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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 1: complex manifolds} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \\[14mm] {\small Misha Verbitsky } \\[20mm] {\tiny\bf HSE, room 306, 16:20, \\[2mm] September 23, 2020 } \end{center} \newpage {\bf \blue Complex structure on a vector space} \definition Let $V$ be a vector space over $\R$, and $I:\; V \arrow V$ an automorphism which satisfies $I^2 = - \Id_V$. Such an automorphism is called {\bf\blue a complex structure operator} on $V$. {\bf \blue We extend the action of $I$ on the tensor spaces $V\otimes V \otimes ... \otimes V \otimes V^*\otimes V^* \otimes ... \otimes V^*$ by multiplicativity:} $I(v_1 \otimes ... \otimes w_1 \otimes ... \otimes w_n)= I(v_1) \otimes ... \otimes I(w_1)\otimes ... \otimes I(w_n)$. {\bf \red Trivial observations:}\\ 1. {\bf \purple The eigenvalues $\alpha_i$ of $I$ are $\pm \1$.} Indeed, $\alpha_i^2=-1$. 2. {\bf \purple $V$ admits an $I$-invariant, positive definite scalar product (``metric'') $g$.} Take any metric $g_0$, and let $g:= g_0 + I(g_0)$. 3. {\bf \purple $I$ is orthogonal for such $g$.}\\ Indeed, $g(Ix,Iy)=g_0(x,y)+ g_0(Ix,Iy)=g(x,y)$. 4. {\bf \purple $I$ diagonalizable over $\C$.} Indeed, any orthogonal matrix is diagonalizable. 5. {\bf \purple There are as many $\1$-eigenvalues as there are $-\1$-eigenvalues.} \newpage {\bf \blue Complex structure operator in coordinates} This implies that in an appropriate basis in $V \otimes_\R \C$, the complex structure operator is diagonal, as follows: \[\left[ \begin{array}{c|c} \begin{array}{cccc} \1 & & &\\ & \1 & & \\ & & \ddots & \\ & & & \1 \end{array} & 0 \\ \hline 0 & \begin{array}{cccc} -\1 & & &\\ & -\1 & & \\ & & \ddots & \\ & & & -\1 \end{array} \end{array}\right] \] We also obtain its normal form in a real basis: \[ \begin{bmatrix} 0 & -1 \\ 1 & 0 \\ & & 0 & -1 \\ & & 1 & 0 \\ & & & & \ddots \\ & & & & & \ddots \\ & & & & & & 0 & -1 \\ & & & & & & 1 & 0 \end{bmatrix} \] \newpage {\bf \blue Hodge decomposition} \definition Let $(V,I)$ be a space equipped with a complex structure. {\bf \blue The Hodge decomposition} $V\otimes_\R \C:= V^{1,0}\oplus V^{0,1}$ is defined in such a way that $V^{1,0}$ is a $\1$-eigenspace of $I$, and $V^{0,1}$ a $-\1$-eigenspace. \remark In the same way one defines the Hodge decomposition on the dual space $V^*$. {\bf \green Remark 1:} The space $V^{1,0} \subset V\otimes_\R \C$ {\bf \purple uniquely determines the operator $I$.} Indeed, $I=\1$ on $V^{1,0}$ and $I=-\1$ on $V^{0,1}$. This gives {\bf \red a bijection between the set of complex structures on $V$ and the set of subspaces $W \subset V\otimes_\R \C$ such that $\dim_\C W = \frac 1 2 \dim_\R V$ and $W \cap \bar W =0$.} \newpage {\bf \blue Hermitian structures} \definition Let $(V,I)$ be a space equipped with a complex structure. {\bf \blue The Hodge decomposition} $V\otimes_\R \C:= V^{1,0}\oplus V^{0,1}$ is defined in such a way that $V^{1,0}$ is a $\1$-eigenspace of $I$, and $V^{0,1}$ a $-\1$-eigenspace. % Denote by $\nu$ the {\bf \blue real structure operator,} %$\nu(\sum \lambda_i w_i)=\sum \bar \lambda_i w_i$, where $w_i\in V$ %is a basis. Then $\nu(I(z))=I(\nu(z))$, that is, {\bf \red $I$ is real}. %For any $\1$-eigenvector %$w$, one has $I(\nu(w))=\nu(I(w))=\nu(\1 w)=-\1 w$, %hence {\bf \purple %$\nu$ exchanges $\1$-eigenvectors and $-\1$-eigenvectors.} \definition An $I$-invariant positive definite scalar product on $(V,I)$ is called {\bf \blue an Hermitian metric}, and $(V,I,g)$ -- an Hermitian space. \remark Let $I$ be a complex structure operator on a real vector space $V$, and $g$ -- a Hermitian metric. Then {\bf \red the bilinear form $\omega(x,y) := g(x, Iy)$ is skew-symmetric.} Indeed, $\omega(x,y) = g(x, Iy) = g(Ix, I^2y) = -g(Ix, y) = -\omega(y, x)$. \definition A skew-symmetric form $\omega(x,y)$ is called {\bf \blue an Hermitian form on $(V,I)$}. \remark In the triple $I, g, \omega$, {\bf \purple each element can recovered from the other two.} \newpage \newpage {\bf \blue Holomorphic functions} \definition Let $I:\; TM \arrow TM$ be an endomorphism of a tangent bundle satisfying $I^2=-\Id$. Then $I$ is called {\bf \blue almost complex structure operator}, and the pair $(M,I)$ {\bf \blue an almost complex manifold}. \example $M=\C^n$, with complex coordinates $z_i=x_i + \1 y_i$, and $I(d/dx_i)=d/dy_i$, $I(d/dy_i)=-d/dx_i$. \example In complex dimension 1, {\bf \purple almost complex structure is the same as conformal structure with orientation} {\bf \red (prove it).} \definition A function $f:\; M \arrow \C$ on an almost complex manifold is called {\bf \blue holomorphic} if $df \in \Lambda^{1,0}(M)$. \remark For some almost complex manifolds, {\bf \red there are no holomorphic functions at all}, even locally. Example: $S^6$ with the unique $G_2$-invariant almost complex structure. \newpage {\bf \blue Holomorphic functions on $\C^n$} \theorem Let $f:\; M \arrow \C$ be a differentiable function on an open subset $M\subset \C^n$, with almost complex structure as above. {\bf \red Then TFAE:}\\ \ \ (1) {\bf \purple $f$ is holomorphic}.\\ \ \ (2) The differential $df:\; TM \arrow \C$, considered as a form on the vector space $T_x M=T_x\C^n=\C^n$ {\bf \purple is $\C$-linear.} \\ \ \ (3) For any complex affine line $L\in \C^n$, the restriction $f\restrict L=\C$ is {\bf \purple holomorphic (complex analytic) as a function of one complex variable.}\\ \ \ (4) $f$ is expressed as a sum of Taylor series around any point $(z_1, ..., z_n)\in M$. %\[ %f(z_1+t_1, z_2+ t_2, ..., z_n +t_n)= %\sum_{i_1, ..., i_n}a_{i_1, ..., i_n} t_1^{i_1}t_2^{i_2}...t_n^{i_n}. %\] %(here we assume that the complex numbers $t_i$ satisfy %$|t_i|<\epsilon$, where $\epsilon$ depends on $f$ and $M$). \proof (1) and (2) are tautologically equivalent. Equivalence of (1) and (3) is also clear, because a restriction of $\theta \in \Lambda^{1,0}(M)$ to a line is a $(1,0)$-form on a line, and, conversely, if $df$ is of type (1,0) on each complex line, it is of type (1,0) on $TM$, which is implied by the following linear-algebraic observation. \lemma Let $\eta\in V^*\otimes \C$ be a complex-valued linear form on a vector space $(V,I)$ equipped with a complex structure. {\bf \purple Then $\eta\in \Lambda^{1,0}(V)$ if and only if its restriction to any $I$-invariant 2-dimensional subspace $L$ belongs to $\Lambda^{1,0}(L)$.} \exercise {\bf \red Prove it.} (4) clearly implies (2). (1) implies (4) by Cauchy formula. %\question Who remembers the proof of Cauchy formula? \newpage {\bf \blue Sheaves} \definition A {\bf\blue presheaf of functions} on a topological space $M$ is a collection of subrings ${\cal F}(U)\subset C(U)$ in the ring $C(U)$ of all functions on $U$, for each open subset $U\subset M$, such that the restriction of every $\gamma\in{\cal F}(U)$ to an open subset $U_1\subset U$ belongs to ${\cal F}(U_1)$. \definition A presheaf of functions ${\cal F}$ is called {\bf\blue a sheaf of functions} if these subrings satisfy the following condition. Let $\{U_i\}$ be a cover of an open subset $U\subset M$ (possibly infinite) and $f_i\in{\cal F}(U_i)$ a family of functions defined on the open sets of the cover and compatible on the pairwise intersections: $$f_i|_{U_i\cap U_j}=f_j|_{U_i\cap U_j}$$ for every pair of members of the cover. {\bf \purple Then there exists $f\in{\cal F}(U)$ such that $f_i$ is the restriction of $f$ to $U_i$ for all $i$.} \newpage {\bf \blue Sheaves and exact sequences} \remark {\bf \purple A presheaf of functions} is a collection of subrings of functions on open subsets, compatible with restrictions. {\bf\purple A sheaf of fuctions is a presheaf allowing ``gluing''} a function on a bigger open set if its restrictions to smaller open sets are compatible. \definition A sequence $A_1 \arrow A_2 \arrow A_3 \arrow ...$ of homomorphisms of abelian groups or vector spaces is called {\bf\blue exact} if the image of each map is the kernel of the next one. \claim A presheaf ${\cal F}$ is a sheaf if and only if for every cover $\{U_i\}$ of an open subset $U\subset M$, {\bf \red the sequence of restriction maps $$0\to{\cal F}(U)\to\prod\limits_i{\cal F}(U_i)\to\prod\limits_{i\ne j}{\cal F}(U_i\cap U_j)$$ is exact,} with $\eta\in {\cal F}(U_i)$ mapped to $\eta\restrict{U_i\cap U_j}$ and $-\eta\restrict{U_j\cap U_i}$. \newpage {\bf \blue Sheaves and presheaves: examples} {\bf \green Examples of sheaves:} * Space of continuous functions * Space of smooth functions, any differentiability class * Space of real analytic functions {\bf \green Examples of presheaves which are not sheaves:} * Space of constant functions {\bf \purple (why?)} * Space of bounded functions {\bf \purple (why?)} \newpage {\bf \blue Ringed spaces} A {\bf\blue ringed space} $(M,{\cal F})$ is a topological space equipped with a sheaf of functions. A~{\bf \blue morphism} $(M,{\cal F})\stackrel\Psi\longrightarrow(N,{\cal F}')$ of ringed spaces is a continuous map $M\stackrel\Psi\longrightarrow N$ such that, for every open subset $U\subset N$ and every function $f\in{\cal F}'(U)$, the function $\psi^* f:=f\circ\Psi$ belongs to the ring ${\cal F}\big(\Psi^{-1}(U)\big)$. An {\bf\blue isomorphism} of ringed spaces is a homeomorphism $\Psi$ such that $\Psi$ and $\Psi^{-1}$ are morphisms of ringed spaces. \example Let $M$ be a manifold of class $C^i$ and let $C^i(U)$ be the space of functions of this class. {\bf \purple Then $C^i$ is a sheaf of functions, and $(M, C^i)$ is a ringed space.} \remark Let $f:\; X \arrow Y$ be a smooth map of smooth manifolds. Since a pullback $f^*\mu$ of a smooth function $\mu\in C^\infty(M)$ is smooth, {\bf \purple a smooth map of smooth manifolds defines a morphism of ringed spaces.} {\bf \red Converse is also true:} \newpage {\bf \blue Ringed spaces and smooth maps} \claim Let $(M, C^i)$ and $(N,C^i)$ be manifolds of class $C^i$. Then {\bf \red there is a bijection between smooth maps $f:\; M \arrow N$ and the morphisms of corresponding ringed spaces.} {\bf \green Proof:} Any smooth map induces a morphism of ringed spaces. Indeed, {\bf \purple a composition of smooth functions is smooth, hence a pullback is also smooth.} Conversely, let $U_i \arrow V_i$ be a restriction of $f$ to some charts; to show that $f$ is smooth, it would suffice to show that $U_i \arrow V_i$ is smooth. However, we know that a pullback of any smooth function is smooth. {\bf \purple Therefore, Claim is implied by the following lemma.} \lemma Let $M,N$ be open subsets in $\Bbb R^n$ and let $f:\; M\to N$ map such that a pullback of any function of class $C^i$ belongs to $C^i$. {\bf \red Then $f$ is of class $C^i$.} {\bf \green Proof:} Apply $f$ to coordinate functions. \endproof \newpage {\bf \blue Smooth manifolds defined through sheaves} As we have just shown, this definition is equivalent to the previous one. \definition Let $(M,{\cal F})$ be a topological manifold equipped with a sheaf of functions. It is said to be a {\bf\blue smooth manifold of class} $C^\infty$ or $C^i$ if every point in $(M,{\cal F})$ has an open neighborhood isomorphic to the ringed space $(\Bbb R^n,{\cal F}')$, where ${\cal F}'$ is a ring of functions on $\Bbb R^n$ of this class. \definition {\bf \blue A chart}, or {\bf\blue a coordinate system} on an open subset $U$ of a manifold $(M,{\cal F})$ is an isomorphism between $(U,{\cal F})$ and an open subset in $(\Bbb R^n,{\cal F}')$, where ${\cal F}'$ are functions of the same class on $\Bbb R^n$. \definition {\bf\blue Diffeomorphism} of smooth manifolds is a homeomorphism $\phi$ which induces an isomorphim of ringed spaces, that is, $\phi$ and $\phi^{-1}$ map (locally defined) smooth functions to smooth functions. \newpage {\bf \blue Complex manifolds} \definition {\bf \blue A holomorphic function} on $\C^n$ is a function $f:\; \C^n \arrow \C$ such that $df$ is complex linear, that is $df\in \Lambda^{1,0}(M)$. \remark Holomorphic functions form a sheaf. \definition {\bf \blue A complex manifold} $M$ is a ringed space which is locally isomorphic to an open ball in $\C^n$ with a sheaf of holomorphic functions. \remark In other words, {\bf \purple $M$ is covered with open balls embedded to $\C^n$} and transition functions (being coordinate functions for one ball considered in other coordinate system) {\bf \purple are holomorphic.} \newpage {\bf \blue Complex manifolds and almost complex manifolds} \definition {\bf \blue Standard almost complex structure} is $I(d/dx_i)=d/dy_i$, $I(d/dy_i)=-d/dx_i$ on $\C^n$ with complex coordinates $z_i=x_i+\1 y_i$. \definition A map $\Psi:\; (M,I)\arrow (N,J)$ from an almost complex manifold to an almost complex manifold is called {\bf \blue holomorphic} if $\Psi^*(\Lambda^{1,0}(N))\subset \Lambda^{1,0}(M)$. \remark This is the same as $d\Psi$ being complex linear; for standard almost complex structures, {\bf \purple this is the same as the coordinate components of $\Psi$ being holomorphic functions.} {\bf \green Another definition:} {\bf \blue A complex manifold} is a manifold equipped with an atlas with charts identified with open subsets of $\C^n$ and transition functions holomorphic. \exercise Prove that {\bf \purple this definition is equivalent to the one with sheaves}. \newpage {\bf \blue Integrability of almost complex structures} \definition An almost complex structure $I$ on a manifold is called {\bf\blue integrable} if any point of $M$ has a neighbourhood $U$ diffeomorphic to an open subset of $\C^n$, in such a way that the almost complex structure $I$ is induced by the standard one on $U\subset \C^n$. \claim {\bf \red Complex structure on a manifold $M$ uniquely determines an integrable almost complex structure, and is determined by it.} \proof Complex structure on a manifold $M$ is determined by the sheaf of holomorphic functions $\calo_M$, and $\calo_M$ is determined by $I$ as explained above. Therefore, an integrable almost complex structure defines a complex structure. Conversely, every complex structure gives a sub-bundle in $\Lambda^{1,0}(M)=d\calo_M\subset \Lambda^1(M,\C)$, and {\bf \purple such a sub-bundle defines an almost complex structure by Remark 1.} \endproof \end{document}