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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 Symplectic geometry\\[15mm] \small lecture 9: pseudo-holomorphic curves and calibrations} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \\[14mm] {\small Misha Verbitsky } \\[20mm] {\tiny\bf HSE, room 306, 16:20, \\[2mm] October 2, 2021 } \end{center} \newpage {\bf \blue Complex manifolds} {\bf\green DEFINITION:} Let $M$ be a smooth manifold. An {\bf \blue almost complex structure} is an operator $I:\; TM \arrow TM$ which satisfies $I^2 = - \Id_{TM}$. The eigenvalues of this operator are $\pm \1$. The corresponding eigenvalue decomposition is denoted $TM=T^{0,1}M\oplus T^{1,0}(M)$. {\bf\green DEFINITION:} An almost complex structure is {\bf \blue integrable} if $\forall X,Y \in T^{1,0}M$, one has $[X,Y]\in T^{1,0}M$. In this case $I$ is called {\bf \blue a complex structure operator}. A manifold with an integrable almost complex structure is called {\bf \blue a complex manifold}. {\bf\green THEOREM:} (Newlander-Nirenberg)\\ {\bf \purple This definition is equivalent to the usual one.} \newpage {\bf \blue Almost complex submanifolds} \definition Let $(M,J)$ be an almost complex manifoldm and $Z\subset M$ a closed subvariety. We allow $Z$ to be singular, but we want the non-singular part to have a finite Riemannian volume in a neighbourhood of any singular point (prove that the {\bf \purple finiteness of the volume is independent of the choice of a Riemannian metric)}. Suppose that $J(T_zZ) \subset T_zZ$ for every smooth point $z\in Z$. Then $Z\subset M$ is called {\bf \blue an almost complex subvariety}. \remark A subvariety of a complex manifold {\bf \red is almost complex if and only if it is complex,} that is, it can be given as the set of common solutions of a system of holomorphic equations. This result follows from Newlander-Nirenberg for smooth subvarieties, and needs a non-trivial argument for singular. {\bf \green FACT:} A general (non-integrable) almost complex manifold {\bf \purple does not admit almost complex subvarieties of dimension $\dim_\C \geq 2$}, even locally. For example, {\bf \red $S^6$ with the standard non-integrable complex structure has no almost complex subvarieties} except 1-dimensional. \newpage {\bf \blue Almost complex symplectic manifolds} \definition Let $(M, \omega)$ be a symplectic manifold, and $I$ an almost complex structure. We say that {\bf \blue $I$ is compatible with the symplectic structure} if $g(x, y):= \omega(Ix, y)$ for some Riemannian form $g$. \remark In the same way one defines {\bf \blue almost complex structures compatible with a non-degenerate 2-form}. \definition {\bf \blue An almost K\"ahler manifold} is a manifold $(M, \omega, I)$ equipped with a symplectic form $\omega$ and an almost complex structure $I$ compatible with $\omega$. \theorem Let $(M, \omega)$ be a manifold equipped with a non-degenerate skew-symmetric 2-form. Then {\bf \red the space of almost complex structures compatible with $\omega$ is contractible.} \newpage {\bf \blue Space of almost complex structures compatible with $\omega$} {\bf \green THEOREM 1:} Let $(M, \omega)$ be a manifold equipped with a non-degenerate skew-symmetric 2-form. Then {\bf \red the space $C$ of almost complex structures compatible with $\omega$ is contractible.} \pstep We identify $C$ with the space of Riemannian metrics $g$ such that $g^{-1}\omega$ is an almost complex structure compatible with the $\omega$. Since the space $R$ of Riemannian metrics is convex, it is contractible, hence {\bf \purple to prove that $C$ is conractible it would suffice to show that $C$ is a deformational retract of $R$.} {\bf \green Step 2:} Let $A:= g^{-1}\omega$, that is, $g(Ax,y)=\omega(x,y)$. {\bf \purple The matrix $A$ us skew-symmetric:} $g(Ax,y)= \omega(x,y)= - \omega(y, x)=-g(Ay,x)=-g(x,Ay)$. A skew-symmetric matrix can be written in some orthonormal basis as \[ \small \omega=\begin{pmatrix} A=\begin{matrix}0 & \alpha_1\\ -\alpha_1 & 0\end{matrix} & & 0 \\ & \ddots & \\ 0 & & \begin{matrix}0 & \alpha_n \\ -\alpha_n & 0\end{matrix} \end{pmatrix} \] Since $g(A^2x,x)=-g(Ax,Ax)$, {\bf \purple the matrix $A^^2$ is symmetric and negative definite.} {\bf \red Then $B_t:=e^{-\frac{t}2\log (-A^2)}$ is correctly defined, symmetric, and continuously depends on $t$ and $A$.} \newpage {\bf \blue Space of almost complex structures compatible with $\omega$ (2)} {\bf \green Step 3:} The operator $AB_1$ is written in the same coordinates as \[ \begin{pmatrix} \omega=\begin{matrix}0 & 1\\ -1 & 0\end{matrix} & & 0 \\ & \ddots & \\ 0 & & \begin{matrix}0 & 1 \\ -1 & 0\end{matrix} \end{pmatrix}, \] that is, defines an almost complex structure on $M$. Since $\omega(AB_1x, y)= g(B_1x, y)$, and $B_1$ is symmetric, {\bf \purple this almost complex structure is compatible with $\omega$.} {\bf \green Step 4:} The map $g,t \stackrel{\Psi_t}\arrow g(B_tx, y)$ for $t=1$ gives the metric $g(B_1\cdot, \cdot)\in C$, and for $t=0$ gives $g$. {\bf\purple Therefore, the map $\Psi_t$ retracts $C$ to $R$.} \endproof \newpage {\bf \blue Conformal structure} \definition Let $h, h'$ be Riemannian structures on $M$. These Riemannian structures are called {\bf \blue conformally equivalent} if $h' = fh$, where $f$ is a positive smooth function. \definition {\bf \blue Conformal structure} on $M$ is a class of conformal equivalence of Riemannian metrics. \exercise Let $I$ be an almost complex structure on a 2-dimensional Riemannian manifold, and $h, h'$ two Hermitian metrics. {\bf \red Prove that $h$ and $h'$ are conformally equivalent}. Prove that any metric conformally equivalent to a Hermitian metric is Hermitian. \remark {\bf \purple The last statement is clear from the definition, and true in any dimension. } \newpage {\bf \blue Stereographic projection} \centerline{\epsfig{file=1280px-Stereographic_projection_in_3D.png,width=0.4\linewidth} \ \ \epsfig{file=Stereographic_projection_SW.JPG,width=0.4\linewidth}} {\bf \blue Stereographic projection} is a light projection from the south pole to a plane tangent to the north pole. {\bf \purple stereographic projection is conformal} {\bf \red (prove it!)} \newpage {\bf \blue Stereographic projection (2)} \ \ \centerline{\epsfig{file=1024px-Stereographic_with_Tissot_Indicatrices.png,width=0.5\linewidth}} {\em \green The stereographic projection with Tissot's indicatrix of deformation. } \newpage {\bf \blue Cylindrical projection} \ \ \centerline{\epsfig{file=cylindrical-proj.png,width=\linewidth}} {\em \green Cylindrical projection is not conformal. However, it is volume-preserving. } \newpage {\bf \blue Conformal structures and almost complex structures} \remark The following theorem implies that almost complex structures on a 2-dimensional oriented manifold are equivalent to conformal structures. \theorem Let $M$ be a 2-dimensional oriented manifold. Given a complex structure $I$, let $\nu$ be the conformal class of its Hermitian metric (it is unique as shown above). {\bf \red Then $\nu$ determines $I$ uniquely.} \proof Choose a Riemannian structure $h$ compatible with the conformal structure $\nu$. Since $M$ is oriented, the group $SO(2)=U(1)$ acts in its tangent bundle in a natural way: $\rho:\; U(1) \arrow GL(TM)$. Rescaling $h$ does not change this action, hence it is determined by $\nu$. Now, define $I$ as $\rho(\1)$; then $I^2= \rho(-1)=-\Id$. Since $U(1)$ acts by isometries, this almost complex structure is compatible with $h$ and with $\nu$. \endproof \definition {\bf \blue A Riemann surface} is a complex manifold of dimension 1, or (equivalently) an oriented 2-manifold equipped with a conformal structure. \exercise Prove that {\bf \purple a smooth map from one Riemann surface to another is holomorphic if and only if it preserves the conformal structure everywhere.} \newpage {\bf \blue Calibrations} \newcommand{\comass}{\operatorname{\sf comass}} \definition (Harvey-Lawson, 1982)\\ Let $W\subset V$ be a $p$-dimensional subspace in a Euclidean space, and $\Vol(W)$ denote the Riemannian volume form of $W \subset V$, defined up to a sign. For any $p$-form $\eta \in \Lambda^p V$, let {\bf\blue comass} $\comass(\eta)$ be the maximum of $\frac{\eta(v_1, v_2, ..., v_p)}{|v_1||v_2|...|v_p|}$, for all $p$-tuples $(v_1, ..., v_p)$ of vectors in $V$ and {\bf\blue face} be the set of planes $W\subset V$ where $\frac\eta {\Vol(W)}=\comass(\eta)$. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \definition\label{_precalibra_Definition_} A {\bf\blue precalibration} on a Riemannian manifold is a differential form with comass $\leq 1$ everywhere. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \definition A {\bf\blue calibration} is a precalibration which is closed. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \definition Let $\eta$ be a $k$-dimensional precalibration on a Riemannian manifold, and $Z\subset M$ a $k$-dimensional subvariety (we always assume that the Hausdorff dimension of the set of singular points of $Z$ is $\leq k-2$, because in this case a compactly supported differential form can be integrated over $Z$). We say that $Z$ is {\bf\blue calibrated by $\eta$} if at any smooth point $z\in Z$, the space $T_zZ$ is a face of the precalibration $\eta$. \newpage {\small \sf \begin{center} \begin{tabular}{cc} \epsfig{file=Blaine_Lawson-1972-Berkeley.jpg,width=0.4\linewidth} & \epsfig{file=Reese_Harvey_1968_Berkeley.jpg,width=0.4\linewidth}\\ H. Blaine Lawson, Jr., & F. Reese Harvey, \\ Berkeley, 1972 & Berkeley, 1968 \end{tabular} Source: George M. Bergman, Berkeley \end{center} } \newpage {\bf \blue Calibrations and minimal submanifolds} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \remark Clearly, for any precalibration $\eta$, one has \[ \Vol(Z) \geq \int_Z\eta,\ \ \ \ \ (*) \] where $\Vol(Z)$ denotes the Riemannian volume of a compact $Z$, and the equality happens iff $Z$ is calibrated by $\eta$. If, in addition, $\eta$ is closed, the number $\int_Z\eta$ is a cohomological invariant. Then, {\bf \purple the inequality (*) implies that $Z$ minimizes the Riemannian volume in its homology class.} \definition A subvariety $Z$ is called {\bf \blue minimal} if for any sufficiently small deformation $Z'$ of $Z$ in class $C^1$, one has $\Vol(Z')\geq \Vol(Z)$. \remark {\bf \red Calibrated subvarieties are obviously minimal.} \definition An almost complex Hermitian manifold $(M, I, h)$ with closed Hermitian form $\omega:= h(I(\cdot), \cdot)$ is called {\bf \blue almost K\"ahler}. \example {\bf \blue (Wirtenger's inequality):} \\ Let $(M, I, \omega)$ be an almost K\"ahler manifold. {\bf \purple Then $\frac{\omega^d}{d! 2^d}$ is a calibration which calibrates $d$-dimensional almost complex subvarieties.} In patricular, {\bf \red almost complex submanifolds in almost K\"ahler manifolds are minimal}. \newpage {\bf \blue Pseudoholomorphic curves} \definition Let $(M,J)$ be an almost complex manifold, $(\Sigma, I)$ a Riemann surface, and $\phi:\; \Sigma \arrow M$ an $I$-holomorphic map, that is, a smooth map with $D\phi(Ix)= J(D\phi(x))$. Then $\phi(\Sigma)$ is called {\bf \blue a pseudo-holomorphic curve}, or {\bf \blue a $J$-holomorphic curve}. \theorem {\bf \blue (Wirtenger's inequality):} \\ Let $(M, I, \omega)$ be an almost K\"ahler manifold. {\bf \purple Then $\frac{1}{2}\omega$ is a calibration which calibrates pseudo-holomorphic curves.} \proof Let $g_S$ be the Riemannian volume form on $S$, and $x, y\in T_sS$ be orthogonal vectors of length 1. Then $g_S(x, y)=1$ and $\omega(x, y)= g(x, Iy)\leq 1$, and the equality is realized if and only if $x=Iy$, by Cauchy-Bunyakovsky-Schwarz inequality. \endproof \corollary {\bf \red Pseudoholomorphic curves are minimal.} \newpage {\bf \blue Symplectic capacity and the pseudoholomorphic curves} {\bf \green THEOREM 2:} Let $M=\C P^1 \times T^{2n}$ be the product of $\C P^1$ and a torus, equipped with the standard symplectic structure, and $J$ a compatible almost complex structure. {\bf \red Then for any $x\in M$ there exists a pseudo-holomorphic curve $S$ homologous to $\C P^1 \times \{m \}$ and passing through $x$.} This theorem implies Gromov's non-squeezing theorem. \theorem {\bf \blue (Gromov)} {\bf \red Symplectic capacity of a symplectic cylinder $\Cyl_1$ is equal to $\pi$.} \end{document} When it is singular, we need to show that the contribution of the singular points vanishes. {\bf \purple This would follow if we prove that $\lim\limits{\epsilon \to 0} \int_{S_\epsilon} \omega=0$,} where $S_\epsilon$ is an intersection of $S$ and an $\epsilon$-ball with the center in a singular point. {\bf\green Step 2:} Since $\int_{S_{\epsilon+\delta}} - \omega \int_{S_\epsilon} \omega = \int_{S_{\epsilon+\delta}\backslash S_\epsilon}\omega >0$, the function $\epsilon \arrow \int_{S_\epsilon}$ is monotonous. Since $\int_{S_{\epsilon}}\omega= \sum_k \int_{S_{{\frac 1 {2^k}\epsilon}\backslash S_{{\frac 1 {2^{k+1}}\epsilon}}\omega$, to prove that it converges to 0 it remains to show that $\int_{S_{\epsilon}}\omega$ is finite. \newpage {\bf \blue K\"ahler manifolds} {\bf\green DEFINITION:} An Riemannian metric $g$ on an almost complex manifiold $M$ is called {\bf \blue Hermitian} if $g(Ix, Iy)= g(x,y)$. In this case, $g(x, Iy)= g(Ix, I^2y) = - g(y, Ix)$, hence $\omega(x,y):= g(x, Iy)$ is skew-symmetric. {\bf\green DEFINITION:} The differential form $\omega\in \Lambda^{1,1}(M)$ is called {\bf \blue the Hermitian form} of $(M,I,g)$. {\bf\green THEOREM:} Let $(M,I,g)$ be an almost complex Hermitian manifold. {\bf \purple Then the following conditions are equivalent.} (i) The complex structure $I$ is integrable, and the Hermitian form $\omega$ is closed. (ii) One has $\nabla(I)=0$, where $\nabla$ is the Levi-Civita connection \[ \nabla:\; \End(TM) \arrow \End(TM)\otimes \Lambda^1(M).\] {\bf\green DEFINITION:} A complex Hermitian manifold $M$ is called {\bf \blue K\"ahler} if either of these conditions hold. The cohomology class $[\omega]\in H^2(M)$ of a form $\omega$ is called {\bf \blue the K\"ahler class} of $M$. The set of all K\"ahler classes is called {\bf \blue the K\"ahler cone}.