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\begin{document}
\setcounter{page}{1}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf Zoll manifolds, conifold transform and symplectic packing }
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\\[14mm]

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

{\small\bf
Semin\'ario de Geometria Diferencial}\\[10mm] \scriptsize
IMPA, June 27, 2023, \\[20mm]

{\bf \red \small \bf 
Joint work with Michael Entov}
\end{center}

\newpage

{\bf \blue Contact manifolds.}

{\bf \green Definition:} Let $M$ be a smooth manifold, $\dim M=2n-1$,
and $\omega$ a symplectic form on $M \times \R^{>0}$.
Suppose that $\omega$ is {\bf \red automorphic}:
$\Psi_q^*\omega=q^2 \omega$, where $\Psi_q(m,t)= (m, qt)$.
Then $M$ is called {\bf\blue contact}.

\definition
{\bf \blue The contact form} on $M$ is defined
as $\theta = i_{v}\omega$, where 
$v=t\frac d {dt}$. Then 
$d\theta = \{d, i_{v}\}\omega=\Lie_{v}\omega = \omega.$
Therefore, {\bf \red the form $(d\theta)^{n-1} \wedge
  \theta= \frac 1 n \Lie_{v}\omega^n$ 
is non-degenerate on $M\times \{ t_0\}\subset M \times \R^{>0} $.}

{\bf \green Remark:} Usually, a contact manifold
is defined as a {\bf \purple ($2n-1$)-manifold with 1-form $\theta$ such that
$d\theta^{n-1} \wedge \theta$ is nowhere degenerate. }

{\bf \green Example:} {\bf \red An odd-dimensional sphere $S^{2n-1}$ is contact.}
Indeed, its cone $S^{2n-1}\times \R^{>0} = \R^{2n}\backslash 0$
has the standard symplectic form $\sum_{i=1}^n dx_{2i-1}\wedge dx_{2i}$
which is obviously homogeneous.

{\bf \purple Contact geometry is an odd-dimensional
counterpart to symplectic geometry}

\definition
{\bf \blue The Reeb field}
on a contact manifold $(M, \theta)$ is a field
$R\in TM$ such that $d\theta(R, \cdot)=0$
and $\langle\theta, R\rangle=1$.


\newpage

{\bf\blue K\"ahler manifolds.}

{\bf \green Definition:} Let $(M,I)$ be a complex manifold, $\dim_\C M=n$,
and $g$ is Riemannian form. Then $g$ is called {\bf \blue Hermitian} if
$g(Ix,Iy)=g(x,y)$.


{\bf \green Remark:} Since $I^2=-\Id$, it
is equivalent to $g(Ix,y) = -g(x, Iy)$. {\bf \purple The form
$\omega(x,y):= g(x,Iy)$ is skew-symmetric.}

{\bf \green Definition:} The differential form $\omega$ is called
{\bf \blue the Hermitian form of $(M,I,g)$}.


{\bf \green Definition:} A complex Hermitian manifold is called
{\bf \blue K\"ahler} if $d\omega=0$.

\newpage

{\bf\blue Sasakian manifolds.}


{\bf \green Definition:} Let $(M,g_M)$ be a Riemannian manifold, $\dim M=2n-1$,
and $(g, \omega,I)$ a Kaehler structure on $M \times \R^{>0}$
with $g=g_M + t^2 dt\otimes dt$.
Suppose that $\omega$ is {\bf \red automorphic}:
$\Psi_q^*\omega=q^2 g$, where $\Psi_q(m,t)= (m, qt)$,
and $I$ is $\Psi_q$-invariant. Then $M$ is called {\bf\blue Sasakian},
and $M \times \R^{>0}$ its {\bf \blue K\"ahler cone}.

{\bf \purple Sasakian geometry is an odd-dimensional
counterpart to K\"ahler geometry}

{\bf \green Remark:} A Sasakian manifold is obviosly contact.
Indeed, {\bf \purple a Sasakian manifold is a contact manifold equipped
with a compatible Riemannian metric.}

{\bf \green Example:} {\bf \red An odd-dimensional sphere $S^{2n-1}$ 
is Sasakian.} Indeed, its cone $S^{2n-1}\times \R^{>0} = \C^{n}\backslash 0$
has the standard K\"ahler form $\1\sum_{i=1}^n dz_{i}\wedge d\bar z_i$
which is obviously automorphic.

{\em \green  S. Sasaki, "On differentiable manifolds with certain
structures which are closely related to almost contact
structure", Tohoku Math. J. 2 (1960), 459-476.}

%\newpage
%
%{\bf\blue K\"ahler potentials on the K\"ahler cones}
%
%\definition
%A {\bf\blue K\"ahler potential} on a K\"ahler manifold $(M,I, \omega)$
%is a function $f$ such that the (1,1)-form $dd^c(f)$ is equal to $\omega$,
%where $dd^c= d I d I^{-1}= \frac{\6\bar\6}{-2\1}$.
%
%\claim
%Let $M$ be a Sasakian manifold and 
%$C(M)=M \times \R^{>0}$ its K\"ahler cone, with $t$ the parameter
%on $\R^{>0}$ and $\omega$ its K\"ahler form.
%{\bf \red 
%Then $dd^c(t^2)=\omega$, in other words, $\frac 1 2 t^2$ is a K\"ahler potential.}
%
%\proof Let $\vec r= t d/dt$ be the radial vector field.
%Cartan's formula $\Lie_X \eta= (d\eta) \cntrct X + d(\eta\cntrct X)$
%gives 
%\[ 
%2\omega=\Lie_{\vec r}\omega= d(\omega\cntrct {\vec r})= d(tI(dt))= 2 dd^c(t^2).
%\]
%\endproof
%
%\corollary The field $I(\vec r)$ {\bf \red acts on $C(M)$ by
%holomorphic isometries. }
%
%\proof Indeed, $\vec r$ is holomorphic, and
%$\Lie_{I\vec r}(t)= \langle dt, I t d/dt\rangle=0$.
%Therefore, $I(\vec t)$ preserves the K\"ahler
%potential and the K\"ahler structure.
%\endproof

\newpage

{\bf\blue Boothby-Wang theorem}

{\em \green W. M. Boothby, H. C. Wang, On Contact Manifolds
Annals of Mathematics, Second Series, Vol. 68, No. 3 (Nov., 1958), pp. 721-734.
}

\definition
A contact manifold $(M, \theta)$ is {\bf \blue normal}
if it is equipped with an $S^1$-action preserving $\theta$
and tangent to the Reeb field.

\remark
Let $(M, \theta)$ be a contact manifold.
{\bf \purple Then the form $d\theta$ is non-degenerate
the bundle $\ker \theta \subset TM$.}

\theorem {\bf \blue (Boothby-Wang, 1958)}\\
Let $(M, \theta)$ be a normal contact manifold.
{\bf \red Then its space $X$ of Reeb orbits is symplectic and
the natural projection $\pi:\; M \arrow X$ induces
a symplectic isomorphism $d\pi:\; \ker \theta\restrict x \arrow T_{\pi(x)}X$.}

\theorem {\bf \blue (Boothby-Wang, 1958)}\\
Let $(M, \theta)$ be a normal contact manifold,
and $(X, \omega)$ the symplectic manifold obtained
as its space of Reeb orbits. {\bf \red Then the cohomology
class of $\omega$ is integral.} Conversely,
for any symplectic manifold $(X, \omega)$
with $[\omega]\in H^2(X, \Z)$,
{\bf \red there exists a principal $S^1$-bundle  $L$ with 
$c_1(L) =[\omega]$ and a normal contact
structure on $M:=\Tot(L)$ such that 
the corresponding Boothby-Wang projection
coincides with the natural map $M \arrow X$.}

\newpage

{\bf\blue Boothby-Wang theorem for Sasakian manifolds}


\remark
Suppose that $M$ is a Sasakian manifold with all  Reeb
orbits closed. Then its space 
$X$ or Reeb orbits is a projective orbifold, 
and the K\"ahler cone $C(M)$ is the affine cone over $X \subset {\C P}^n$.
The converse is also true: {\bf \purple whenever $X$ is K\"ahler,
the corresponding Boothby-Wang contact manifold is Sasakian.}


\newpage

{\bf\blue Hypersurfaces of contact type}

\definition
A vector field $v$ on a symplectic manifold $(M, \omega)$ is
called {\bf \blue Liouville} if $\Lie_v \omega = \omega$.
 We say that a smooth hypersurface
$S\subset M$ is {\bf \blue a hypersurface of contact type} if there
exists a Liouville vector field $v\in TM$ defined
in a neighbourhood $U \supset S$ and transversal to $S$.

\claim {\bf \red A hypersurface of contact type
is contact,} with the contact form given by
$\alpha:= i_v \omega\restrict S$.

\pstep
Since $v$ is transversal to $S$, its orbit space is $S$.
This can be used to identify its tubular neighbourhood 
$U$ with $S\times I$, where $I$ is an interval, in such a way that
the shift in $I$ multiplies the symplectic
form $\omega$ by a scalar. 

{\bf \green Step 2:}
We write $\omega= t dt \wedge \alpha + t^2\omega_0$,
where $\omega_0$ is a 2-form on $S$ and
$t$ the coordinate on $I$. 
In these notations, $v= t \frac{d}{dt}$.
Then  $t^2\alpha= i_v \omega$. Therefore, $d\omega=0$
implies that $d\alpha = - \omega_0$,
hence 
$i_v \omega^n =\alpha\wedge (\omega_0)^n= \alpha\wedge (d\alpha)^n$
is non-degenerate on $S$.
\endproof

\remark
{\bf \blue Gray stability theorem} claims that {\bf \red
 a smooth deformation of a compact contact manifold $S$
is contactomorphic to $S$. } The space of Liouville
vector fields in a  neighbourhood of $S\subset M$ is convex, hence
contractible. This implies that {\bf \purple the contact
structure on a hypersurface of contact type is unique}
up to a contact diffeomorphism.

\newpage

{\bf\blue Reeb field and almost complex structures}

\definition
An almost complex structure $I$ 
is {\bf \blue compatible with a symplectic structure $\omega$}
if $\omega(Ix, Iy) = \omega(x, y)$ and $\omega(x, Ix) >0$
for any $x\neq 0$. {\bf \purple In this case, 
$g(x, y):= \omega(x, Iy)$ is a positive definite scalar product.}

{\bf \green PROPOSITION 1:}
Let $S$ be a contact manifold and $(C(S), \omega)$
its symplectic cone, equipped with the symplectic homothety
diffeomorphism $\Psi_t$. Consider an $\Psi_t$-invariant almost
complex structure $I$ on $C(S)$. 
Assume that
the vector field $d/dt$ satisfies $|d/dt|=1$ and is
orthogonal to $S\subset C(S)$ embedded as $S\times \{r\}$.
 {\bf \red Then the Reeb field
can be expressed as $R=I(d/dt)$,} where $t$ is the 
coordinate on $\R^{>0}$, considered as a function
on $C(S)= S \times \R^{>0}$.

\proof
For any $x\in TC(S)$, we have $x\in TS$ if and only if
$x\bot \frac d{dt}$. Therefore, the symplectic orthogonal to $TS$
is $I (d/dt)$; this vector field clearly has constant length.
\endproof

\newpage

{\bf\blue Geodesic flow on a Riemannian manifold 
as a Hamiltonian flow}

\remark
Recall that {\bf \blue a Hamiltonian vector field}
on a symplectic manifold $M$ is a vector field $v$ which
is symplectically dual to $dH$, where $H$ is a smooth
function, called {\bf \blue the Hamiltonian} of $v$.

\definition
Let $M$ be a complete Riemannian manifold, 
$(m, v)\in TM$ a point in its tangent space, and
$\gamma_{(m, v)} (t)$ the geodesic starting in $m$
and tangent to $v$. 
{\bf \blue The geodesic flow}  is a diffeomorphism 
flow $\Psi_t, t\in \R$ on the tangent bundle $TM$ taking
$(m, v)\in TM$ to 
$(\gamma_{(m, v)} (t), \dot\gamma_{(m, v)} (t))\in T_{\gamma_{(m, v)} (t)}M$.


\claim 
Let $M$ be a Riemannian manifold. We use the Riemannian
metric to identify $TM$ and $T^*M$. 
This identification gives a symplectic structure on $TM$.
Denote by
$H$ the function $H(v)=|v|^2$. {\bf \red Then the
geodesic flow on $TM$ is the Hamiltonian flow
associated with the function $H$.}

\proof See
{\green \em V. I. Arnold, Mathematical Methods of Classical Mechanics.}
\endproof


\newpage

{\bf\blue Contact manifold associated with the cotangent bundle}

\claim 
Let $M$ be a smooth manifold, and $\omega$
the Hamilton symplectic form on $T^* M$. Then {\bf \red the set
$ST^* M:=\{v\in T^* M\ \ |\ \ |v|=1\}$ is a hypersurface
of contact type.}

\proof
In coordinates the form $\omega$ can be written as
$\sum dp_i \wedge dq_i$, where $p_i$ are coordinates
on $M$ and $q_i$ the corresponding coordinates 
on the fibers of the bundle $T^* M \arrow M$.
The fiberwise homothety vector field $\sum q_i d/dq_i$
is a Liouville field, transversal to $ST^* M$. \endproof

\newpage

{\bf\blue Reeb orbits and the geodesic flow}

\proposition
In the above assumptions, the Reeb field of
$ST^* M$ is equal to the symplectic dual of $d H$,
that is, {\bf \red
it is the vector field generating the geodesic flow.}

\pstep
To use Proposition 1, we need to construct a compatible
almost complex structure which is invariant with respect
to the homothety map.  Let $\pi:\; T^*M \arrow M$ be the projection.
Using the Levi-Civita connection, we
obtain a decomposition $TT^* M = \pi^* T^*M \oplus \pi^* TM$.
The symplectic structure on $T^* M$ is induced by the
natural pairing of these two factors.

The metric on $M$ induces a Riemannian metric on $TT^* M$,
called {\bf \blue the Sasaki metric}. 
The corresponding almost complex structure
uses the decomposition 
$TT^* M = \pi^* T^*M \oplus \pi^* TM$,
with the first term $T_{vert} T^*M$ consisting of
fiberwise tangent vector fields, and the
second term  $T_{hor} T^*M$ the ``horizontal
sub-bundle'', obtained using the connection.
The almost complex structure exchanges
$T_{vert} T^*M$ identified with $\pi^* TM$
using the metric and $T_{hor} T^*M=\pi^* TM$.
Under this identification the radial
vector field becomes the vector field
which is horizontal and equal to $v$
in $(m, v)\in TM=T^*M$; this is precisely
the vector field tangent to the geodesic
flow.

{\bf \green Step 2:}
By Proposition 1, the Reeb field is $F(v)$,
where $v$ is the radial vector field tangent to the homothety.
\endproof

\newpage

{\bf\blue Zoll manifolds and contact manifolds}


\definition
A compact Riemannian manifold $M$ is called {\bf \blue Zoll}
if all its geodesics are compact and the geodesics have constant length

\proposition
Let $M$ be a compact Riemannian manifold, and
$ST^* M$ the manifold of unit cotangent vectors,
considered as a contact manifold. {\bf \red Then
$ST^* M$ is a normal contact manifold if and only if
$M$ is Zoll.}

\pstep
Let $Z$ be a Riemannian manifold 
equipped with a rank 1 foliation ${\cal F}$ with compact fibers,
and $Z \arrow Z/{\cal F}$ be the projection to the leaf space.
{\bf \purple Then $Z/{\cal F}$ is smooth if and only if length of an orbit
is a continuous function on $Z/{\cal F}$.} This is left as an exercise.

{\bf \green Step 2:} We apply this observation to the geodesic
flow on $ST^* M$. If $M$ is Zoll, this implies that the
the projection from $ST^* M$ to the space of Reeb orbits is
smooth, hence $ST^* M$  is normal; converse follows from
Wadsley theorem. \endproof

\newpage

{\bf\blue Zoll manifolds and K\"ahler geometry}

\example
In dimension 2, there are many non-trivial metrics on 2-spheres
(``Zoll spheres''), and a symmetric metric on $\R P^2$. In bigger dimension, 
the only known Zoll manifolds are  $S^n$, $\R P^n$, $\C P^n$, 
${\Bbb H} P^n$, and the Cayley projective plane ${\Bbb C}a P^2$.

\proposition
Let $M$ be $S^n$, $\R P^n$, $\C P^n$, 
${\Bbb H} P^n$, or ${\Bbb C}a P^2$.
{\bf \red Then the contact manifold $S T^* M$ is Sasakian},
and its space of Reeb orbits is K\"ahler.

\remark
For $S^n$ the space of Reeb orbits is $\Gr_2(\R^{n+1})$,
with its natural structure of the K\"ahler symmetric space.
For $\C P^n$ it is the space of $1,2$-flags (point and a line)
in $\C P^n$. For  ${\Bbb H} P^n$ and ${\Bbb C}a P^2$ we don't know.

\newpage

{\bf\blue The conifold transform for Calabi-Yau threefolds}

{\em \green 
Smith, I.; Thomas, R. P.; Yau, S.-T.,
Symplectic conifold transitions,
J. Differential Geom. 62 (2002), no. 2, 209-242.}

%Symplectic surgeries from singularities
%Smith, Ivan; Thomas, Richard
%Turkish J. Math. 27 (2003), no. 1, 231-250.}
\definition
{\bf \blue A Calabi-Yau threefold} is a complex 3-dimensional
compact K\"ahler manifold with $c_1(M)=0$.

\remark
Let $S\subset M$ be a smooth $\C P^1$ on a Calabi-Yau manifold. 
%By adjunction formula, 
%$NS \cong\calo(i) \oplus \calo(j)$, where $i+j=-2$.
In the typical situation, $NS \cong\calo(-1) \oplus
\calo(-1)$. By Grauert theorem, such a rational
curve can be blown down, defining a map
$M \arrow M_0$, where $M_0$ is a singular
complex variety. It is not hard to see that
{\bf \red this singularity is locally biholomorphic to an
affine cone over $\C P^1\times \C P^1$.}


\proposition
Let $S\subset M$ be a smooth rational curve 
on a Calabi-Yau manifold, with  
$NS \cong\calo(-1) \oplus \calo(-1)$, 
and $M_0$ the singular variety obtained
by a blow-down of $S$. {\bf \red Then $M_0$
admits a smooth deformation $M_1$.}

\definition
The transition from $M$ to 
$M_1$  is called {\bf \blue the conifold transition.}

\newpage

{\bf\blue Miles Reed's conjecture}

\remark
The manifold $M_1$ has trivial canonical bundle.
If it satisfies $b_2(M_1)=0$, this manifold
is diffeomorphic to $\#_i S^3 \times S^3$
(connected sum of several copies of $S^3 \times S^3$).
Miles Reed {\bf \purple conjectured that  any Calabi-Yau can be devolved
to such a manifold after a sequence of conifold
transitions.}


Note that the conifold transform in 
complex-analytic setup {\bf \red is emphatically
non-invertible:} {\bf \purple there exists a Calabi-Yau manifold
with a Lagrangian $S^3$ which cannot be blown down.}


{\em \green Smith-Thomas and Yau described the
topology of conifold transition symplectically.}


\newpage

{\bf\blue Gluing symplectic manifolds over contact hypersurfaces}

\definition
Let $S\subset M$  be a
contact-type boundary hypersurface in a symplectic
manifold. We say that $S$ is {\bf \blue convex} if
the Liouville field is directed from $M$ to $S$, and
{\bf \blue concave} otherwise.

\proposition
Let $S_1 \subset M_1$ be a concave component of a boundary
of a symplectic manifold, and $S_2 \subset M_2$
a convex component. Assume that $S_1$ is isomorphic
to $S_2$ as contact manifold. {\bf \red Then one
can glue $M_1$ to $M_2$ by taking 
an appropriate contact diffeomorphism,
identifying $S_1$ with $S_2$.}

\proof
Since $M_i$ is symplectomorphic to a symplectic cone
in a neighbourhood of $S_i$, it would suffice to
pick a contactomorphism $S_1\arrow S_2$ which 
can be extended to a cone. However, any contactomorphism
can be extended to a cone, by construction.
\endproof

\newpage

{\bf\blue Symplectic manifolds with normal contact boundary}

\example
Let $M$ be a symplectic threefold,
and $S^3\subset M$ a Lagrangian 3-sphere.
By Weinstein neighbourhood theorem, 
there is a neighbourhood of $S$ in $M$
which is symplectomorphic to a manifold of open balls in
$T^* S^3$ (for sufficiently small radius of the ball).
{\bf \purple Its boundary $Z$ is a Boothby-Wang contact manifold, 
which is $S^1$-fibered over the space of
geodesics in $S^3$,} identified with $\C P^1\times \C P^1$
Then $Z$ is a boundary of the cone over
$\C P^1\times \C P^1$ associated with the ample
bundle $\calo(1,1)$.

We glue the corresponding singular complex
variety in $Z$, using the gluing theorem.
This replaces the Lagrangian $S^3$ with
a neighbourhood of zero in the affine cone over
$\C P^1\times \C P^1$.

\claim
A small resolution of this conical singularity {\bf \red is
biholomorphic to a neighbourhood of a rational curve $C$}
in a Calabi-Yau manifold, with $NC = \calo(-1) \oplus \calo(-1)$.

\remark
Unlike the construction with complex deformations, 
{\bf \purple this construction is invertible}:
by Weinstein neighbourhood theorem, {\bf \red any symplectic
submanifold in a symplectic manifold has a neighbourhood
symplectomorphic to its neighbourhood in the normal bundle.}


\newpage

{\bf\blue The conifold transition, a general definition}

\definition
Let $M$ be a symplectic manifold,
$L\subset M$ a Lagrangian submanifold admitting
a Zoll metric.  Consider a Weinstein neighbourhood
$U_L$ with normal (in Boothby-Wang sense) contact boundary.
Denote by $Z_L$ the corresponding singular complex
variety, $\6 Z_L=\6 U_L$, and let $V_L$ be a complex
resolution of $Z_L$. {\bf \blue The direct conifold
  transform} is obtained by replacing $U_L$ with $V_L$;
it replaces a Lagrangian submanifold by a  symplectic
submanifold.

\definition
Its inverse is defined in a similar way: given
a smooth symplectic submanifold $X\subset M$,
isomorphic to the preimage of the singularity
in the resolution map $V_L \arrow Z_L$,
with normal bundle isomorphic to the normal 
bundle of $X\subset V_L$, {\bf \blue the inverse 
conifold transform uses the Weinstein
neighbourhood theorem to replace $V_L$ with
$U_L$. }

\example The symplectic conifold transition of
 Smith-Thomas and Yau is an example of this construction.

\newpage

{\bf\blue The conifold transition, examples and applications}

\example
For dimension $n\geq 4$ the cone over
$\Gr_2(\R^n)$ does not have partial resoltions,
this means that {\bf \purple we can only replace a Lagrangian sphere $S^n$
with a Grassmannian $\Gr_2(\R^n)$,
symplectically embedded to an
ambient manifoldm, and vice versa.}

\example
The cone over (1,2)-flags (the space of geodesics in $\C P^n$)
{\bf \purple has a partial resolution which is bimeromorphic to a
holomorphic cotangent bundle to $\C P^n$.}
The corresponding conifold transform replaces
a Lagrangian $\C P^n$ by a symplectic $\C P^n$
and vice versa, similar to the hyperk\"ahler rotation.

{\bf \purple This construction is useful for symplectic packing:}
if we have a control over the volume of $V_L$, 
we obtain control over volume of $U_L$ and vice versa.

This brings the following result about K3 surfaces.

\theorem
Let $L_1, ..., L_n$ be a collection of non-intersecting
special Lagrangian 2-spheres in a K3 surface $M$,
and $u_1, ..., u_n$ a numbers which satisfy
$\sum u_i \leq \Vol_\omega M$.
Denote by $U_{L_i}$ the Weinstein neighbourhood
of $S^2$ in $T^* S^2$ which is invariant
under rotations and has symplectic volume $u_i$;
such a neighbourhood is clearly unique.
{\bf \red Then there exists a collection of
symplectic embeddings $\phi_i:\; U_{L_i} \arrow M$,
with non-intersecting images, taking the zero
section $S^2 \subset U_{L_i} \subset T^* S^2$
to $L_i \subset M$.}



\end{document}








Conifold transform replaces U by the resolved cone, yielding \tM with a fixed symplectic sbmfd \tL.

Examples: 
(-2)-Lagr. spheres in K3, or, more generally, Lagr.  \C P^n in a hyperkahler mfd

From now on assume L is symmetric. 

For a positive v, define: 

Teich := {(\omega, L, v) s.t. the max volume of a symmetric Weinstein nbhd sympl. of L embedded into M is v^n}/Diff_0 (M)

TTeich := {(\tomega, \tL,v) s.t. \tomega|_\tL =  v (stand. form on a nbhd of \tL)}/\Diff_0 (\tM)

Thm.: Conifold transform gives a diff-m Teich->TTeich.

Full packing by Weinstein nbhds of Lagr. CP^n for a sympl. form on a complex Kummer K3.

Any K.-type sympl. form on K3 is diffeomorphic to one compatible with the complex Kummer structure.








\newpage

{\bf\blue Reeb field}

{\bf \green Definition:}
Given a contact manifold $(M,\theta)$, a vector field $R$
is called {\bf \blue the Reeb field} of $(M,\theta)$, if
$d\theta\cntrct R=0$ and $\theta(R)=1$.

\definition
Let $C(M):=M \times \R^{>0}$ be the cone of a Sasakian manifold $M$.
The vector field $\vec r :=t d/dt$ is called {\bf the Lee field} on 
$C(M)$. Clearly, $\vec r $ acts on $C(M)$ by holomorphic homotheties.

\remark On a Sasakian manifold, $d\theta$ is restriction of
the K\"ahler form to $M=M\times \{1\} \subset M \times \R^{>0}$,
hence $\ker d\theta\restrict M= I(R^c)$. Also, $\theta(I(\vec r))\restrict M=1$.
{\bf \purple Therefore, $R:=I(\vec r)$ is the Reeb field of $M$.}

\corollary The Reeb field of a Sasakian manifold {\bf
  \purple is the Riemannian dual to its 
contact form $\theta=\omega(\vec r, \cdot)$.}

\corollary For any Sasakian manifold,
{\bf \purple the Reeb field generates a flow of diffeomorphisms acting
on $M$ by contact isometries}. 

\proof The Reeb field 
$\theta^\sharp = I (\vec r)$
acts holomorphically because $\vec r$ acts holomorphically,
and preserves the K\"ahler potentialm as shown above.
\endproof

\newpage

{\bf\blue Quasiregular Sasakian manifolds}

{\bf \green Definition:} A Sasakian manifold $M$ is called
{\bf \blue quasiregular} if all orbits of the Reeb flow are 
compact. 

The following claim is 
{\bf \purple the Sasakian version of Boothby-Wang theorem.}

\claim {\bf \red Every quasiregular Sasakian manifold
is a total space of $S^1$-bundle over a K\"ahler orbifold.}

\proof
The quotient of $M$ over the Reeb flow
is the same as the quotient of its cone $C(M)$ over its complexification,
generated by $\vec r$ and $R=I(\vec r)$.  \endproof

\remark The space of Reeb orbits $X$ of a quasiregular
Sasakian manifold {\bf \purple is in fact projective,}
as follows from Kodaira theorem. Indeed, $C(M)$
is $\C^*$-bundle over $X$, and the corresponding line
bundle has positive curvature which can be expressed
as $dd^c\log(\phi)$, where $\phi=r^2$ is the K\"ahler
potential of $C(M)$.

