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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:\ }} \begin{document} \setcounter{page}{1} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{center} {\Large\bf Holography principle and Moishezon twistor spaces } %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \\[14mm] {\small Misha Verbitsky } \\[20mm] {\tiny\bf Conference on The Geometry, Topology and Physics \\ of Moduli Spaces of Higgs Bundles\\ National University of Singapore, 4-8 Aug 2014 } \end{center} \newpage {\bf \blue Plan} 1. Hyperk\"ahler manifolds. Twistor spaces. 2. Holography principle 3. Moishezon twistor spaces. Applications of the holography principle to the local structure of twistor spaces. 4. Proof of holography principle. * 5. Hyperk\"ahler reduction. \newpage {\bf \blue Questions} \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 $I,J,K$. \remark Any compact, K\"ahler, holomorphically symplectic manifold, such as a K3 surface, admits a hyperk\"ahler metric, which is unique in is K\"ahler class. \question Let $(M,I,J,K)$ be a hyperk\"ahler manifold. {\bf \purple Define when \\ $(M,I,J,K)$ is ``algebraic''.} \remark This question is motivated by [HKLR] definition of hyperk\"ahler manifolds in terms of twistor spaces, which makes it possible to define a general ``hyperk\"ahler space'' with singularities and nilpotents. \question {\bf \purple Can an open ball in a K3 surface (or some other compact hyperk\"ahler manifold) with its hyperk\"ahler structure be equivalent to an open ball in a quiver space or a Hitchin moduli space with its hyperk\"ahler structure?} \newpage {\bf \blue Twistor spaces} \definition {\bf \color{blue} Induced complex structures} on a hyperk\"ahler manifold are complex structures of form $L:= aI + bJ +c K$. {\bf \purple It is a sphere: $S^2 \cong \{ L:= aI + bJ +c K, \ \ \ a^2+b^2+c^2=1.\}$} \definition A {\bf\blue twistor space} $\Tw(M)$ of a hyperk\"ahler manifold is {\bf \green a complex manifold obtained by gluing these complex structures into a holomorphic family over $\C P^1$.} More formally: Let $\Tw(M) := M \times S^2$. Consider the complex structure $I_m:T_mM \to T_mM$ on $M$ induced by $J \in S^2 \subset {\Bbb H}$. Let $I_J$ denote the complex structure on $S^2 = \C P^1$. The operator $I_{\Tw} = I_m \oplus I_J:T_x\Tw(M) \to T_x\Tw(M)$ satisfies $I_{\Tw} ^2 = -\Id$. {\bf \purple It defines an almost complex structure on $\Tw(M)$.} This almost complex structure is known to be integrable (Obata) \newpage {\bf \blue Geometry of twistor spaces} \example If $M={\Bbb H^n}$, $\Tw(M)= \Tot (\calo(1)^{\oplus n}) \cong \C P^{2n+1} \backslash \C P^{2n-1}$ (total space of a vector bundle $(\calo(1)^{\oplus n}$). \remark (Deligne, Simpson)\\ The quaternionic structure on a hyperk\"ahler manifold {\bf \purple can be reconstructed from the geometry of rational lines on its twistor space}. \remark Twistor spaces can be defined for any conformally semiflat 4-dimensional Riemannian manifold $M$. These twistor space are often Moishezon (birational to projective). Results of today's talk (``holography principle'') are true for such twistor spaces. \theorem {\bf \blue (Hitchin)} For compact 4-dimensional $M$, its twistor space is K\"ahler only if $M=S^4$ or $M=\C P^2$. \newpage {\bf \blue Twistor spaces are non-K\"ahler} \claim Suppose that $(M,I,J,K)$ is a hyperk\"ahler manifold such that $(M,I)$ contains a compact, odd-dimensional complex subvariety $Z$. {\bf \red Then $\Tw(M)$ is non-K\"ahler.} {\bf \green Proof:} Consider the fundamental class $[Z_I]\in H_*(M)$ of $Z$ in $(M,I)\subset \Tw(M)$. Then $[Z_I]+ [Z_{-I}]=0$, giving $\int_{Z_I\cup Z_{-I}}\alpha=0$ for each closed form $\alpha\in H^2(M)$. This is impossible if $\alpha$ is a K\"ahler form, because this integral is Riemannian volume of $Z_I\cup Z_{-I}$. \endproof \claim {\bf \red When $M$ is compact and hyperk\"ahler, $\Tw(M)$ never admits a K\"ahler structure.} {\bf\blue Proof:} Let $\omega$ be the standard Hermitian form of $\Tw(M)$. Then $dd^c\omega$ is a positive (2,2)-form (a calculation due to Kaledin-V.) For any K\"ahler form $\omega_0$, {\bf \purple this would imply \[ \int_{\Tw(M)}d\left(\omega_0^{\dim_\C M-1} \wedge d^c\omega\right)= \int_{\Tw(M)}\omega_0^{\dim_\C M-1} \wedge dd^c\omega>0, \] which is impossible by Stokes' theorem}. \endproof \newpage {\bf \blue Rational curves on $\Tw(M)$.} \definition {\bf\blue An ample rational curve} on a complex manifold $M$ is a smooth curve $S \cong \C P^1\subset M$ such that $NS=\bigoplus_{k=1}^{n-1}\calo(i_k)$, with $i_k >0$. It is called {\bf \blue a quasiline} if all $i_k=1$. \claim Let $M$ be a compact complex manifold containing a an ample rational line. {\bf \purple Then any $N$ points $z_1, ..., z_N$ can be connected by an ample rational curve.} \claim Let $M$ be a hyperk\"ahler manifold, $\Tw(M)\stackrel \sigma\arrow M$ its twistor space, $m\in M$ a point, and $S_m=\C P^1 \times \{m\}$ the corresponding rational curve in $\Tw(M)$. {\bf \red Then $S_m$ is a quasiline.} {\bf \blue Proof:} Since the claim is essentially infinitesimal, it suffices to check it when $M$ is flat. {\bf \purple Then $\Tw(M)= \Tot (\calo(1)^{\oplus 2p}) \cong \C P^{2p+1} \backslash \C P^{2p-1}$, and $S_m$ is a section of $\calo(1)^{\oplus 2p}$.} \endproof \newpage {\bf \blue Holography principle} {\bf \green Theorem 1:} {\blue \bf (Holography principle for line bundles)}\\ Let $S\subset M$ be an ample curve in a simply connected, connected complex manifold, which is covered by deformations of $S$. Consider a tubular neighbourhood $U\supset S$. Then, {\bf \red for any holomorphic line bundle $L$ on $M$, the space $H^0(U,L)$ is independent from the choice of $U$ and $S$ in its deformation class.} \remark In these assumptions, {\bf \purple $H^0(U,L)$ is always finite-dimensional} (Hartshorne). \theorem {\blue \bf (Holography principle for meromorphic functions)}\\ In assumptions of Theorem 1, {\bf \red the space of meromorphic functions on $U$ is equal to the space of meromorphic functions on $M$.} \definition Given a complex manifold $Z$, denote by $\Mer(Z)$ the field of global meromorphic functions on $Z$, and let {\bf\blue the algebraic dimension} $a(Z)$ be the transcendence degree of $\Mer(Z)$. \remark {\bf \purple It could be infinite!} \corollary {\bf \purple $a(\Tw(M))=a(U)$ for any connected neighbourhood $U$ of a quasiline} in a connected, simply connected $M$. \newpage {\bf \blue Moishezon twistor spaces} \definition A compact complex variety $Z$ is called {\bf \blue Moishezon} if the ring of meromorphic functions on $Z$ has algebraic dimension $\dim Z$, or, equivalently, if $Z$ is bimeromorphic to a projective manifold. \claim Let $M$ be a simply connected hyperka\"ahler manifold, and $\Tw(M)$ its twistor space. {\bf \purple Then $a(\Tw(M))\leq \dim_\C \Tw(M)$.} \definition A twistor space satisfying $a(\Tw(M))=\dim_\C \Tw(M)$ is called {\bf \blue Moishezon}. \claim {\bf \purple All Moishezon twistor spaces are bimeromorphic to open subsets of projective manifolds.} \theorem Let $V$ be a quaterionic Hermitian vector space, and $G\subset \Sp(V)$ a compact Lie group acting on $V$ by quaternionic isometries. Denote by $M$ the hyperk\"ahler reduction of $V$. {\bf \red Then $\Tw(M)$ is Moishezon.} \newpage {\bf \blue Local structure of hyperk\"ahler manifolds} \theorem (Fujiki, 1987) Let $M$ be a compact hyperk\"ahler manifold, and $L\in S^2 \subset {\Bbb H}$ a generic induced complex structure. {\bf \red Then $M$ contains no divisors.} \corollary Let $M$ be a compact hyperk\"ahler manifold, and $D\subset \Tw(M)$ a divisor. {\bf \purple Then $D$ is a union of several fibers of the projection $\pi:\; \Tw(M) \arrow \C P^1$.} {\bf \green Proof:} Suppose that $D$ contains a component which intersects generic fiber of $\pi$. By transversality, this intersection is a divisor, contradicting Fujiki's theorem. \endproof \corollary Let $M$ be a compact, simply connected hyperk\"ahler manifold, $M'$ a hyperk\"ahler manifold obtained by hyperk\"ahler reduction (such as Nakajima quiver variety), and $U\subset M$, $U'\subset M'$ open subsets. {\bf \red Then $U$ is never equivalent to $U'$ as a hyperk\"ahler manifold.} {\bf \green Proof:} $a(\Tw(U))=a(\Tw(M))=1$, and\\ $a(\Tw(U'))=a(\Tw(M'))=\dim(\Tw(M'))$. \endproof \newpage {\bf \blue Complex manifolds and quasilines} \remark Let $S\subset M$ be a quasiline. Then, for an appropriate tubular neighbourhood $U\subset M$ of $S$, {\bf \red ``for every two points $x, y\in U$ close to $S$ and far from each other, there is a unique deformation of $S$ containing $X$ and $Y$.''} More precisely: \claim Let $S\subset M$ be a quasi-line. Then, for any sufficiently small tubular neighbourhood $U\subset M$ of $S$, there exists a smaller tubular neighbourhood $W\subset U$, satisfying the following condition. Let $\Delta_S$ be the image of the diagonal embedding $\Delta_S:\; S \arrow W \times W$. Then there exists an open neighbourhood $V$ of $\Delta_S$, properly contained in $W\times W$, such that {\bf \purple for any pair $(x,y) \in W\times W \backslash V$, there exists a unique deformation $S'\subset U$ of $S$ containing $x$ and $y$.} \corollary For any quasiline in $M$, its deformation space is $2(\dim M-1)$-dimensional. \newpage {\bf \blue Proof of holography principle} \theorem {\bf \blue (Holography Principle)}\\ Let $S\subset M$ be a quasiline in a simply connected complex manifold, which is covered by deformations of $S$. Assume that $M$ is equipped with a projection $\pi:\; M \arrow S$ inducing identity on $S$. Consider a tubular neighbourhood $U\supset S$, and assume that $\pi:\; U \arrow S$ has connected fibers. Then, {\bf \purple for any holomorphic line bundle $L$ on $M$, the space $H^0(U,L)$ is independent from the choice of $U$ and $S$ in its deformation class.} {\bf \green A (slighly) weaker statement.} {\bf \green THEOREM 1:} Let $S\subset M$ be a quasiline, and $L$ a holomorphic bundle on $M$. Assume that $M$ is equipped with a projection $\pi:\; M \arrow S$ inducing identity on $S$. Consider a sufficiently small tubular neighbourhood $U\supset S$, and a smaller tubular neighbourhood $V\subset U$. {\bf \red Then the restriction map $H^0(U,L)\arrow H^0(V,L)$ is an isomorphism.} {\bf \blue We deduce the ``Holography principle'' from Theorem 1.} \newpage {\bf \blue Deducing the holography principle from Theorem 1} {\bf \green Step 1:} Choose a continuous, connected family $S_b$ of quasilines parametrized by $B$ such that $\bigcup_{b\in B} S_b = M$. Find a tubular neighbourhood $U_b$ for each $S_b$ in such a way that an intersection $U_b\cap U_{b'}$ for sufficiently close $b,b'$ always contains $S_b$ and $S_{b'}$. {\bf \purple By Theorem 1, $H^0(U_b\cap U_{b'},L)=H^0(U_b,L)=H^0( U_{b'},L)$.} {\bf \green Step 2:} Since $B$ is connected, {\bf \purple all the spaces $H^0(U_b,L)$ are isomorphic,} and these isomorphisms are compatible with the restrictions to the intersections $U_b\cap U_{b'}$. {\bf \green Step 3:} Let now $f\in H^0(U_b,L)$, and let $\tilde M_f$ be the {\bf \blue domain of holomorphy} for $f$, that is, a maximal domain (non-ramified over $M$) such that $f$ admits a holomorphic extension to $\tilde M_f$. Since $\cup U_b=M$, and $f$ can be holomorphically extended to any $U_b$, the domain $\tilde M_f$ is a covering of $M$. {\bf \purple Now, ``holography principle'' follows, because $M$ is simply connected.} \endproof \newpage {\bf \blue Proof of Theorem 1} {\bf \green THEOREM 1:} Let $S\subset M$ be a quasiline, and $L$ a holomorphic bundle on $M$. Assume that $M$ is equipped with a projection $\pi:\; M \arrow S$ inducing identity on $S$. Consider a sufficiently small tubular neighbourhood $U\supset S$, and a smaller tubular neighbourhood $V\subset U$. {\bf \red Then the restriction map $H^0(U,L)\arrow H^0(V,L)$ is an isomorphism.} {\bf \green Proof. Step 1:} Fix a point $x_0=\infty$ in $\C P^1$. A section of $L\restrict{\C P^1}=\calo(d)$ is the same as meromorphic function having a pole of degree $\leq d$ at $\pi^{-1}(\infty)$. {\bf \green Step 2:} For each section $S_1:\; \C P^1 \arrow M$ of $\pi$, a degree $d$ meromorphic function on $S_1$ is uniquely determined by its values at any $d+1$ points of $S_1$. {\bf \green Step 3:} Given a meromorphic function $f\in H^0(V,L)$ and a quasiline $S_1$ intersecting $V$ in an open set, we can extend $f$ to a meromorphic function $\tilde f$ on $S_1$ by computing its values at $d+1$ distinct points $z_1, ..., z_{d+1}$ of $S_1\cap V$. Whenever $S_1$ is in $V$, this procedure gives $f\restrict {S_1}$. {\bf \purple By analytic continuation, the values of $\tilde f(z)$ at any $z\in S_1$ are independent from the choice of $z_i$ and $S_1$.} {\bf \green Step 4:} Therefore, {\bf \purple $\tilde f$ is a well-defined meromorphic function on the union $V_1$ of all quasilines intersecting $V$. } For $U$ sufficiently small, $V_1$ contains $U$. Therefore, any $f \in H^0(V,L)$ can be extended to $U$. \endproof \newpage {\bf \blue Hamiltonians} {\bf \green Let's define the hyperk\"ahler reduction.} We denote the Lie derivative along a vector field as $\Lie_x:\; \Lambda^i M \arrow \Lambda^i M$, and contraction with a vector field by $i_x:\; \Lambda^i M \arrow \Lambda^{i-1} M$. {\bf \blue Cartan's formula:} $d\circ i_x + i_x \circ d =\Lie_x$. \remark Let $(M,\omega)$ be a symplectic manifold, $G$ a Lie group acting on $M$ by symplectomorphisms, and $\goth g$ its Lie algebra. For any $g\in {\goth g}$, denote by $\rho_g$ the corresponding vector field. Then $\Lie_{\rho_g}\omega=0$, giving $d(i_{\rho_g}(\omega))=0$. {\bf \purple We obtain that $i_{\rho_g}(\omega)$ is closed, for any $g\in {\goth g}$.} \definition {\bf\blue A Hamiltonian} of $g\in {\goth g}$ is a function $h$ on $M$ such that $dh=i_{\rho_g}(\omega)$. \newpage {\bf \blue Moment maps} \definition $(M,\omega)$ be a symplectic manifold, $G$ a Lie group acting on $M$ by symplectomorphisms. {\bf \blue A moment map} $\mu$ of this action is a linear map ${\goth g}\arrow C^\infty M$ associating to each $g\in G$ its Hamiltonian. \remark It is more convenient to consider $\mu$ as an element of ${\goth g}^* \otimes_\R C^\infty M$, or (and this is most standard) {\bf \red as a function with values in ${\goth g}^*$}. \remark Moment map {\bf \purple always exists} if $M$ is simply connected. \definition A moment map $M \arrow {\goth g}^*$ is called {\bf \blue equivariant} if it is equivariant with respect to the coadjoint action of $G$ on ${\goth g}^*$. \remark $M\stackrel\mu \arrow {\goth g}^*$ is a moment map iff for all $g\in {\goth g}$, $\langle d\mu,g\rangle= i_{\rho_g}(\omega)$. Therefore, {\bf \purple a moment map is defined up to a constant ${\goth g}^*$-valued function.} An equivariant moment map is is defined up to {\bf \purple a constant ${\goth g}^*$-valued function which is $G$-invariant}. \definition A $G$-invariant $c\in \goth g^*$ is called {\bf\blue central}. \claim {\bf \red An equivariant moment map exists whenever $H^1(G, {\goth g}^*)=0$.} In particular, when $G$ is reductive and $M$ is simply connected, an equivariant moment map exists. \newpage \newcommand{\3}{{/\!\!/\!\!/}} \newcommand{\2}{{/\!\!/}} \newpage {\bf\blue Hyperk\"ahler reduction} \definition Let $G$ be a compact Lie group, $\rho$ its action on a hyperk\"ahler manifold $M$ by hyperk\"ahler isometries, and $\g^*$ a dual space to its Lie algebra. {\bf \blue A hyperk\"ahler moment map} is a $G$-equivariant smooth map $\mu: M\to\g^*\otimes\R^3$ such that $\langle \mu_i(v),g \rangle = \omega_i(v,d\rho(g))$, for every $v\in TM$, $g\in\g$ and $i=1,2,3$, where $\omega_i$ is one three K\"ahler forms associated with the hyperk\"ahler structure. \definition Let $\xi_1,\xi_2,\xi_3$ be three $G$-invariant vectors in $\g^*$. The quotient manifold $M\3 G := \mu^{-1}(\xi_1,\xi_2,\xi_3)/G$ is called {\bf\blue the hyperk\"ahler quotient} of $M$. \theorem (Hitchin, Karlhede, Lindstr\"om, Ro\v cek)\\ {\bf \red The quotient $M\3 G$ is hyperkaehler}. \newpage {\bf \blue Holomorphic moment map} Let $\Omega:=\omega_J+ \1\omega_K$. This is a holomorphic symplectic (2,0)-form on $(M,I)$. {\bf \green The proof of HKLR theorem. Step 1:} Let $\mu_J, \mu_K$ be the moment map associated with $\omega_J, \omega_K$, and $\mu_\C:= \mu_J+ \1\mu_K$. Then $\langle d\mu_\C,g\rangle= i_{\rho_g}(\Omega)$ Therefore, $d\mu_\C\in \Lambda^{1,0}(M,I)\otimes {\goth g}^*$. {\bf \green Step 2:} This implies that the map $\mu_\C$ is holomorphic. It is called {\bf \blue the holomorphic moment map.} {\bf \green Step 3:} By definition, $M\3 G=\mu_{\C}^{-1}(c)\2 G$, where $c\in {\goth g}^*\otimes_\R \C$ is a central element. {\bf \red This is a K\"ahler manifold,} because it is a K\"ahler quotient of a K\"ahler manifold. {\bf \green Step 4:} We obtain 3 complex structures $I,J,K$ on the hyperk\"ahler quotient $M\3 G$. {\bf \purple They are compatible in the usual way} (an easy exercise). \endproof \newpage {\bf \blue Twistor spaces and hyperk\"ahler reduction} \theorem Let $V$ be a quaterionic Hermitian vector space, and $G\subset \Sp(V)$ a compact Lie group acting on $V$ by quaternionic isometries. Denote by $M$ the hyperk\"ahler reduction of $V$. {\bf \red Then $\Tw(M)$ is Moishezon.} {\bf \green Proof:} $\Tw(M)$ is obtained as the space of stable $G_\C$-orbits in $\mu_\C^{-1}(0)\subset \Tw(V)$. The space $\Tw(V)=\C P^{2n+1}\backslash \C P^{2n-1}$ is algebraic. Averaging over $G$, we obtain that the field $G_\C$-invariant rational functions on $\Tw(M)$ has dimension $\dim \Tw(M)$, and $\Tw(M)$ is Moishezon. \endproof \corollary Let $U$ be an open subset of a compact, simply connected hyperk\"ahler manifold, and $U'$ an open subset of a hyperk\"ahler manifold obtained as $V\3 G$, where $V$ is flat and $G$ reductive. {\bf \red Then $U$ is not isomorphic to $U'$ as hyperk\"ahler manifold}. {\bf \green Proof:} $\Tw(U')$ has many meromorphic functions, and $a(\Tw(U))=1$. \endproof %\newpage % %{\bf \blue Infinite-dimensional vector bundles} % %\question %Is there a ``normal form'' for a %formal neighbourhood of a quasiline? % %{\bf \purple Apparently, no.} % %\definition %{\bf \blue An infinite-dimensional vector bundle} on $S$ %is a $\calo_S$-sheaf $A$ equipped with a filtration %$F^iA\subset F^{i-1}A \subset ... \subset F^0 A=A$ %with all associated graded components $F^iA/F^{i+1}A$ %vector bundles. % %\definition %{\bf \blue A morphism of infinite-dimensional vector bundles} %is a map $A\arrow B$ of the corresponding $\calo_S$-sheaves, %mapping $F^iA$ to $F^{i+j}B$. % %\example %Let $S\subset M$ be a quasiline, %$U$ its tubular neighbourhood, and %$\pi:\; U \arrow S$ a projection with connected Stein fibers. %Then $\pi_*\calo_U$ is a ring object in the category of %infinite-dimensional vector bundles. % %Indeed: let $I_S$ be the ideal sheaf of $S\subset U$. %Then $I^k_S/I^{k+1}_S$ is $\Sym^k (NS)$, hence %the quotients $\pi_*\calo_U/\pi_* I^{k+1}_S$ are vector %bundles, and we can take the filtration %$F^k\pi_*\calo_U=\pi_*I_s^k$. % %\remark We are interested in the local structure %of a twistor space around a quasiline. It turns %out that the infinite-dimensional vector bundle %$\pi_*\calo_U$ with its ring structure is very %rich in invariants, and uniquely determines the %twistor space if it is Moishezon. % % %\newpage % %{\bf \blue Formal hyperk\"ahler manifold} % % % %\definition %Let $A$ be an infinite-dimensional vector bundle over $A$. %The inverse limit $\lim\limits_{\leftarrow} (A/F^iA)$ %is called {\bf\blue the formal completion of $A$}. % \end{document}