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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf Birational contractions of hyperkaehler manifolds are
diffeomorphic}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\\[14mm]

{\small Misha Verbitsky } 
\\[20mm]

{\small\bf 
Seminar of Laboratory of Algebraic Geometry\\[2mm]
Moscow, HSE, July 22, 2016
}
\end{center}

\newpage

{\bf \blue Hyperk\"ahler manifolds}

\def\Teich{\operatorname{Teich}}
\newcommand{\Comp}{\operatorname{Comp}}
\newcommand{\Kah}{\operatorname{\sf Kah}}
\newcommand{\Mon}{\operatorname{\sf Mon}}
\newcommand{\Pos}{\operatorname{\sf Pos}}
\newcommand{\Per}{\operatorname{\sf Per}}
\newcommand{\Perspace}{\operatorname{{\Bbb P}\sf er}}
\newcommand{\Gr}{\operatorname{\sf Gr}}

\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 A hyperk\"ahler manifold {\bf \purple has three symplectic forms\\
$\omega_I:=  g(I\cdot, \cdot)$, $\omega_J:=  g(J\cdot, \cdot)$,
$\omega_K:=  g(K\cdot, \cdot)$.}


{\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
$I,J,K$).

\newpage


{\bf \blue Holomorphically symplectic manifolds}

{\bf\green DEFINITION:} A holomorphically symplectic manifold 
is a complex manifold equipped with non-degenerate, holomorphic
$(2,0)$-form.

\remark Hyperk\"ahler manifolds are holomorphically symplectic.
Indeed, $\Omega:=\omega_J+\1\omega_K$ is a holomorphic symplectic
form on $(M,I)$.

{\bf\green THEOREM:} (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 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 simple} 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 
simple hyperk\"ahler manifolds.


{\bf \red Further on, all hyperk\"ahler manifolds
are assumed to be simple}.

\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 $x,y$. Then it is not ``minimal'' (in the usual sense:
minimal curve is 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 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, K\"ahler cone is locally polyhedral (with
some round pieces in the boundary), and its faces are
orthogonal complements to the extremal curves.

This result follows from Kawamata base point free theorem (see below).


\newpage

{\bf \blue Birational contractions and Kawamata bpf}

\definition
{\bf \blue Base point set} of a holomorphic line bundle is 
an intersection of all zero divisors of its sections.
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 $\int_M c_1(L)^{\dim_\C M} >0$.

\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

\definition
A {\bf \blue Calabi-Yau manifold} is a compact, K\"ahler manifold $M$
with $c_1(M)=0$.


{\bf \green Theorem 1:}
 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 (Shokurov and others). \endproof

\remark Let $M$ be a 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 $c_q(L)=\omega_0$. Then $L$ is big and nef.
Then the corresponding birational contraction {\bf \purple contracts
all curves $C$ with $[C]=\lambda \eta$.} Indeed, $\langle L_1 ,C\rangle=0$.


\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 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}.


\newpage

{\bf \blue Faces of the K\"ahler cone}

\definition
Let $M$ be a hyperk\"ahler manifold.
{\bf \blue A codimension 1 face}, or (sometimes) just 
{\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 $h^{1,1}-1$. 
{\bf \blue A face of codimension $k$} of the K\"ahler cone is
an intersection of $k$ adjacent codimension 1 faces.

\theorem (Huybrechts, Boucksom)
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 in intersection of the boundary and
$\eta^\bot$.


\newpage

{\bf \blue Faces of the K\"ahler cone and birational contractions}

\theorem
Let $M$ be a projective hyperk\"ahler manifold,
and $\pi:\; M \arrow M_1$ a birational contraction.
Consider the set $[C_1], ..., [C_k]$ of cohomology classes
of all extremal curves which are contracted by $\pi$.
{\bf \red Then $\bigcap_i [C_i]^\bot\cap\6\Kah$ is a codimension $k$ face of
the K\"ahler cone.} Moreover, {\bf \red faces are in bijective correspondence
with birational contractions.}

\proof
Let $L_1$ be an ample bundle on $M_1$. Then $L:=\pi^* L_1$ is a big,
nef bundle with  $c_1(L)\in \bigcap_i [C_i]^\bot$,
hence the set $\bigcap_i [C_i]^\bot$ is a non-empty face.
Conversely, for any face $F:=\bigcap_i [C_i]^\bot\cap\6\Kah$, and
any line bundle with $c_1(L)$ in interior of $F$, the bundle
$L$ is big and nef and the corresponding contraction
contracts curves in $[C_i]$ and only them.
\endproof 

\remark
Define {\bf \blue nef cone} of a projective variety
as the set of all (1,1)-classes which are non-negative on curves.
{\bf \purple Then the nef cone of $M_1$ is identified with the
interior of the face $F$.}

\newpage

{\bf \blue Centers of birational contraction are diffeomorphic}

\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 $M$}.

The main result of this talk:

\theorem (Ekaterina Amerik, V.)
Let $F$ be a codimension $k$ face of a K\"ahler cone of a 
hyperk\"ahler manifold, $F=\bigcap [C_i]^\bot\cap \6\Kah$, and
$(M_1,F_1)$ be obtained as a deformation of $(M,F)$
in a continuous family such that all $[C_i]$ 
remain 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 diffeomorphism
$\Psi:\; M \arrow M_1$ identifying the corresponding
contraction centers and the contracted extremal curves.}

\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 Teichm\"uller spaces}


{\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 Computation of the mapping class group}




\theorem (Sullivan) 
Let $M$ be a compact, simply connected 
K\"ahler manifold, $\dim_\C M\geq 3$. Denote by $\Gamma_0$ the group
of automorphisms of an algebra $H^*(M, \Z)$
preserving the Pontryagin classes $p_i(M)$. 
Then {\bf \red the natural map 
$\Diff_+(M)/\Diff_0\arrow \Gamma_0$ has finite kernel,
and its image has finite index in $\Gamma_0$.}

\theorem
Let $M$ be a simple hyperk\"ahler manifold,
and $\Gamma_0$ as above.  Then \\
(i)  $\Gamma_0\restrict{H^2(M,\Z)}$ {\bf \blue is a finite index 
subgroup of $O(H^2(M, \Z), q)$.}\\
(ii) The map $\Gamma_0\arrow O(H^2(M, \Z), q)$
{\bf \blue has finite kernel.}

\remark
Sullivan's theorem implies that the mapping class group for $\dim_\C M\geq 3$,
$\pi_1(M)=0$, {\bf\purple is an arithmetic lattice}. Very much unlike
the mapping class group for curves!


\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
(see below for a more detailed argument).

\theorem {\bf \blue (Bogomolov)} 
For any hyperk\"ahler manifold, 
{\bf \red period map is locally a diffeomorphism.}



\newpage

{\bf \blue Period space as a Grassmannian of positive 2-planes}

\proposition
The period space 
\[
{\Bbb Per}:= \{l\in {\Bbb P}H^2(M, \C)\ \ | \ \  q(l,l)=0, q(l, \bar l) >0\}.
\]
{\bf \red is identified with $SO(b_2-3,3)/SO(2) \times SO(b_2-3,1)$,} which
is a Grassmannian of positive oriented 2-planes in $H^2(M,\R)$.

{\bf \green 
Proof. Step 1:} Given $l\in {\Bbb P}H^2(M, \C)$, {\bf \purple the space
generated by $\Im l, \Re l$ is 2-dimensional,} because 
$q(l,l)=0, q(l, \bar l)$ implies that $l \cap H^2(M,\R)=0$. \\
\phantom{XXXXy} {\bf \green  Step 2:} {\bf \purple This 2-dimensional plane is 
positive,} because 
 $q(\Re l, \Re l) = q(l+ \bar l, l+ \bar l) = 2 q(l, \bar l)>0$. \\
\phantom{XXXXy}  {\bf\green  Step 3:} Conversely, for any 2-dimensional positive
plane  $V\in H^2(M,\R)$, {\bf \purple 
the quadric $\{l\in V \otimes_\R \C\ \ |\ \ q(l,l)=0\}$
consists of two lines;} a choice of a line is determined by orientation.
\endproof


\newpage

{\bf \blue Birational Teichm\"uller moduli space}

\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 Ergodic complex structures}

\definition
Let $M$ be a complex manifold, $\Teich$ its Techm\"uller
space, and $\Gamma$ the mapping group acting on $\Teich$.
{\bf\blue An ergodic complex structure} is a complex
structure with dense $\Gamma$-orbit.

\definition
Let $(M,\mu)$ be a space with measure,
and $G$ a group acting on $M$ preserving measure.
This action is {\bf\blue ergodic} if all
$G$-invariant measurable subsets $M'\subset M$
satisfy $\mu(M')=0$ or $\mu(M\backslash M')=0$.

\claim
Let $M$ be a manifold, $\mu$ a Lebesgue measure, and
$G$ a group acting on $M$ ergodically. {\bf \red Then the 
set of non-dense orbits has measure 0.}


{\bf\green Proof:}
Consider a non-empty open subset $U\subset M$. 
Then $\mu(U)>0$, hence $M':=G\cdot U$ satisfies 
$\mu(M\backslash M')=0$. For any orbit $G\cdot x$
not intersecting $U$, $x\in M\backslash M'$.
Therefore, the set $Z_U$ of such orbits has measure 0. 

 {\bf\green Step 2:} Choose a countable base
$\{U_i\}$ of topology on $M$. Then the set of 
points in dense orbits is $M \backslash \bigcup_i Z_{U_i}$.
\endproof


\newpage

{\bf \blue Ergodicity of the mapping class group action}

\definition
{\bf\blue A lattice} in a Lie group is a discrete
subgroup $\Gamma\subset G$ such that $G/\Gamma$ has finite
volume with respect to Haar measure.

\theorem (Calvin C. Moore, 1966)
Let $\Gamma$ be a lattice in a non-compact 
simple Lie group $G$ with finite center, and $H\subset G$ a 
non-compact subgroup. {\bf \red Then the left action of $\Gamma$
on $G/H$ is ergodic.}

\theorem Let ${\Perspace}$ be a component of 
a birational Teichm\"uller space, and
$\Gamma$ its monodromy group. Let $\Perspace_e$ be
a set of all points $L\subset \Perspace$
such that the orbit $\Gamma\cdot L$ is dense. {\bf \red Then 
$Z:=\Perspace\backslash \Perspace_e$ has measure 0.} 

{\bf \green Proof. Step 1:}
 Let $G=SO(b_2-3,3)$, $H=SO(2) \times SO(b_2-3,1)$.
{\bf \purple Then $\Gamma$-action on $G/H$ is ergodic,}
by Moore's theorem. \\
\phantom{XXXXy} 
{\bf \green Step 2:}
Ergodic orbits are dense, becuse the union of non-ergodic orbits has measure
0.
\endproof


\theorem
Let $(M,I)$ be a hyperk\"ahler manifold with $b_2>3$ 
or a compact torus of dimension $>1$.
{\bf \red Then $I$ ergodic (has dense $\Gamma$-orbit in $\Teich$) 
if and only if $\rk\Pic(M,I)$ is not maximal.}

\proof Follows from M. Ratner's theorems.
\endproof

\newpage

{\bf \blue The space $\Teich_F$}

\definition
Let $M$ be a hyperk\"ahler manifold, $F:=\bigcap_i [C_i]^\bot\cap \6\Kah$
a face of the K\"ahler cone, and
$\Teich_F$ denote the Teichm\"uller space of all deformations
of $M$ such that all $C_i$ remain extremal.
Denote by $\Perspace_F$ the corresponding period space,
\[
{\Bbb Per}:= \{l\in {\Bbb P}H^2(M, \C)\ \ | \ \  q(l,l)=0, q(l, \bar l) >0,
q(l, [C_i])=0\}.
\]
Clearly, $\Perspace_F=\Gr_{++}(\bigcap_i [C_i]^\bot)$.

\remark
Global Torelli theorem and stability of extremal curves
under deformations imply that 
{\bf \red $\Per:\; \Teich_F\arrow \Perspace_F$
is the Hausdorff reduction map (gluing together
all non-separable points).}

Then the same Ratner theorem argument as above gives the following
result.

\theorem
Let $F$ be a face of the K\"ahler
cone, and $\Gamma_F$ be a subgroup of the mapping class group
$\Gamma$ preserving the connected component of $\Teich_F$.
{\bf \red Then an orbit $\Gamma_F\cdot I $ of $\Gamma_F$ 
on $\Teich_F$ is dense if and only if $b_2(M)-\codim F > 3$ and
 $(M,I)$ has non-maximal Picard
lattice.}

\newpage

{\bf \blue The main result deduced from ergodicity}

\theorem (Whitney)\\
Let ${\cal X} \supset {\cal M}\stackrel \Psi \arrow B$ 
be a holomorphic family of pairs $X\subset M$ of compact complex varieties,
with $M$ smooth. Consider the set $B_0\subset B$ of all points
$b\in B$ such that the family $\Psi$ admits a smooth
trivialization in a neighbourhood of $b$, in such a way
that all fibers of $\Psi \restrict{\cal X}$ are identified.
{\bf \red Then $B_0$ is open in $B$.}


\theorem 
Let $F=\bigcap_i [C_i]^\bot\cap \6\Kah$ be a face of the K\"ahler 
cone of a hyperk\"ahler manifold, and
$(M_1,F)$ be obtained as a deformation of $(M,F)$
in a continuous family such that all $C_i$ remain of type $(1,1)$
for all fibers of this family, and 
$F=\bigcap_i [C_i]^\bot\cap \6\Kah$ is a face on
$M_1$. Assume that neither $M$ nor
$M_1$ has maximal Picard rank. {\bf \red Then
there exists a diffeomorphism
$\Psi:\; M \arrow M_1$ identifying the centers of corresponding 
birational contractions, and minimal curves in the
cogomology classes $[C_i]$.}

\proof Let ${\cal U} \arrow \Teich_F$ be the corresponding
universal family over the Teichm\"uller space, and 
$\Teich_F^0 \subset\Teich_F $ the subset consisting of all points $I$
such that the universal family admits a smooth trivialization
in a neighbourhood of $I$ compatible with centers of 
contraction as in Whitley theorem.  Since $\Teich_F^0$
is non-empty, open and $\Gamma_F$-invariant, 
it contains all dense orbits. \endproof



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