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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 Contraction loci in hyperk\"ahler manifolds} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \\[14mm] {\small Misha Verbitsky } \\[20mm] {\scriptsize\bf Automorphic Forms and Algebraic Geometry, \\ May 17, 2018\\POMI, SPb\\[10mm] {\bf \red Joint work with Ekaterina Amerik} } \end{center} \newpage {\bf \blue Holomorphically symplectic manifolds} {\bf\green DEFINITION:} {\bf \blue A holomorphically symplectic manifol} is a complex manifold equipped with non-degenerate, holomorphic $(2,0)$-form. {\bf\green THEOREM:} {\bf \blue (Calabi-Yau)} A compact, K\"ahler, holomorphically symplectic manifold {\bf \red admits a unique hyperk\"ahler metric in any K\"ahler class.} {\bf\green DEFINITION:} For the rest of this talk, {\bf \red a hyperk\"ahler manifold is a compact, K\"ahler, holomorphically symplectic manifold.} {\bf\green DEFINITION:} A hyperk\"ahler manifold $M$ is called {\bf \blue of maximal holonomy} (also: simple, or IHS) if $\pi_1(M)=0$, $H^{2,0}(M)=\C$. {\bf \purple Bogomolov's decomposition:} Any hyperk\"ahler manifold admits a finite covering which is a product of a torus and several hyperk\"ahler manifolds of maximal holonomy. {\bf \red Further on, all hyperk\"ahler manifolds are assumed to be of maximal holonomy}. \newpage {\bf \blue The Bogomolov-Beauville-Fujiki form} {\bf \green THEOREM:} (Fujiki). Let $\eta\in H^2(M)$, and $\dim M=2n$, where $M$ is hyperk\"ahler. Then $\int_M \eta^{2n}=cq(\eta,\eta)^n$, for some primitive integer quadratic form $q$ on $H^2(M,\Z)$, and $c>0$ a rational number. {\bf \green Definition:} This form is called {\bf \blue Bogomolov-Beauville-Fujiki form}. {\bf \purple It is defined by the Fujiki's relation uniquely, up to a sign.} The sign is determined from the following formula (Bogomolov, Beauville) \begin{align*} \lambda q(\eta,\eta) &= \int_X \eta\wedge\eta \wedge \Omega^{n-1} \wedge \bar \Omega^{n-1} -\\ &-\frac {n-1}{n}\left(\int_X \eta \wedge \Omega^{n-1}\wedge \bar \Omega^{n}\right) \left(\int_X \eta \wedge \Omega^{n}\wedge \bar \Omega^{n-1}\right) \end{align*} where $\Omega$ is the holomorphic symplectic form, and $\lambda>0$. {\bf \green Remark:} {\bf \red $q$ has signature $(3,b_2-3)$.} It is negative definite on primitive forms, and positive definite on $\langle \Omega, \bar \Omega, \omega\rangle$, where $\omega$ is a K\"ahler form. \newpage {\bf \blue Extremal curves} \definition A rational curve on a complex variety is called {\bf \blue minimal} if for any two distinct points $x, y$ on $C$, the space of deformations $Z_{x,y}$ of $C$ passing through $x,y$ is 0-dimensional. \remark If $\dim Z_{x,y}>0$, $C$ can be deformed to a union of two curves, with one of them still passing through $x,y$. {\bf \purple Then it is not ``minimal''} (in the usual sense: minimal curve is the one which cannot be deformed to a non-irreducible curve). \remark Let $C$ be a curve which can be deformed to a non-irreducible curve $C'$. On a compact K\"ahler manifold, degree of each components of $C'$ is strictly smaller than degree of $C$, hence {\bf \purple any rational curve is cohomologous to a sum of minimal curves.} \remark {\bf \purple Using the BBF form, we shall identify $H_2(M,\Q)$ and $H^2(M,\Q)$}. This allows us to consider the BBF form on $H_2(M,\Q)$. \definition A rational curve is called {\bf \blue extremal} if it is minimal, and its homology class has negative self-intersection. \newpage {\bf \blue Characterization of the K\"ahler cone} \definition The BBF form on $H^{1,1}(M,\R)$ has signature $(1, b_2-2)$. This means that the set $\{\eta \in H^{1,1}(M,\R)\ \ |\ \ (\eta,\eta)>0\}$ has two connected components. The component which contains the K\"ahler cone $\Kah(M)$ is called {\bf \blue the positive cone}, denoted $\Pos(M)$. \theorem (Huybrechts, Boucksom)\\ {\bf \red The K\"ahler cone of $M$ is the set of all $\eta\in \Pos(M)$ such that $(\eta, C)>0$ for all extremal curves $C$.} In other words, {\bf \purple the K\"ahler cone is locally polyhedral} (with some round pieces in the boundary), and its faces are orthogonal complements to the extremal curves. \newpage {\bf \blue MBM classes} \remark Suppose that $M$ and $M'$ are two birational Calabi-Yau manifolds (e. g. holomorphically symplectic manifolds). {\bf \red Then $H^2(M)$ is naturally identified with $H^2(M')$.} Indeed, $M'$ is obtained from $M$ by a sequence of blow-ups and blow-downs, but since their canonical bundles are trivial, all blown up divisors are blown down in the end, and $M$ is identified with $M'$ outside of real codimension 4. \theorem (Ekaterina Amerik, V.) Let $\eta$ be a cohomology class of an extremal curve on a hyperk\"ahler manifold, and $(M_1,\eta)$ be obtained as a deformation of $(M,\eta)$ in a continuous family such that $\eta$ remains of type $(1,1)$ for all fibers of this family. {\bf \red Then $M_1$ is birational to a hyperk\"ahler manifold $M_1'$ such that $\eta$ is a class of an extremal curve on $M_1'$.} \definition A class $\eta\in H^2(M,\Z)$ which can be represented by an extremal curve for some complex holomorphically symplectic structure on $M$ is called {\bf \blue an MBM class}. \remark Equivalent definition: {\bf \purple An MBM class $\eta\in H^2(M)$ is a class which can be represented by a rational curve in $(M,I)$ when $(M,I)$ is a non-algebraic deformation of $M$ with $\Pic(M, I)_\Q =\langle \eta\rangle$.} \newpage {\bf \blue Kawamata bpf} \definition {\bf \blue Base point set} of a holomorphic line bundle is an intersection of all zero divisors of all sections of its tensor powers. A line bundle with trivial base point set is called {\bf \blue base point free} (bpf). A line bundle $L$ with $nL$ bpf is called {\bf \blue semiample}. \claim Let $L$ be a semiample line bundle on a compact complex variety $M$. {\bf \purple Then $M$ is equipped with a holomorphic map $\phi:\; M \arrow X$ such that $L=\phi^* L_0$, where $L_0$ is an ample bundle on $X$.} \definition A line bundle $L$ is {\bf \blue nef} if $c_1(L)$ lies in the closure of the K\"ahler cone, and {\bf \blue big} if $H^0(M, L^N)= O({\dim M}^N)$. \theorem {\bf \blue (Kawamata bpf theorem; very weak form)}\\ Let $L$ be a nef line bundle on $M$ such that $nL -K_M$ is big. {\bf \red Then $L$ is semiample.} For Calabi-Yau manifolds this means just that {\bf \purple big and nef bundles are semiample.} \newpage {\bf \blue Birational contractions} \definition {\bf \blue Birational contraction} of a complex manifold is a holomorphic birational map $M\arrow X$ to a complex variety $X$. \remark From Kawamata bpf it follows that {\bf \red any big and nef bundle $L$ on Calabi-Yau is obtained as $L=\phi^* L_0$, where $\phi:\; M \arrow X$ is a birational contraction and $L_0$ an ample bundle on $X$.} \definition A variety is called {\bf \blue rationally connected} if any two of its points can be connected by a sequence of rational curves \theorem Let $\phi:\; M \arrow X$ be a birational contraction of a Calabi-Yau manifold. {\bf \purple Then any fiber $\phi^{-1}(x)$ is rationally connected. }\\ \proof Highly non-trivial (Kawamata, Shokurov, Hacon-McKernan). \endproof \remark Let $M$ be a projective hyperk\"ahler manifold, $\eta$ the cohomology class of an extremal curve, $\omega_0$ an integer point on the corresponding face of the K\"ahler cone, and $L$ the holomorphic line bundle with $c_1(L)=\omega_0$. Then $L$ is big (by Grauert-Riemenschneider conjecture, proven by Siu and Demailly) and nef. The corresponding birational contraction {\bf \purple contracts all curves $C$ with $[C]=\lambda \eta$.} Indeed, $\langle L_1 ,C\rangle=0$. \newpage {\bf \blue Faces of the K\"ahler cone} \definition Let $M$ be a hyperk\"ahler manifold. {\bf \blue A codimension 1 face}, or {\bf \blue a face} of the K\"ahler cone is a subset of its boundary obtained as an intersection of this boundary and a hyperplane which has dimension $h^{1,1}-1$. \theorem Codimension 1 faces of a K\"ahler cone {\bf \red are in bijective correspondence with cohomology classes $\eta$ of extremal curves.} Each such face is obtained as an intersection of the boundary and $\eta^\bot$. \theorem Let $(M,I)$ be a hyperk\"ahler manifold and $S$ the set of all MBM classes of type (1,1) on $(M,I)$. Let $S^\bot$ the union of all orthogonal complement to all $s\in S$. Then $\Kah(M,I)$ is a connected component of $\Pos(M)\backslash S^\bot$, and {\bf \red each connected component of $\Pos(M)\backslash S^\bot$ can be realized as a K\"ahler cone of some birational model of $(M,I)$.} \proof Follows from the deformational stability of MBM curves and the global Torelli theorem. \endproof \newpage {\bf \blue Centers of birational contraction are homeomorphic} \remark Clearly, $H^{1,1}(M)$ is obtained as orthogonal complement to the 2-dimensional space $\langle \Re\Omega, \Im \Omega\rangle$, where $\Omega$ is the cohomology class of the holomorphic symplectic form. Then $\Pic(M) = H^{1,1}(M)\cap H^2(M,\Z)$ has maximal rank only if the plane $\langle \Re\Omega, \Im \Omega\rangle$ is rational. {\bf \red There is at most countable number of such $M$}. \theorem (Ekaterina Amerik, V.) Let $F$ be a face of a K\"ahler cone of a hyperk\"ahler manifold, $C$ the corresponding rational curve, and $(M_1,F_1)$ be obtained as a deformation of $(M,F)$ in a continuous family such that the cohomology class $[C]$ remains of type (1,1). Assume that neither $M$ nor $M_1$ has maximal Picard rank, and $b_2(M)-k>3$. {\bf \red Then there exists a homeomorphism $\Psi:\; M \arrow M_1$ identifying the corresponding contraction centers and the contracted extremal curves.} \remark Existence of a homeomorphism follows from ergodicity of mapping group action, global Torelli theorem, and Thom-Mathers stratification of proper real analytic maps. \remark Stability of minimal rational curves under deformations of hyperk\"ahler manifolds is in essentially due to Ziv Ran and Claire Voisin: {\bf \purple if you deform a hyperk\"ahler manifold with a minimal rational curve, and its cohomology class remains of type (1,1), the curve also deforms.} \newpage {\bf \blue Centers of birational contraction for K3${}^{[2]}$} The homeomorphism theorem allows one to give a classification of birational contraction centers in terms of the period spaces and lattices. \theorem Let $M$ be a deformation of K3${}^{[2]}$, and $Z$ a birational contraction center associated with a face of a K\"ahler cone. {\bf \red Then one of the following three cases occurs:} (a) $Z$ is Lagrangian $\C P^2$ obtained as a deformation of $C^{[2]}\subset M^{[2]}$, where $C\subset M$ is a smooth $-2$-curve on a K3 surface $M$. (b) $Z$ is a deformation of the exceptional divisor on $M^{[2]}$. (c) $Z$ is a singular divisor obtained as a deformation of $Z_C\subset M^{[2]}$, where $Z_C$ is the set of all length 2 ideals on $M$ with support intersecting $C$. In all three cases $Z$ is homeomorphic to its model in $M^{[2]}$. \newpage {\bf \blue Centers of birational contraction for K3${}^{[2]}$ and lattices} This result is implied by the following lattice-theoretic result. {\bf \green THEOREM 1:} Let $M$ be a K3 and $\Lambda$ be the lattice $H^2(M^{[2]}, \Z)$ with its BBF form. Denote by $\Gamma$ the group of isometries of $\Lambda$ generated by reflections $x\arrow x- 2\frac{(x,z)}{(z,z)}z$ with negative $z$ {\bf \blue (``negative reflections'')} and let $E\in \Lambda$ be the exceptional divisor of $E$. {\bf \red Then for any $x\in \Lambda$ with $(x, x)<0$, there exists $\gamma\in \Gamma$ such that the rank 2 lattice $\langle \gamma(x), E\rangle$ has no positive vectors.} \proof Follows from the classification of orbits of $\Gamma$ on $\Lambda$, due to Gritsenko, Hulek, Sankaran: they prove that there are at most 2 orbits of $\Gamma$ action on the set $\{x\in \Lambda\ \ |\ \ (x, x)=w\}$ for any given $w$. One of these orbits intersects $H^2(M)\subset \Lambda$ and for such an orbit Theorem is obvious. Another orbit is $E$ and then it starts from $w\leq -8$. The later contains a vector $\gamma(x)=E+y$ where $y\in H^2(M)$, and $\langle \gamma(x), E\rangle$ is negative definite. \endproof \conjecture {\bf \red This result is true for $M^{[n]}$ for all $n$.} \remark If this is true, we have a similar simple classification of contraction centers for all deformations of Hilbert schemes on K3. \newpage {\bf \blue Teichm\"uller spaces} \def\Teich{\operatorname{Teich}} \newcommand{\Comp}{\operatorname{Comp}} \newcommand{\Per}{\operatorname{\sf Per}} \newcommand{\Perspace}{\operatorname{{\Bbb P}\sf er}} \newcommand{\Gr}{\operatorname{\sf Gr}} {\bf \green Definition:} Let $M$ be a compact complex manifold, and $\Diff_0(M)$ a connected component of its diffeomorphism group ({\bf\blue the group of isotopies}). Denote by $\Comp$ the space of complex structures on $M$, and let $\Teich:=\Comp/\Diff_0(M)$. We call it {\bf \blue the Teichm\"uller space.} \remark In all known cases $\Teich$ is {\bf \purple a finite-dimensional complex space} (Kodaira-Spencer-Kuranishi-Douady), but often {\bf \red non-Hausdorff}. \theorem {\bf \blue (Bogomolov-Tian-Todorov)} {\bf \purple $\Teich$ is a complex manifold when $M$ is Calabi-Yau}. {\bf \green Definition:} Let $\Diff(M)$ be the group of diffeomorphisms of $M$. We call $\Gamma:=\Diff(M)/\Diff_0(M)$ {\bf \blue the mapping class group}. \remark The quotient $\Teich/\Gamma$ {\bf \purple is identified with the set of equivalence classes of complex structures.} \newpage {\bf \blue The period map} {\bf \green Remark:} For any $J\in \Teich$, $(M,J)$ is also a simple hyperk\"ahler manifold, hence $H^{2,0}(M,J)$ is one-dimensional. {\bf \green Definition:} Let $\Per:\; \Teich \arrow {\Bbb P}H^2(M, \C)$ map $J$ to a line $H^{2,0}(M,J)\in {\Bbb P}H^2(M, \C)$. The map $\Per:\; \Teich \arrow {\Bbb P}H^2(M, \C)$ is called {\bf\blue the period map}. \remark {\bf \purple $\Per$ maps $\Teich$ into an open subset of a quadric,} defined by \[ \Perspace:= \{l\in {\Bbb P}H^2(M, \C)\ \ | \ \ q(l,l)=0, q(l, \bar l) >0\}. \] It is called {\bf \blue the period space} of $M$. \remark {\bf \red ${\Bbb Per}=SO(b_2-3,3)/SO(2) \times SO(b_2-3,1)=\Gr_{++}(H^2(M,\R)$.} Indeed, the group $SO(H^2(M,\R),q)=SO(b_2-3,3)$ acts transitively on $\Perspace$, and $SO(2) \times SO(b_2-3,1)$ is a stabilizer of a point. \newpage {\bf \blue Global Torelli theorem} \definition Let $M$ be a topological space. We say that $x, y \in M$ are {\bf \blue non-separable} (denoted by $x\sim y$) if for any open sets $V\ni x, U\ni y$, $U \cap V\neq \emptyset$. \theorem (Huybrechts) Two points $I,I'\in \Teich$ {\bf \purple are non-separable if and only if there exists a bimeromorphism $(M,I)\arrow (M,I')$ which is non-singular in codimension 2 and acts as identity on $H^2(M)$.} \remark This is possible only if $(M,I)$ and $(M,I')$ contain a rational curve. {\bf \purple General hyperk\"ahler manifold has no curves;} ones which have curves belong to a countable union of divisors in $\Teich$. \definition The space $\Teich_b:= \Teich/\sim$ is called {\bf \blue the birational Teichm\"uller space} of $M$. Since $\Teich_b$ is obtained by gluing together all non-separable points, it is also called {\bf \blue Hausdorff reduction} of $\Teich$. \theorem {\bf \blue (Torelli theorem for hyperk\"ahler manifolds)}\\ {\bf \red The period map $\Teich_b\stackrel \Per \arrow \Perspace$ is a diffeomorphism,} for each connected component of $\Teich_b$. \newpage {\bf \blue Mapping class group for Hilbert scheme of a K3} \theorem {\bf \blue (E. Markman)} Let $X$ be a deformation of the Hilbert scheme $M^{[n]}$ of K3, $\Gamma$ a subgroup of its mapping class group fixing a connected component of $\Teich_b$, and $\Lambda= H^2(X,\Z)$ with its BBF form. {\bf \blue Then $\Gamma$ is the subgroup of $O(\Lambda)$ generated by negative reflections.} \theorem Let $Z\subset X$ be a birational contraction center on a deformation $X$ of a Hilbert scheme $M^{[n]}$ of K3. Suppose that Theorem 1 is true for $M^{[n]}$. {\bf \red Then the pair $(X, Z)$ can be smoothly deformed to $(M^{[n]},Z')$, where $M$ is a non-algebraic K3.} \proof Let $\eta$ be the MBM class associated with $Z$ and $\Teich_\eta$ the Teichm\"uller space of deformations of $X$ such that $\eta$ remains of type (1,1). Then any lattice $\Lambda_1\subset \Lambda$ with $\Lambda_1^\bot$ containing a positive 2-plane and $\Lambda_1 \ni \eta$ can be realized as a Picard lattice of $I\in \Teich_\eta$. Applying this to $\langle \gamma(\eta), E\rangle$ from Theorem 1, we obtain a deformation of $(X, Z)$ with a Picard lattice $\langle \gamma(\eta), E\rangle$, and this is a Hilbert scheme for a non-algebraic K3. \endproof \remark {\bf \purple All curves on a Hilbert scheme $M^{[n]}$ of a non-algebraic K3 $M$ can be contracted.} This contraction gives a symmetric power of a singular K3 which has no curves at all. This gives an explicit description of aqll contraction centers on $M^{[n]}$. \end{document}