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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 for geometric structures,\\ lecture 1} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \\[14mm] {\scriptsize Misha Verbitsky } \\[20mm] {\tiny\bf Conference - Teichm\"uller Theory in Higher Dimension and Mirror Symmetry \\ April 24-28, 2017} \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. Hyperk\"ahler maifolds (introduction) \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 the same group $G\subset GL(d,\R)$, $d=\dim_\R M$. 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. \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. {\bf \purple 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 $P:\; S\arrow H^2(M,\R)$ 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 Non-Hausdorff points on symplectic Teichm\"uller space} Example of D. McDuff found in Salamon, Dietmar, {\em\green Uniqueness of symplectic structures}, Acta Math. Vietnam. 38 (2013), no. 1, 123-144. Let $M= S^1\times S^1 \times S^2 \times S^2$ with coordinates $\theta_1, \theta_2 \in S^1\subset \C^*$ and $z_1, z_2\in S^2$. Let $\phi_{\theta, z}\; \C P^1 \arrow \C P^1$ be a rotation around the axis $z\in \C P^1$ by the angle $\theta$. {\bf \blue Consider the diffeomorphism $\Psi:\; M \arrow M$ mapping $(\theta_1, \theta_2, z_1, z_2)$ to $(\theta_1, \theta_2, z_1, \phi_{\theta_1, z_1}(z_2))$.} \theorem Let $\omega_\lambda$ be the product symplectic form on $M = T^2 \times \C P^1 \times \C P^1$ obtained as a product of symplectic forms of volume 1, 1, $\lambda$ on $T^2$, $\C P^1$, $\C P^1$. {\bf \red The form $\Psi^*(\omega_1)$ is homologous, but not diffeomorphic to $\omega_1$.} However, {\bf \red the form $\Psi^*(\omega_\lambda)$ is diffeomorphic to $\omega_\lambda$ for any $\lambda\neq 1$.} {\small (D. McDuff, {\em\green Examples of symplectic structures}, Invent. Math. 89 (1987), 13-36.)} \newpage {\bf \blue The space of standard symplectic forms on a torus} \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 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 K\"ahler manifolds} {\bf\green DEFINITION:} A Riemannian metric $g$ on a complex manifold $(M,I)$ 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 DEFINITION:} A complex Hermitian manifold $(M,I,\omega)$ is called {\bf \blue K\"ahler} if $d\omega=0$. The cohomology class $[\omega]\in H^2(M)$ of a form $\omega$ is called {\bf \blue the K\"ahler class} of $M$, and $\omega$ {\bf \blue the K\"ahler form}. \remark {\bf \purple This is equivalent to $\nabla\omega=0$,} where $\nabla$ is Levi-Civita connection. \newpage {\bf \blue Hyperk\"ahler manifolds} \definition A {\bf \blue hyperk\"ahler structure} on a manifold $M$ is a Riemannian structure $g$ and a triple of complex structures $I,J,K$, satisfying quaternionic relations $I\circ J = - J \circ I =K$, such that $g$ is K\"ahler for $I,J,K$. {\bf\green REMARK:} This is equivalent to $\nabla I=\nabla J = \nabla K=0$: the parallel translation along the connection preserves $I, J,K$. {\bf\green DEFINITION:} Let $M$ be a Riemannian manifold, $x\in M$ a point. The subgroup of $GL(T_xM)$ generated by parallel translations (along all paths) is called {\bf \blue the holonomy group} of $M$. {\bf\green REMARK:} {\bf \purple A hyperk\"ahler manifold can be defined as a manifold which has holonomy in $Sp(n)$ } (the group of all endomorphisms preserving $I,J,K$). \claim A compact hyperk\"ahler manifold $M$ {\bf \purple has maximal holonomy of Levi-Civita connection $Sp(n)$} if and only if {\bf \purple $\pi_1(M)=0$, $h^{2,0}(M)=1$.} \theorem {\bf \blue (Bogomolov decomposition)}\\ {\bf \red Any compact hyperk\"ahler manifold has a finite covering isometric to a product of a torus and several maximal holonomy hyperk\"ahler manifolds.} \newpage {\bf \blue Holomorphically symplectic manifolds} {\bf\green DEFINITION:} {\bf \blue A holomorphically symplectic manifold} is a complex manifold equipped with non-degenerate, holomorphic $(2,0)$-form. \remark In these lectures , all holomorphically symplectic manifolds are assumed to be K\"ahler and compact. \remark A hyperk\"ahler manifold {\bf \purple has three symplectic forms\\ $\omega_I:= g(I\cdot, \cdot)$, $\omega_J:= g(J\cdot, \cdot)$, $\omega_K:= g(K\cdot, \cdot)$.} \claim In these assumptions, $\omega_J + \1\omega_K$ is holomorphic symplectic on $(M,I)$. {\bf\green THEOREM:} {\bf \blue (Calabi-Yau)}\\ A compact, K\"ahler, holomorphically symplectic manifold admits a unique hyperk\"ahler metric in any K\"ahler class. {\bf\green DEFINITION:} For the rest of these lectures, {\bf \red a hyperk\"ahler manifold is a compact, K\"ahler, holomorphically symplectic manifold of maximal holonomy.} \newpage {\bf \blue EXAMPLES.} \example An even-dimensional complex torus. \example {\bf \purple A non-compact example:} $T^* \C P^n$ (Calabi). \remark $T^*\C P^1$ {\bf \blue is a resolution of a singularity $\C^2/{\pm1}$.} \example Take a 2-dimensional complex torus $T$, then the singular locus of $T/{\pm1}$ is of form $(\C^2/{\pm1}) \times T$. Its resolution $\widetilde {T/{\pm1}}$ is called {\bf \green a Kummer surface}. {\bf \red It is holomorphically symplectic}. \remark Take a symmetric square $\Sym^2 T$, with a natural action of $T$, and let $T^{[2]}$ be a blow-up of a singular divisor. {\bf \purple Then $T^{[2]}$ is naturally isomorphic to the Kummer surface $\widetilde {T/{\pm1}}$.} \definition A complex surface is called {\bf \blue K3 surface} if it a deformation of the Kummer surface. \theorem {\bf \blue (a special case of Enriques-Kodaira classification)}\\ Let $M$ be a compact complex surface which is hyperk\"ahler. {\bf \red Then $M$ is either a torus or a K3 surface.} \newpage {\bf \blue Hilbert schemes} \definition A {\bf\blue Hilbert scheme} $M^{[n]}$ of a complex surface $M$ is a classifying space of all ideal sheaves $I\subset \calo_M$ for which the quotient $\calo_M/I$ has dimension $n$ over $\C$. \remark A Hilbert scheme {\bf \purple is obtained as a resolution of singularities} of the symmetric power $\Sym^n M$. \theorem (Fujiki, Beauville) {\bf \red A Hilbert scheme of a hyperk\"ahler surface is hyperk\"ahler.} \example {\bf\blue A Hilbert scheme of K3} is hyperk\"ahler. \example Let $T$ be a torus. Then it acts on its Hilbert scheme freely and properly by translations. For $n=2$, the quotient $T^{[n]}/T$ is a Kummer K3-surface. For $n>2$, a universal covering of $T^{[n]}/T$ is called {\bf \blue a generalized Kummer variety}. \remark There are 2 more ``sporadic'' examples of compact hyperk\"ahler manifolds, constructed by K. O'Grady. {\bf \purple All known compact hyperkaehler manifolds are these 2 and the two series:} Hilbert schemes of K3, and generalized Kummer. \newpage {\bf \blue Global Torelli theorem.} \theorem Let $M$ be a simple hyperk\"ahler manifold, and $\Gamma:= \Diff/\Diff_0$ its mapping class group. Then $H^2(M,\Z)$ admits a $\Gamma$-invariant, non-degenerate integer quadratic form $q$ such that the natural action of $\Gamma$ on $H^2(M,\Z)$ {\bf \red induces a homomorphism $\Gamma\arrow SO(H^2(M,\Z), q)$ with finite index and finite kernel.} \remark Suppose that $\phi:\; M \arrow M'$ is a bimeromorphic map of Calabi-Yau manifolds. Then the exceptional set of $\phi$ has codimension $\geq 2$, {\bf \red hence $H^2(M)=H^2(M')$.} \definition Let $\Teich$ be the Teichm\"uller space of complex structures of hyperk\"ahler type on $M$, and $\Teich_b$ the quotient of $\Teich$ by an equivalence relation induced by bimeromorphic maps $(M,I)\arrow (M,I')$ inducing identity on $H^2(M)$. Then $\Teich_b$ is called {\bf \blue birational Teichm\"uller space} of $M$. As we shall see, $\Teich_b$ is a smooth, Hausdorff complex manifold. \theorem Consider the space \[ \Perspace:= \{l\in {\Bbb P}H^2(M, \C)\ \ | \ \ q(l,l)=0, q(l, \bar l) >0\} \] and let the {\bf \blue period map} $\Per:\; \Teich \arrow {\Bbb P}H^2(M, \C)$ map a complex structure $I$ to a line $H^{2,0}(M,I)\in {\Bbb P}H^2(M, \C)$. {\bf \red Then $\Per:\; \Teich \arrow {\Bbb P}H^2(M, \C)$ induces a bijective map $\Teich_b\arrow \Perspace$} on any connected component of $\Teich_b$. \end{document}