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\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:\ }} \newcommand{\question}{% {\bf \green QUESTION:\ }} \newcommand{\proof}{{\bf \green Proof:\ }} \newcommand{\pstep}{{\bf \green Proof. Step 1:\ }} \newcommand{\ps@verbit}{% \renewcommand{\@oddhead}{% \scriptsize {\it \small Teichmuller space of geometric structures 1 \hfil \tiny M. Verbitsky }} \renewcommand{\@evenhead}{\@oddhead} \renewcommand{\@oddfoot}{\hfil\thepage\hfil} \renewcommand{\@evenfoot}{\@oddfoot}} \pagestyle{verbit} \begin{document} \setcounter{page}{1} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{center} {\Large\bf Teichm\"uller spaces and moduli of geometric structures (1)} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \\[14mm] {\scriptsize Misha Verbitsky } \\[20mm] {\tiny\bf TWENTY-THIRD G\"OKOVA GEOMETRY / TOPOLOGY CONFERENCE\\ May 30 - June 4 (2016)\\ G\"okova, Turkey} \end{center} \newcommand{\Per}{\operatorname{\sf Per}} \newcommand{\Gr}{\operatorname{\sf Gr}} \newcommand{\Perspace}{\operatorname{{\Bbb P}\sf er}} \newcommand{\Teich}{\operatorname{Teich}} \newcommand{\Comp}{\operatorname{Comp}} \newcommand{\Symp}{\operatorname{Symp}} \newcommand{\Hyp}{\operatorname{Hyp}} \newpage {\bf \blue Plan:} 1. Set-up: Teichm\"uller space and the moduli space of geometric structures. 2. Moser's theorem. Teichm\"uller space of symplectic structures on a torus. 3. Other geometric structures and their Teichm\"uller spaces. $G_2$ structures, holomorphically symplectic structures. \newpage {\bf \blue Geometric structures} \definition {\bf \blue ``Geometric structure''} on a manifold $M$ is a reduction of its structure group $GL(n,\R)$ to a subgroup $G\subset GL(n,\R)$. However, it is easier to define it by a collection of tensors $\Psi_1, ..., \Psi_n$ such that the stabilizer $\St_{\langle \Psi_1, ..., \Psi_n\rangle}$ of $\Psi_1, ..., \Psi_n$ at each point $x\in M$ is conjugate to $G\subset GL(T_xM)$. Let me give some examples. {\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 $TM\otimes \C=T^{0,1}M\oplus T^{1,0}(M)$. \definition {\bf \blue Symplectic form} on a manifold is a non-degenerate differential 2-form $\omega$ satisfying $d\omega=0$. \newpage {\bf \blue Teichm\"uller space of geometric structures} Let ${\cal C}$ be the set of all geometric structures of a given type, say, complex, or symplectic. We put topology of uniform convergence with all derivatives on ${\cal C}$. Let $\Diff_0(M)$ be the connected component of its diffeomorphism group $\Diff(M)$ ({\bf\blue the group of isotopies}). \definition The quotient ${\cal C}/\Diff_0$ is called {\bf \blue Teichm\"uller space} of geometric strictures of this type. \definition The group $\Gamma:=\Diff(M)/\Diff_0(M)$ is called {\bf\blue the mapping class group} of $M$. It acts on $\Teich$ by homeomorphisms. \definition The orbit space ${\cal C}/\Diff=\Teich/\Gamma$ is called {\bf \blue the moduli space} of geometric structure of this type. Today I will describe $\Teich$ and $\Gamma$ in some interesting cases and explain some important concepts, such as {\bf \red ergodicity of $\Gamma$-action.} \newpage {\bf \blue Teichm\"uller space for symplectic structures} \definition Let $\Gamma(\Lambda^2 M)$ be the space of all 2-forms on a manifold $M$, and $\Symp\subset \Gamma(\Lambda^2 M)$ the space of all symplectic 2-forms. We equip $\Gamma(\Lambda^2 M)$ with $C^\infty$-topology of uniform convergence on compacts with all derivatives. Then $\Gamma(\Lambda^2 M)$ is a Frechet vector space, and $\Symp$ a Frechet manifold. \definition Consider the group of diffeomorphisms, denoted $\Diff$ or $\Diff (M)$ as a Frechet Lie group, and denote its connected component (``group of isotopies'') by $\Diff_0$. The quotient group $\Gamma:=\Diff/\Diff_0$ is called {\bf\blue the mapping class group} of $M$. \definition {\bf \blue Teichm\"uller space of symplectic structures on $M$} is defined as a quotient $\Teich_s:= \Symp/\Diff_0$. The quotient $\Teich_s/\Gamma=\Symp/\Diff$, is called {\bf\blue the moduli space of symplectic structures.} \remark In many cases $\Gamma$ acts on $\Teich_s$ with dense orbits, hence {\bf \red the moduli space is not always well defined.} \definition Two symplectic structures are called {\bf \blue isotopic} if they lie in the same orbit of $\Diff_0$, and {\bf \blue diffeomorphic} is they lie in the same orbit of $\Diff$. \newpage {\bf \blue Moser's theorem} \definition Define {\bf \blue the period map} $\Per:\; \Teich_s \arrow H^2(M,\R)$ mapping a symplectic structure to its cohomology class. \theorem {\bf \blue (Moser, 1965)}\\ The {\bf \red Teichm\"uler space $\Teich_s$ is a manifold} (possibly, non-Hausdorff), and the {\bf \red period map $\Per:\; \Teich_s \arrow H^2(M,\R)$ is locally a diffeomorphism.} The proof is based on another theorem of Moser. {\bf \green Theorem 1:} {\bf \blue (Moser)}\\ Let $\omega_t$, $t\in S$ be a smooth family of symplectic structures, parametrized by a connected manifold $S$. Assume that the cohomology class $[\omega_t]\in H^2(M)$ is constant in $t$. {\bf \purple Then all $\omega_t$ are diffeomorphic.} \newpage {\bf \blue The proof of Moser's theorem} \theorem {\bf \blue (Moser)}\\ The {\bf \red Teichm\"uler space $\Teich_s$ is a manifold} (possibly, non-Hausdorff), and the {\bf \red period map $\Per:\; \Teich_s \arrow H^2(M,\R)$ is locally a diffeomorphism.} {\bf \green Proof. Step 1:} We can locally find a section $S$ for the $\Diff_0$-action on $\Symp$, producing a local decomposition $\Symp=O\times S$, where $O$ is a $\Diff_0$-orbit. Here $O$ and $S$ are both Frechet manifolds. {\bf \green Step 2:} The period map $P:\; U\arrow H^2(M,\R)$ is a smooth submersion. By Theorem 1, the fibers of $P$ are 0-dimensional. Therefore, $P$ is locally a diffeomorphism. \endproof \newpage {\bf \blue Symplectic structures on a compact torus} \definition A symplectic structure $\omega$ on a torus is called {\bf \blue standard} if there exists a flat torsion-free connection preserving $\omega$. \remark Moser's theorem immediately implies that {\bf\purple the set $\Teich_{st}$ of standard symplectic structures is open in the Teichm\"uller space.} Indeed, the period map from $\Teich_{st}$ to $H^2(M)$ is also locally a diffeomorphism. \remark {\bf \red It is not known if any non-standard symplectic structures exist} (even in dimension =4). \theorem Let $\Lambda^2_{nd}(H_1(T))\subset H^2(T)$ be the space of symplectic forms on $H_1(T)$, where $T$ is an even-dimensional torus. Consider the period map $\Per:\; \Teich_{st}\arrow \Lambda^2_{nd}(H_1(T))\subset H^2(T)$, where $\Teich_{st}$ is the Teichm\"uller space of standard symplectic structures on $T$. {\bf \red Then $\Per$ is a diffeomorphism.} \newpage {\bf \blue The space of flat Hermitian metrics} \theorem Let $\Lambda^2_{nd}(H_1(T))\subset H^2(T)$ be the space of symplectic forms on $H_1(T)$, where $T$ is an even-dimensional torus. Consider the period map $\Per:\; \Teich_{st}\arrow \Lambda^2_{nd}(H_1(T))\subset H^2(T)$, where $\Teich_{st}$ is the Teichm\"uller space of standard symplectic structures on $T$. {\bf \red Then $\Per$ is a diffeomorphism.} \pstep Let $\Teich_{h}$ be the Teichm\"uller space of flat Hermitian metrics on $T$. Clearly, $\Teich_h= GL(2n, \R)/U(n)$. Moreover, {\bf \purple the natural forgetful map $\Teich_h\arrow\Teich_{st}$ is surjective}. {\bf \green Step 2:} The fibers of the natural projection $\Teich_{h}\arrow\Lambda^2_{nd}(H_1(T))$ are connected. Using the diagram \[\begin{CD} \Teich_{h}@>>> \Lambda^2_{nd}(H_1(T))\\ @VVV @VV\Id V\\ \Teich_{st}@>>> \Lambda^2_{nd}(H_1(T))\\ \end{CD}\] we obtain that $\Per:\; \Teich_{st}\arrow\Lambda^2_{nd}(H_1(T))$ has connected fibers. By Moser's theorem, this map is a diffeomorphism. \endproof \newpage {\bf \blue $G_2$-structures} \definition Let $\rho\in \Lambda^2 \R^7$ be a 3-form on $\R^7$. We say that $\rho$ is {\bf\blue non-degenerate} if the dimension of its stabilizer is maximal: \[ \dim St_{GL(7)}\rho = \dim GL(7) - \dim \Lambda^3(\R^7) = 49-35 =14. \] In this case, $St(\rho)$ is one of two real forms of a 14-dimensional Lie group $G_2(\C)$. We say that $\rho$ is {\bf \blue non-split} if it satisfies $St(\rho|_x)\cong G_2$, where $G_2$ denotes the compact real form of $G_2(\C)$. {\bf\blue A $G_2$-structure} on a 7-manifold is a 3-form $\rho \in \Lambda^3(M)$, which is non-degenerate and non-split at each point $x\in M$ (``stable'', in the sense of Hitchin). \remark A form $\rho$ defines a $\Lambda^7 M$-valued metric on $M$: \[ g(x,y) = (\rho\cntrct x)\wedge (\rho \cntrct y) \wedge \rho \] This defines a conformal structure on $M$. The conformal factor is fixed if we want $|\rho|=1$. Therefore, {\bf \red every $G_2$-manifold is equipped with a natural Riemannian structure.} \newpage {\bf \blue Holonomy $G_2$-manifolds} \definition An $G_2$-manifold is called {\bf \blue a holonomy $G_2$-manifold} if $\rho$ is preserved by the corresponding Levi-Civita connection. \remark $(M,\rho)$ {\bf \purple is a holonomy $G_2$-manifold if and only if the form $\rho$ and its Hodge dual $*\rho$ are closed.} \theorem {\bf \blue (Joyce)} Let $\Teich_{G_2}$ be the Teichm\"uller space of $G_2$-structures on a compact 7-manifold $M$, and $\Per \Teich_{G_2}\arrow H^3(M)$ the period map associating to $\rho$ its cohomology class. {\bf \red Then $\Per$ is locally a diffeomorphism.} \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 $TM\otimes \C=T^{0,1}M\oplus T^{1,0}(M)$. {\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 Holomorphic symplectic form} \definition {\bf\blue Holomorphic symplectic form} on an almost complex manifold $(M,I)$ is a non-degenerate differential 2-form $\omega\in \Lambda^2(M,\C)$ satisfying $d\omega=0$ and $\Omega(Ix, y)=\1 \Omega(x,y)$. \remark This is the same as to say that {\bf \purple $\Omega(X, \cdot)=0$ for all $X\in T^{1,0}(M)$, and $\Omega$ is non-degenerate on $T^{0,1}(M)$.} \definition Let $\Omega$ be a differential form on $M$. The {\bf \blue kernel}, or {\bf \blue the null-space} $\ker(\Omega)\subset TM$ of $\Omega$ is the space of all vector fields $X\in TM$ such that the contraction $\Omega\cntrct X$ vanishes. \pstep Let $X,X_1\in \ker(\Omega)$, and $X_2, ..., X_p$ any vector fields. Cartan's formula implies that $\Lie_X(\Omega) = d(i_X(\Omega))+ i_x(d\Omega)=0$, hence $\Lie_X(\Omega)=0$. {\bf \green Step 2:} $\Lie_X(\Omega)(X_1, ..., X_p)= \Lie_X(\Omega(X_1, ..., X_p))- \sum_{i=1}^p\Omega(X_1, ..., [X, X_i], ... X_p). $ All terms of this sum, except $\Omega([X,X_1], X_2, ..., X_p)$, vanish, because $X_1 \in \ker(\Omega)$. Since $\Lie_X(\Omega)=0$, we have $\Omega([X,X_1], X_2, ..., X_p)=0$ for all $X_2, ..., X_p$. Therefore, $[X, X_1]\in \ker(\Omega)$. \endproof \corollary Let $(M,I)$ be an almost complex manifold admitting a holomorphic symplectic form. {\bf \red Then $I$ is integrable.} \newpage {\bf \blue Teichm\"uller space for holomorphic symplectic structures} \theorem {\bf \blue (Kaledin, V.)} Let $(M,\Omega)$ be a holomorphic symplectic manifold, $\Teich_\Omega$ the Teichm\"uller space of holomorphic symplectic structures on $M$, and $\Per:\; \Teich_\Omega\arrow H^2(M,\C)$ the map associating to $(M,\Omega)$ the cohomology class of $\Omega$, {\bf \red Then $\Per$ is locally an embedding.} \end{document}