all its limit points. \remark Usually $E({\cal F})$ {\bf \red is non-Hausdorff}. \newpage {\bf \blue \'Etal\'e space of a constant sheaf} \claim Let ${\cal F}= {\Bbb V}_M$ be a constant sheaf on a manifold, and $x\in M$ a connected subset. {\bf \red Then the space of germs of ${\cal F}$ in $x$ is equal to ${\Bbb V}$.} \proof Since ${\cal F}$ is constant, the set of its sections on any connected open set is equal to ${\Bbb V}$. This gives a natural map $r_x:={\cal F}(U) \arrow {\Bbb V}$: we restrict $f\in {\cal F}(U)$ to a connected component $U_1$ of $U$ containing $x$, and obtain an element of ${\Bbb V}$. {\bf \purple Clearly, two sections $f, g$ are equivalent in $K$ if and only if $r_x(f)=r_x(g)$.} This identifies ${\Bbb V}$ with the set of equivalence classes of sections in $x$. \endproof {\bf \green Corollary 1:} Let ${\cal F}= {\Bbb V}_M$ be a constant sheaf on a manifold. {\bf \red Then the \'etal\'e space $E({\cal F})$ of ${\cal F}$ is identified with ${\Bbb V}$ disconnected copies of $M$.} \proof Indeed, a sequence $f_i \in {\cal F}_{m_i}$ converges to $f$ if $\lim_i m_i =m$ and $r_{m_i}(f_i)= r_m(f)$ for almost all $i$. \endproof \newpage {\bf \blue Local systems} \definition {\bf \blue Category of coverings} of $M$ is category $\cac$ with $\Ob(\cac)$ all coverings and morphisms continuous maps of coverings compatible with projections to $M$. \definition Let $\pi_1:\; M_1 \arrow M$, $\pi_2:\; M_2 \arrow M$ be continuous maps. {\bf\blue Fibered product} $M_1 \times_M M_2$ is the subset of $M_1 \times M_2$ defined as $M_1 \times_M M_2:= \{(x, y)\in M_1 \times M_2\ \ |\ \ \pi_1(x)=\pi_2(y)\}$, with induced topology. \exercise Prove that {\bf \purple a fibered product of coverings is a covering}. \definition {\bf \blue An abelian group structure on a covering $\pi_1:\; M_1 \arrow M$} is a morphism of coverings $\mu:\; M_1 \times_M M_1\arrow M_1$ together with a morphism $e:\; M \arrow M_1$ from a trivial covering to $M_1$ and $\in \Hom_M(M_1)$ such that $\mu$ defines an additive structure of an abelian group on the set $\pi_1^{-1}(x)$ for each $x\in M$, with $e(x)$ a unit in this group and $a$ the inverse. \remark If, in addition, we have a group homomorphism $\R^* \arrow \Aut_M(M_1, M_1)$ which equips each $\pi_1^{-1}(x)$ with a structure of a vector space, we obtain {\bf\blue a structure of a vector space on a covering}. \definition {\bf \blue A local system} is a covering with a structure of an abelian group or a vector space. \newpage {\bf \blue \'Etal\'e space of a locally constant sheaf} \theorem Let ${\cal F}= {\Bbb V}_M$ be a locally constant sheaf on a manifold. {\bf \red Then its \'etal\'e space $E({\cal F})$ is a covering of $M$.} \proof Immediately follows from Corollary 1. \endproof \theorem {\bf \red Category of locally constant sheaves is equivalent to the category of local systems.} \proof Let ${\cal F}$ be a locally constant sheaf, and $E({\cal F})$ its etale space. Then $E({\cal F})$ is a covering of $M$. The structure of vector space on germs defines the structure of vector space on $E({\cal F})$. {\bf \purple This gives a functor from locally constant sheaves to local systems.} Conversely, let $\pi:\; M_1 \arrow M$ be a local system, and ${\cal F}(U)$ be the space of the sections of $\pi^{-1}(U)\stackrel \pi \arrow U$. Then ${\cal F}(U)$ is a vector space. The correspondence $U \arrow {\cal F}(U)$ gives a sheaf, which is clearly locally constant. \endproof \newpage {\bf \blue Connections (reminder)} {\bf \green Notation:} Let $M$ be a smooth manifold, $TM$ its tangent bundle, $\Lambda^iM$ the bundle of differential $i$-forms, $C^\infty M$ the smooth functions. {\bf \purple The space of sections of a bundle $B$ is denoted by $B$.} \definition A {\bf\blue connection} on a vector bundle $B$ is an operator $\nabla:\; B \arrow B \otimes \Lambda^1 M$ satisfying $\nabla(fb) = b \otimes df + f \nabla(b)$, where $f \arrow df$ is de Rham differential. When $X$ is a vector field, we denote by $\nabla_X(b)\in B$ the term $\langle \nabla(b), X\rangle$. \remark When $M=[0,a]$ is an interval, any bunlde $B$ on $M$ is trivial. Let $b_1, ..., b_n$ be a basis in $B$. Then $\nabla$ can be written as \[ \nabla_{d/dt}\left(\sum f_i b_i\right)= \sum_i \frac{df_i}{dt} b_i + \sum f_i \nabla_{d/dt} b_i \] with the last term linear on $f$. Therefore, the equation $\nabla_{d/dt}(b)=0$ is a first order ODE, and {\bf \purple it has a unique solution for any initial value $b_0=b\restrict{\{0\}}$.} \newpage {\bf \blue Curvature} Let $\nabla:\; B \arrow B \otimes \Lambda^1 M$ be a connection on a vector bundle $B$. {\bf \purple We extend $\nabla$ to an operator \[ B \stackrel{\nabla}\arrow \Lambda^{1}(M)\otimes B \stackrel{\nabla}\arrow \Lambda^{2}(M)\otimes B \stackrel{\nabla}\arrow \Lambda^{3}(M)\otimes B \stackrel{\nabla}\arrow ... \] using the Leibnitz identity $\nabla(\eta \otimes b) = d\eta\otimes b + (-1)^{\tilde \eta} \eta \wedge \nabla b$.} \remark This operation is well defined, because \begin{multline*} \nabla(\eta \otimes fb)= d\eta\otimes fb + (-1)^{\tilde \eta} \eta \wedge \nabla (f b)= \\ d\eta\otimes fb + (-1)^{\tilde \eta} \eta \wedge df\otimes b + f \eta \wedge \nabla b = d(f\eta) \otimes b + f \eta \wedge \nabla b= \nabla(f\eta\otimes b) \end{multline*} \remark Sometimes $\Lambda^{2}(M)\otimes B \stackrel{\nabla}\arrow \Lambda^{3}(M)\otimes B$ is denoted $d_\nabla$. \definition The operator $\nabla^2:\; B \arrow B\otimes \Lambda^{2}(M)$ is called {\bf \blue the curvature} of $\nabla$. \remark {\bf \purple The algebra of differential forms with coefficients in $\End B$ acts on $\Lambda^* M \otimes B$} via $\eta \otimes a (\eta' \otimes b) = \eta \wedge \eta' \otimes a(b)$, where $a\in \End(B)$, $\eta, \eta'\in \Lambda^* M$, and $b\in B$. {\bf \red This is the formula expressing the action of $\nabla^2$ on $\Lambda^* M \otimes B$}. \newpage {\bf \blue Riemann-Hilbert correspondence} \definition A connection is {\bf \blue flat} if its curvature vanishes. \theorem Let $M$ be a connected manifold, $\cac_1$ the category of representations of $\pi_1(M)$, and $\cac_2$ the category of local systems. {\bf \red Then the categories $\cac_1$ and $\cac_2$ are naturally equivalent.} \proof Follows from the equivalence between locally constant sheaves and local systems. \endproof \theorem The categories $\cac_1$ and $\cac_2$ {\bf \red are naturally equivalent to the category of vector bundles on $M$ equipped with flat connection.} {\bf \green We prove it later in this lecture.} \newpage {\bf \blue Curvature and commutators} \claim Let $X, Y\in TM$ be vector fields, $(B, \nabla)$ a bundle with connection, and $b\in B$ its section. Consider the operator \[ \Theta_B^* (X, Y, b):= \nabla_X\nabla_Yb-\nabla_Y\nabla_Xb-\nabla_{[X,Y]}b \] {\bf \red Then $\Theta_B^* (X, Y, b)$ is linear in all three arguments.} \pstep The term $\Theta_B^* (X, Y, fb)$ {\bf \purple has 3 components: one which is $C^\infty$-linear in $f$, one which takes first derivative and one which takes the second derivative.} The first derivative part is \[ \Lie_Yf\nabla_X b+ \Lie_Xf\nabla_Y b-\Lie_Yf\nabla_X b-\Lie_Xf\nabla_Y b- \Lie_{[X,Y]}f b=-\Lie_{[X,Y]}f b,\] the second derivative part is $\Lie_X\Lie_Y (f)b- \Lie_Y\Lie_X (f)b=\Lie_{[X,Y]}f$, they cancel. Therefore, {\bf \purple $\Theta_B^* (X, Y, b)$ is $C^\infty$-linear in $b$.} {\bf \green Step 2:} Since $[X, fY]= \Lie_X fY + f[X,Y]$, we have $\nabla_{[X,fY]}b=f\nabla_{[X,Y]}b + \Lie_X f\nabla_Y b$. {\bf \green Step 4:} The term $\Theta_B^* (X, fY, b)$ has two components, $f$-linear and the component with first derivatives in $f$. Step 2 implies that the component with derivative of first order is $\Lie_X f\nabla_Y b- \Lie_X f\nabla_Y b=0$. \endproof \newpage {\bf \blue Curvature and commutators (2)} \remark \[ \Theta_B^* (X, Y, b):= \nabla_X\nabla_Yb-\nabla_Y\nabla_Xb-\nabla_{[X,Y]}b \] is another definition of the curvature. {\bf \red The following theorem shows that it is equivalent to the usual definition}. \theorem Consider $\Theta_B^*:\; TM \otimes TM \otimes B \arrow B$ as a 2-form with coefficients in $\End(B)$. {\bf \red Then $\Theta^*_B=\Theta_B$,} where $\Theta_B=\nabla^2$ is the usual curvature. \pstep Since $\Theta^*_B(X,Y)$, $\Theta_B(X,Y)$ are linear in $X, Y$, it would suffice to prove this equality for coordinate vector fields $X, Y$. {\bf \green Step 2:} Consider the operator $i_X:\; \Lambda^iM \otimes B \arrow \Lambda^{i-1}M \otimes B$ of convolution with a vector field $X$. Writing $\nabla = d+ A$, where $A\in \Lambda^1 M \otimes \End B$, we obtain $\nabla_X = \Lie_X + A(X)$, which gives $[\nabla_X, i_Y]= [\Lie_X, i_Y]= 0$ when $X, Y$ are coordinate vector fields. {\bf \green Step 3:} \[ \nabla^2(b)(X,Y)= (i_Xi_Y -i_Xi_Y) \nabla^2(b)= i_Y\nabla_X \nabla b- i_X\nabla_Y \nabla b= \nabla_X\nabla_Y b- \nabla_Y \nabla_X b. \] \endproof \newpage {\bf \blue Parallel transport along the connection (reminder)} \remark When $M=[0,a]$ is an interval, any bundle $B$ on $M$ is trivial. Let $b_1, ..., b_n$ be a basis in $B$. Then $\nabla$ can be written as \[ \nabla_{d/dt}\left(\sum f_i b_i\right)= \sum_i \frac{df_i}{dt} b_i + \sum f_i \nabla_{d/dt} b_i \] with the last term linear on $f$. \theorem Let $B$ be a vector bundle with connection over $\R$. Then for each $x\in \R$ and each vector $b_x \in B\restrict x$ {\bf \red there exists a unique section $b\in B$ such that $\nabla b=0$, $b\restrict x= b_x$.} \proof This is existence and uniqueness of solutions of an ODE $\frac {db}{dt} + A(b)=0$. \endproof \definition Let $\gamma:\; [0, 1] \arrow M$ be a smooth path in $M$ connecting $x$ and $y$, and $(B, \nabla)$ a vector bundle with connection. Restricting $(B, \nabla)$ to $\gamma([0,1])$, we obtain a bundle with connection on an interval. Solve an equation $\nabla(b)=0$ for $b\in B\restrict{\gamma([0,1])}$ and initial condition $b\restrict x= b_x$. This process is called {\bf\blue parallel transport} along the path via the connection. The vector $b_y:= b\restrict y$ is called {\bf\blue vector obtained by parallel transport of $b_x$ along $\gamma$}. \newpage {\bf \blue Holonomy group} \definition (Cartan, 1923) Let $(B,\nabla)$ be a vector bundle with connection over $M$. For each loop $\gamma$ based in $x\in M$, let $V_{\gamma, \nabla}:\; B\restrict x \arrow B\restrict x$ be the corresponding parallel transport along the connection. The {\bf \blue holonomy group} of $(B,\nabla)$ is a group generated by $V_{\gamma, \nabla}$, for all loops $\gamma$. If one takes all contractible loops instead, $V_{\gamma, \nabla}$ generates {\bf \blue the local holonomy}, or {\bf \blue the restricted holonomy} group. \remark Let $B_1= B^{\otimes n} \otimes (B^*)^{\otimes m}$ be a tensor power of $B$. The connection on $B$ gives the connection on $B_1$. Since parallel transport is compatible with the tensor product, {\bf \purple the holonomy representation, associated with $B_1$, is the corresponding tensor power of $B\restrict x$.} \definition Let $B$ be a vector bundle, and $\Psi$ a section of its tensor power. We say that {\bf \blue connection $\nabla$ preserves $\Psi$} if $\nabla(\Psi)=0$. In this case we also say that the tensor $\Psi$ is {\bf \blue parallel} with respect to the connection. \newpage {\bf \blue Flat bundles} \remark $\nabla(\Psi)=0$ is equivalent to $\Psi$ being a solution of $\nabla(\Psi)=0$ on each path $\gamma$. This means that {\bf \purple parallel transport preserves $\Psi$.} We obtained \corollary {\bf \red A section of the tensor power of $B$ is parallel if and only if it is holonomy invariant.} \definition A bundle is {\bf \blue flat} if its curvature vanishes. The following theorem will be proven later today. \theorem Let $(B, \nabla)$ be a vector bundle with connection over a simply connected manifold. {\bf \red Then $B$ is flat if and only if its holonomy group is trivial}. \newpage {\bf \blue Fiber of a locally free sheaf} \definition Recall that {\bf\blue a vector bundle} is a locally free sheaf of modules over $C^\infty M$. A vector bundle is called {\bf \blue trivial} if it is isomorphic to $(C^\infty M)^n$. \definition Let ${\cal B}$ be an $n$-dimensional locally free sheaf of $C^\infty$-modules on $M$, $x\in M$ a point, ${\goth m}_x\subset C^\infty M$ an ideal of $x\in M$ in $C^\infty M$. Define {\bf\blue the fiber} of ${\cal B}$ in $x$ as a quotient ${\cal B}(M)/{\goth m}{\cal B}$. A fiber of ${\cal B}$ is denoted ${\cal B}\restrict x$. \remark {\bf \purple A fiber of a vector bundle of rank $n$ is an $n$-dimensional vector space.} \remark Let ${\cal B}= C^\infty M^n$, and $b\in {\cal B}\restrict x$ a point of a fiber, represented by a germ $\phi \in {\cal B}_x=C^\infty_m M^n$, $\phi=(f_1, ..., f_n)$. Consider a map $\Psi$ from the set of all fibers ${\cal B}$ to $M \times \R^n$, mapping $(x, \phi=(f_1, ..., f_n))$ to $(f_1(x), ..., f_n(x))$. {\bf \red Then $\Psi$ is bijective.} Indeed, ${\cal B}\restrict x=\R^n$. \newpage {\bf \blue Total space of a vector bundle} \definition Let ${\cal B}$ be an $n$-dimensional locally free sheaf of $C^\infty$-modules. Denote the set of all vectors in all fibers of ${\cal B}$ over all points of $M$ by $\Tot {\cal B}$. Let $U\subset M$ be an open subset of $M$, with ${\cal B}\restrict U$ a trivial bundle. Using the local bijection $\Tot {\cal B}(U)=U \times \R^n$ we consider topology on $\Tot {\cal B}$ induced by open subsets in $\Tot {\cal B}(U)=U \times \R^n$ for all open subsets $U\subset M$ and all trivializations of ${\cal B}\restrict U$. Then $\Tot {\cal B}$ is called {\bf \blue a total space of a vector bundle ${\cal B}$.} \claim The space $\Tot {\cal B}$ with this topology {\bf \purple is a locally trivial fibration over $M$, with fiber $\R^n$.} \remark Let $B$ be a vector bundle on $M$, and $\psi\in B^*$ a section of its dual. Then $\psi$ defines a function $x\arrow \langle \psi, x\rangle$ on its total space $\Tot(B)\stackrel \pi \arrow M,$ linear on fibers of $\pi$. This gives a {\bf \purple bijective correspondence between sections of $B^*$ and functions on $\Tot(B)$ linear on fibers.} This gives the following claim \claim Let $B$ be a vector bundle and $\Sym^* B^*$ the direct sum of all symmetric tensor powers of $B^*$. {\bf \red Then the ring of sections of $\Sym^* B^*$ is identified with the ring of all smooth functions on $\Tot B\stackrel \pi\arrow M$ which are polynomial on fibers of $\pi$.} \endproof \newpage {\bf \blue Polynomial functions on $\Tot(B)$ } \claim Let $D$ be the space of derivations $\delta:\; \R[x_1, ..., x_n] \arrow \C^\infty \R^n$. {\bf \red Then $D$ is the space of derivations of the ring $\C^\infty \R^n$.} \endproof \exercise {\bf \purple Prove it.} The same argument brings the following {\bf \green CLAIM 1:} Let $D$ be the space of derivations $\delta:\; \Sym^* B^* \arrow \C^\infty (\Tot B)$. {\bf \red Then $D$ is the space of derivations of the ring $\C^\infty (\Tot B)$.} \proof Indeed, any derivation which vanishes on fiberwise polynomial functions vanishes everywhere on $\C^\infty (\Tot B)$. \endproof \newpage {\bf \blue Vector fields on $\Tot(B)$ } \theorem Let $(B,\nabla)$ be a bundle on $M$ with connection, and $X\in TM$ a vector field. {\bf \red Then there exists a vector field $\tau_\nabla(X)$ on $\Tot(B)$ mapping a section $u\in \Sym^* B^*$ to $\nabla_X u$.} \proof Let $u, v \in \Sym^* B^*$, and $uv\in \Sym^* B^*$ their product. Then $\nabla_x(uv)= u \nabla_x v + v \nabla_x u$ because $\nabla(b_1 \otimes b_2) = \nabla(b_1) \otimes b_2 + b_1 \otimes \nabla(b_2)$. Therefore, $\tau_\nabla(X)(u):=\nabla_x(u)$ is a derivation of the ring of functions on $\Tot(B)$ which are polynomial on fibers. By Claim 1, any such derivation can be uniquely extended to a vector field on $\Tot(B)$. \endproof \definition Let $(B,\nabla)$ be a bundle with connection on $M$. The corresponding {\bf \blue Ehresmann connection} on $\Tot(B)$ is the distribution $E_\nabla\subset T \Tot(B)$ obtained as $\tau_\nabla(TM)$. \newpage {\bf \blue Vector fields on $\Tot(B)$ and parallel sections} {\bf \green CLAIM 2:} Let $(B,\nabla)$ be a bundle with connection, and $\pi:\; \Tot(B)\arrow M$ the standard projection, and $T_\pi\Tot(B)=\ker D\pi$ is the vertical tangent space (Lecture 14). \\ \phantom{x}\ \ \ (i) {\bf \red Then $T\Tot B = E_\nabla\oplus T_\pi\Tot(B)$, where $E_\nabla$ is the Ehresmann connection. } \\ \phantom{x}\ \ \ (ii) Moreover, {\bf \red a section $f$ of $B$ is parallel if an only if its image $f(M)\subset \Tot(B)$ is tangent to $E_\nabla$.} \proof The second assertion is clear from the definition: {\bf \purple a section $b$ is tangent to $E_\nabla$ if it is preserved by all vector fields $a=\tau_\nabla(X)$ generating $E_\nabla$.} In this case $\Lie_a(\tilde b)=0$, where $\tilde b$ is a function on $\Tot(B^*)$ defined by $b$. However, $\Lie_a(\tilde b)=\widetilde{\nabla_X(b)}$ where $\widetilde{\nabla_X(b)}$ is a function on $\Tot(B^*)$ associated with $\nabla_X(b)$. Therefore, $\Lie_a(\tilde b)=0$ $\Leftrightarrow$ $\nabla_X(b)=0$. To prove (i), we notice that $D\pi\restrict{E_\nabla}:\; E_\nabla \arrow TM$ is an isomorphism at every point of $\Tot B$. Indeed, these bundles have the same rank, and for each $\tau_\nabla(X)\in E_\nabla$, this vector field acts on functions pulled back from $M$ as $\Lie_X$, hence $D\pi\restrict{E_\nabla}$ is injective. \endproof {\bf \blue Bundles with trivial holonomy } \theorem Let $(B, \nabla)$ be a vector bundle with connection over a simply connected manifold. {\bf \red Then $B$ is flat if and only if its holonomy group is trivial}.\\ \proof Let $B$ be a flat bundle on $M$, and $X, Y \in TM$ commuting vector fields. Then $\nabla_X:\; B \arrow B$ commutes with $\nabla_Y$. Then the Ehresmann connection bundle $E_\nabla\subset T\Tot B$ is generated by commuting vector fields $\tau_\nabla(X)$, $\tau_\nabla(Y)$, ..., hence it is involutive. By Frobenius theorem, every point $b\in \Tot(B)$ is contained in a leaf $\cal E$ of the corresponding foliation, tangent to $E_\nabla$. By Claim 2, such a leaf is a parallel section of $B$. The projection from $\cal E$ to $M$ is a covering. Since $M$ is simply connected, $\cal E=M$, and $B$ is trivialized by parallel sections. Conversely, assume that $B$ has trivial holonomy. Then $\Tot(B)= M \times B\restrict x$ because each point is contained in a unique parallel section, hence the bundle $E_\nabla$ is involutive. Then $[\nabla_X, \nabla_Y]=0$ for any commuting $X, Y\in TM$, and the curvature vanishes. \endproof {\bf \green Corollary 1:} Let $B$ be a flat vector bundle on a simply connected, connected manifold $M$. {\bf \red Then for each $x\in M$ and each $b \in B\restrict x$, there exists a unique parallel section of $B$ passing through $b$.} \endproof \newpage {\bf \blue Riemann-Hilbert correspondence} \theorem The category of locally constant sheaves of vector spaces {\bf \red is naturally equivalent to the category of vector bundles on $M$ equipped with flat connection.} \\ \pstep Consider a constant sheaf $\R_M$ on $M$. This is a sheaf of rings, and any locally constant sheaf is a sheaf of $\R_M$-modules. Let ${\Bbb V}$ be a locally constant sheaf, and $B:= {\Bbb V}\otimes_{\R_M} \C^\infty M$. Since ${\Bbb V}$ is locally constant, the sheaf $B$ is a locally free sheaf of $C^\infty$-modules, that is, a vector bundle. Let $U\subset M$ be an open set such that ${\Bbb V}\restrict U$ is constant. If $v_1, ..., v_n$ is a basis in ${\Bbb V}(U)$, all sections of $B(U)$ have a form $\sum_{i=1}^n f_i v_i$, where $f_i \in C^\infty U$. Define the connection $\nabla$ by $\nabla\left (\sum_{i=1}^n f_i v_i\right) = \sum df_i \otimes v_i$. This connection is flat because $d^2=0$. It is independent from the choice of $v_i$ because $v_i$ is defined canonically up to a matrix with constant coefficients. {\bf \purple We have constructed a functor from locally constant sheaves to flat vector bundles}. {\bf \green Step 2:} Let now $(B, \nabla)$ be a flat bundle over $M$. The functor to locally constant sheaves takes $U\subset M$ and maps it to the space of parallel sections of $B$ over $U$. This defines a sheaf ${\Bbb B}(U)$. For any simply connected $U$, and any $x\in M$, the space ${\Bbb B}(U)$ is identified with a vector space $B\restrict x$ (Corollary 1), hence ${\Bbb B}(U)$ is locally constant. Clearly, $B= {\Bbb B}\otimes_{\R_M} \C^\infty M$, hence {\bf \purple this construction gives an inverse functor to ${\Bbb V}\mapsto {\Bbb V}\otimes_{\R_M} \C^\infty M$.} \endproof \end{document} \end{document}