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\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}




