\documentclass{slides} \usepackage{amssymb, amsmath, amscd, color, epsfig} %\usepackage[matrix,arrow]{xy} \newcommand{\green}{\color[rgb]{0,0.4,0}} \newcommand{\purple}{\color[rgb]{0.4,0,0.4}} \newcommand{\red}{\color[rgb]{0.7,0,0}} \newcommand{\blue}{\color{blue}} \def\eqref#1{(\ref{#1})} \newcommand{\goth}{\mathfrak} \newcommand{\g}{{\frak g}} \renewcommand{\a}{{\frak a}} \newcommand{\arrow}{{\:\longrightarrow\:}} \newcommand{\Z}{{\Bbb Z}} \def\C{{\Bbb C}} \newcommand{\R}{{\Bbb R}} \newcommand{\Char}{\operatorname{\sf char}} \newcommand{\Q}{{\Bbb Q}} \renewcommand{\H}{{\Bbb H}} \newcommand{\6}{\partial} \def\1{\sqrt{-1}\:} \newcommand{\restrict}[1]{{\left|_{{#1}}\right.}} \newcommand{\cntrct} % contraction with a vector field {\hspace{2pt}\raisebox{1pt}{\text{$\lrcorner$}}\hspace{2pt}} \def\Bbb#1{\mathbb #1} \newcommand{\calo}{{\cal O}} \newcommand{\cac}{{\cal C}} % Correcting TeX... %\let\oldtilde=\tilde %\renewcommand{\tilde}{\widetilde} \renewcommand{\bar}{\overline} \renewcommand{\phi}{\varphi} \renewcommand{\epsilon}{\varepsilon} \renewcommand{\geq}{\geqslant} \renewcommand{\leq}{\leqslant} % Operatornames \newcommand{\even}{{\rm even}} \newcommand{\ev}{{\rm even}} \newcommand{\odd}{{\rm odd}} \newcommand{\const}{{\it const}} \newcommand{\fl}{{\rm fl}} \newcommand{\im}{\operatorname{im}} \newcommand{\inv}{{\operatorname{\sf inv}}} \newcommand{\End}{\operatorname{End}} \newcommand{\Sym}{\operatorname{Sym}} \newcommand{\Hol}{\operatorname{{\cal H}ol}} \newcommand{\Tot}{\operatorname{Tot}} \newcommand{\Id}{\operatorname{Id}} \newcommand{\id}{\operatorname{\text{\sf id}}} \newcommand{\Vol}{\operatorname{Vol}} \newcommand{\Hom}{\operatorname{Hom}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\Alt}{\operatorname{Alt}} \newcommand{\Alb}{\operatorname{Alb}} \newcommand{\Iso}{\operatorname{Iso}} \newcommand{\Sec}{\operatorname{Sec}} \newcommand{\Gr}{\operatorname{Gr}} \newcommand{\Av}{\operatorname{Av}} \newcommand{\Sing}{\operatorname{Sing}} \newcommand{\Spin}{\operatorname{Spin}} \newcommand{\codim}{\operatorname{codim}} \newcommand{\coim}{\operatorname{coim}} \newcommand{\coker}{\operatorname{coker}} \newcommand{\slope}{\operatorname{slope}} \newcommand{\rk}{\operatorname{rk}} \newcommand{\Def}{\operatorname{Def}} \newcommand{\Lie}{\operatorname{Lie}} \newcommand{\Tw}{\operatorname{Tw}} \newcommand{\Tr}{\operatorname{Tr}} \newcommand{\Spec}{\operatorname{Spec}} \newcommand{\Diff}{\operatorname{Diff}} \renewcommand{\Re}{\operatorname{Re}} \renewcommand{\Im}{\operatorname{Im}} \newcommand{\inbfpare}[1]{{% \mbox{\tt (}\hspace{-5pt}\mbox{\tt (} #1 % \mbox{\tt )}\hspace{-5pt}\mbox{\tt )}% }} \newcommand{\comment}[1]{{}} \def\blacksquare{\hbox{\vrule width 10pt height 10pt depth 0pt}} \def\endproof{\blacksquare} \def\shortdash{\mbox{\vrule width 4.5pt height 0.55ex depth -0.5ex}} \makeatletter %\@ifundefined{Bbb} % {\newcommand{\Bbb}[1]{{\mathbb #1}}}% %{}% {\edef\Bbb#1{{\Bbb #1}}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Pagestyle % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newcommand{\ps@verbit}{% \renewcommand{\@oddhead}{% \scriptsize {\it \small Conifold transform\hfil \tiny Misha 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: % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newcommand{\exercise}{% {\bf \green EXERCISE:\ }} \newcommand{\lemma}{% {\bf \green LEMMA:\ }} \newcommand{\claim}{% {\bf \green CLAIM:\ }} \newcommand{\corollary}{% {\bf \green COROLLARY:\ }} \newcommand{\theorem}{% {\bf \green THEOREM:\ }} \newcommand{\conjecture}{% {\bf \green CONJECTURE:\ }} \newcommand{\proposition}{% {\bf \green PROPOSITION:\ }} \newcommand{\proof}{% {\bf \green Proof:\ }} \newcommand{\pstep}{{\bf \green Proof. Step 1: }} \newcommand{\definition}{% {\bf \green DEFINITION:\ }} \newcommand{\example}{% {\bf \green EXAMPLE:\ }} \newcommand{\remark}{% {\bf \green REMARK:\ }} \newcommand{\question}{% {\bf \green QUESTION:\ }} \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 $C(M)$.