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\begin{document}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf  Subtwistor metric \\[4mm] on the moduli of hyperk\"ahler manifolds}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\\[14mm]

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

{\tiny\bf International conference\\[2mm]
``Geometry and analysis on metric structrures''\\[2mm]
Geometric control theory laboratory
\\[2mm] December 4-7, 2013.
}
\end{center}



\newpage

{\bf \blue Sub-Riemannian structures}

\definition
Let $M$ be a Riemannian manifold and $B\subset TM$
a sub-bundle. 
A {\bf\blue horizontal path} is a piecewise smooth
path $\gamma:\; [b,a] \arrow M$ tangent to $B$ everywhere.
A {\bf \blue sub-Riemannian}, or {\bf\blue Carno-Carath\'eodory}
metric $M$ is \[ d_B(x,y):=\inf_{\gamma \text{\ horizontal}}L(\gamma):\]
the infimum of the length $L(\gamma)$
for all horizontal paths connecting $x$ to $y$.

\theorem
{\bf \blue (Chow-Rashevskii theorem; 1938, 1939)}\\
Consider {\bf \blue the Frobenius form}
 $\Phi:\; \Lambda^2 B \arrow TM/B$ mapping vector fields $X, Y
\in B$ to an image of $[X,Y]$ modulo $B$. Suppose that
$\Phi$ is surjective. {\bf \red Then any two points can be
connected by a horizontal path,} and the sub-Riemannian
metric $d_B$ is finite.


\newpage

{\bf \blue Properties of sub-Riemannian metrics}

Let $(M,B,g)$ be a sub-Riemannian manifold.

\claim
{\bf \red Every two points $x, y\in M$ are connected by a smooth,
horizontal path $\gamma$.}
Moreover, $d_B(x,y)=\inf_{\gamma\text{\ horizontal, smooth}}L(\gamma)$:
the sub-Riemannian distance can be taken as 
infimum of the length for smooth horizontal paths
connecting $x$ to $y$.

\theorem {\bf \blue (ball-box theorem)}
An $\epsilon$-ball in $d_B$ is {\bf \red asymptotically
equivalent to a product of $\epsilon$-ball in direction of
$B$ and $\epsilon^2$-ball in orthogonal direction.}

\corollary 
The sub-Riemannian metric {\bf \red 
induces the standard topology on $M$.}

\corollary 
The Hausdorff dimension of a sub-Riemannian manifold
is integer, {\bf \purple and strictly bigger than $\dim M$.}



\newpage

\newcommand{\Gr}{\operatorname{Gr}}
{\bf \blue Subtwistor metric}

Throughout this talk, 
$H$ is a real vector space with non-degenerate 
scalar product of signature $(3,b-3)$, and $\Gr_{++}(H)$ --
Grassmannian of 2-dimensional positive oriented planes
in $H$. The space $\Gr_{++}(H)$ is in fact a complex
manifold, and it is called {\bf \blue the period space
of weight 2 Hodge structures on $H$.}

\definition
Let $W\subset V$ be a positive 3-dimensional subspace,
and $S_W=\Gr_{++}(W)\subset \Gr_{++}(H)$ a 2-dimensional sphere consisting
all 2-dimensional oriented planes in $W$. Then $S_w$ is
called {\bf \blue a twistor line}.

\claim 
{\bf \purple Each pair $x, y \in \Gr_{++}(H)$ can be connected by
an intersecting chain $S_{W_1}, S_{W_2}, ..., S_{W_n}$ of
twistor lines;} moreover, $n\leq 3$.

\definition
{\bf\blue A twistor path} on $\Gr_{++}(H)$
is a piecewise smooth path $\gamma:\; [a,b]\arrow \Gr_{++}(H)$
with each smooth component sitting on a twistor line.

\definition 
Fix a Euclidean structure on $H$, and let 
$g$ be the corresponding Riemannian metric on
$\Gr_{++}(H)$. {\bf \blue Subtwistor metric}
$d_{tw}(x,y)$ on $\Gr_{++}(H)$ is defined as
$d_{tw}(x,y):=\inf_\gamma L(\gamma)$
where $L(\gamma)$ is a length of the 
path $\gamma$ taken with respect to $g$,
and infimum is taken over all
subtwistor paths connecting $x$ to $y$.


\newpage

{\bf \blue Properties of subtwistor metric}

\question
Can we connect any pair $x, y \in \Gr_{++}(H)$
with a smooth path tangent to twistor line at each point?
Would the infimum of its length give the same metric?

\question
What about the ball-box theorem? What is a shape of a
small $\epsilon$-ball in $d_{tw}$?

\question What us the Hausdorff dimension
$(\Gr_{++}(H), d_{tw})$? 

\question
The definition I gave obviously can be generalized.
What is an appropriate generality?

\theorem
{\bf \red The subtwistor metric $d_{tw}$ induces the standard
topology on $\Gr_{++}(H)$.}

\remark Its proof is highly non-trivial; uses 
a solution of Hilbert's fifth problem on continuous groups.

\newpage

{\bf \blue Hilbert's 5 problem}

\question {\bf \blue (Hilbert, 1900)}\\
{\bf \purple ``How is Lie's concept of continuous groups of 
transformations of manifolds approachable in our 
investigation without the assumption of
differentiability?''}


{\bf \red Answered affirmative} by
von Neumann, Gleason, Montgomery-Zippin.

\theorem
Let $M$ be a topological manifold equipped with
a continuous group structure. {\bf \red Then $M$ admits 
a smooth structure compatible with the group action.}


I will state the Gleason-Palais refinement of this
theorem.




\newpage

{\bf \blue Gleason-Palais theorem}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\definition
Let $M$ be a topological space. We say that $M$ {\bf\blue has
  Lebesgue covering dimension $\leq n$} if every
open covering of $M$ has a refinement $\{U_i\}$
such that each point of $M$ belongs to at most 
$n+1$ element of $\{U_i\}$. A {\bf\blue  Lebesgue covering
dimension} of $M$ (denoted by $\dim M$) is an infimum
of all such $n$. 

\example 
If $M$ is an $n$-manifold, $\dim M=n$.

\claim If $X\subset M$ is a subset of a
topological space, with induced topology, 
one has $\dim X \leq \dim M$. 

\theorem {\bf \blue (Gleason-Palais)}\\
Let $G$ be a topological group, which is
locally path connected, and has $\dim K < \infty$ for
each compact, metrizable subset $K\subset G$. {\bf \red Then  $G$ is
homeomorphic to a Lie group. }



\newpage

{\bf \blue Subtwistor norm on a Lie group}


\remark We define a norm on the group $SO(H)$
compatible with the subtwistor metric on $\Gr_{++}(H)$.

\definition
Let $G$ be a connected component of $SO(H)$ acting on 
$\Gr_{++}(H)$ in a susual way.
We define {\bf \blue subtwistor norm} 
on $G$ in such a way that {\bf \purple 
the bijective map $(G/G_0, \|\cdot\|_{tw})\arrow (\Gr_{++}(H), d_{tw})$
is continuous}, where $G_0\subset G$ is a stabilizer of a point
$V\in \Gr_{++}(H)$.

\definition
An {\bf \blue elementary transform} is an element $h\in G$ 
fixing a codimension 2 subspace $V_1\subset V$ of
signature $(1,n-3)$. {\bf\blue An elementary decomposition}
of $h\in G$ is a decomposition $h=h_1h_2...h_n$,
where $h_i$ are elementary transforms. 
Define the {\bf\blue subtwistor norm} on $G$
as $\|h\|_{tw}:= \inf(\|h_1\|+\|h_2\|+...+\|h_n\|)$,
where the infinum is taken over all elementary
decompositions $h=h_1 h_2 ... h_n$.

\claim
{\bf \purple The action of $(G, \|\cdot\|_{tw})$ 
on $(\Gr_{++}(H), d_{tw})$ is continuous}, and induces
a homeomorphism $(G/G_0, \|\cdot\|_{tw})\arrow (\Gr_{++}(H), d_{tw})$.

\newpage

{\bf \blue Transformation groups and subtwistor metrics}


\theorem
{\bf \red The subtwistor metric $d_{tw}$ induces the standard
topology on $\Gr_{++}(H)$.}


{\bf \green Step 1:}
Since $\Gr_{++}(H)\cong (G/G_0, \|\cdot\|_{tw})$,
it suffices to show that the subtwistor norm defines the
usual topology on $G$.

{\bf \green Step 2:} Let $\|\cdot\|$ be the usual
norm on $G$. Since $\|\cdot\|_{tw}\geq \|\cdot\|$,
{\bf \purple the identity map $(G,\|\cdot\|_{tw}) \arrow (G,\|\cdot\|)$
is  continuous.}

{\bf \green Step 3:} {\bf \blue 
(Brouwer's invariance of domain theorem):}\\
{\bf \red Let $X\stackrel f\arrow Y$ be a continuous, bijective
map of Hausdorff manifolds.} Then $f$ is a homeomorphism.
Apply this to the identity map  
$(G, \|\cdot\|_{tw})\arrow G$.
To prove that it is a homeomorphism, {\bf \purple it remains to show
that $(G, \|\cdot\|_{tw})$ is a manifold.}

{\bf \green Step 4:}  Since a bijective continuous
map from a compact is a homeomorphism, the identity map
$(G,\|\cdot\|_{tw}) \arrow (G,\|\cdot\|)$ is a
homeomorphism on compacts. Therefore, the Lebesgue 
covering dimension of any compact is the same in
$(G,\|\cdot\|_{tw})$ and in $(G,\|\cdot\|)$, hence
finite. Path connectedness of $(G,\|\cdot\|_{tw})$ is
clear from its construction.
Then {\bf \purple Gleason-Palais implies
that $G,\|\cdot\|_{tw}$ is a manifold.} \endproof

\newpage


\newcommand{\Teich}{\operatorname{Teich}}
\newcommand{\Comp}{\operatorname{Comp}}
\newcommand{\Per}{\operatorname{\sf Per}}
\newcommand{\Perspace}{\operatorname{{\Bbb P}\sf er}}

{\bf\blue Teichm\"uller space}


{\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 $\Teich$ is {\bf \blue a finite-dimensional
complex space} (Kodaira-Spencer-Kuranishi-Douady),
but often {\bf \red non-Hausdorff}.

\definition Let $\Diff_+(M)$ be the group of oriented
diffeomorphisms of $M$. We call $\Gamma:=\Diff_+(M)/\Diff_0(M)$ {\bf \blue the
mapping class group}. The {\bf \blue moduli space of complex
structures on $M$} is a connected component of $\Teich/\Gamma$.

\remark This terminology is {\bf \purple standard for curves.}


\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 $I,J,K$.

\remark {\bf \purple A hyperk\"ahler manifold is holomorphically
symplectic:} $\omega_J+\1 \omega_K$ is a holomorphic
symplectic form on $(M,I)$. 

{\bf\green THEOREM:} (Calabi-Yau) 
A compact, K\"ahler, holomorphically symplectic manifold
{\bf \red admits a unique hyperk\"ahler metric in any K\"ahler class.}

\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 \blue a Kummer surface}. {\bf \red
It is holomorphically symplectic}.

\definition A complex surface is called {\bf \blue a K3 surface}
if it a deformation of a Kummer surface. K3 surface is
also hyperk\"ahler.


\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-form.


{\bf\green THEOREM:} (Calabi-Yau) 
A compact, K\"ahler, holomorphically symplectic manifold
{\bf \red admits a unique hyperk\"ahler metric in any K\"ahler class.}

\remark Usually, one says ``hyperk\"ahler manifold''
meaning ``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 {\bf \red admits a finite covering
which is a product of a torus and several 
simple hyperk\"ahler manifolds.}


{\bf \green THEOREM:} (Fujiki). 
Let $\eta\in H^2(M)$, and $\dim M=2n$, where $M$ is
simple and hyperk\"ahler. Then $C\int_M \eta^{2n}=q(\eta,\eta)^n$,
for some primitive integer quadratic form $q$ on $H^2(M,\Z)$ and $C>0$.


{\bf \green Definition:} This form is called
{\bf \blue Bogomolov-Beauville-Fujiki form}. {\bf \purple It is defined
by this relation uniquely, up to a sign.} 

\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 
$P:\; \Teich \arrow {\Bbb P}H^2(M, \C)$
map $J$ to a line $H^{2,0}(M,J)\in {\Bbb P}H^2(M, \C)$.
The map $P:\; \Teich \arrow {\Bbb P}H^2(M, \C)$ is 
called {\bf\blue the period map}.

\remark 
{\purple \bf $P$ maps $\Teich$ into an open subset of a 
quadric,} defined by
\[
{\Bbb Per}:= \{l\in {\Bbb P}H^2(M, \C)\ \ | \ \  q(l,l)=0, \ \ q(l, \bar l) >0.
\]
It is called {\bf \blue the period space} of $M$.

\remark 
${\Bbb Per}=\Gr_{++}(H^2(M,\R), q)$

{\bf \green THEOREM:}  (Bogomolov) 
Let $M$ be a simple hyperk\"ahler manifold,
and $\Teich$ its Teichm\"uller space. 
{\bf \red Then the period map $P:\; \Teich \arrow {\Bbb Per}$ is 
locally a diffeomorphism.}

\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
Let $M$ be a hyperk\"ahler manifold,
$\Teich$ its Teichm\"uller space, and
$\Teich_b$ the quotient of $\Teich$ by
$\sim$. {\bf \red Then the period map
$P:\; \Teich_b \arrow {\Bbb Per}$
induces a diffeomorphism on each connected
component.}

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


\newpage

{\bf \blue Proof of Global Torelli theorem}


\definition
Let $(M,I,J,K)$ be a hyperk\"ahler manifold.
{\bf \blue A hyperk\"ahler 3-plane} in $H^2(M,\R)$ is a positive oriented
3-dimensional subspace $W$, generated by $\omega_I, \omega_J, \omega_K$.

\remark
The set of oriented 2-dimensional planes in $W$ is identified 
with $S^2 =\C P^1$. It is called {\bf \blue the twistor family}
of a hyperk\"ahler structure. A point in the twistor family
corresponds to a complex structure $aI + bJ + cK \in {\Bbb H}$,
with $a^2+b^2+c^2=1$. We call the corresponding $\C P^1\subset \Teich$
{\bf \blue the twistor lines}.

\definition We call a subspace $R\subset H^2(M,\R)$
{\bf \blue irrational} if $R^\bot \cap H^2(M,\Q)$ is empty.

\theorem
Let $S\subset \Perspace$ be a twistor line
corresponding to an irrational plane 
$\Gr_{+++}(H^2(M,\R)$. {\bf \red Then 
it can be lifted to $\Teich$ with each 
of the irrational point in its preimage. }

\corollary
The period map $\Teich_b\arrow \Perspace$ {\bf \red is an isometry
with respect to the subtwistor metrics.}

\remark Now the global Torelli follows, because (being an
isometry) it is also a covering.


\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 $l \cap H^2(M,\R)=0$.

{\bf \green  Step 2:} {\bf \purple This 2-dimensional plane is 
positive,} because 
 $q(\Re l, \Re l) = q(l+ \bar l, l+ \bar l) = 2 q(l, \bar l)>0$.

{\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




\end{document}

