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\begin{document}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf Symplectic geometry\\[15mm]
\small lecture 9: pseudo-holomorphic curves and calibrations}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\\[14mm]

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

{\tiny\bf HSE, room 306, 16:20,
\\[2mm] 
October 2, 2021
}
\end{center}


\newpage


{\bf \blue Complex manifolds}

{\bf\green DEFINITION:} Let $M$ be a smooth manifold. 
An {\bf \blue almost complex structure} is an operator
$I:\; TM \arrow TM$ which satisfies $I^2 = - \Id_{TM}$.

The eigenvalues of this operator are $\pm \1$.
The corresponding eigenvalue 
decomposition is denoted $TM=T^{0,1}M\oplus T^{1,0}(M)$.


{\bf\green DEFINITION:}
An almost complex structure is {\bf \blue integrable}
if $\forall X,Y \in T^{1,0}M$, one has $[X,Y]\in T^{1,0}M$.
In this case $I$ is called {\bf \blue a complex structure operator}.
A manifold with an integrable almost complex structure
is called {\bf \blue a complex manifold}. 

{\bf\green THEOREM:} (Newlander-Nirenberg)\\
{\bf \purple This definition is equivalent to the usual one.}

\newpage


{\bf \blue Almost complex submanifolds}

\definition
Let $(M,J)$ be an almost complex manifoldm
and $Z\subset M$ a closed subvariety.
We allow $Z$ to be singular, but 
we want the non-singular part to have
a finite Riemannian volume in a neighbourhood
of any singular point (prove that the
{\bf \purple finiteness of the volume is independent of the choice
of a Riemannian metric)}.
Suppose that $J(T_zZ) \subset T_zZ$ for every
smooth point $z\in Z$. Then $Z\subset M$
is called {\bf \blue an almost complex subvariety}.

\remark 
A subvariety of a complex manifold {\bf \red is 
almost complex if and only if it is complex,}
that is, it can be given as the set of common
solutions of a system of holomorphic equations.
This result follows from Newlander-Nirenberg
for smooth subvarieties, and needs a non-trivial
argument for singular.

{\bf \green FACT:} A general (non-integrable)
almost complex manifold {\bf \purple
does not admit almost complex subvarieties
of dimension $\dim_\C \geq 2$}, even locally.
For example, {\bf \red $S^6$ with the standard
non-integrable complex structure has no 
almost complex subvarieties} except 
1-dimensional.

\newpage

{\bf \blue Almost complex symplectic manifolds}

\definition
Let $(M, \omega)$ be a symplectic manifold, and
$I$ an almost complex structure. We say that
{\bf \blue $I$ is compatible with the symplectic
structure} if $g(x, y):= \omega(Ix, y)$
for some Riemannian form $g$.

\remark In the same way one defines
{\bf \blue almost complex structures compatible
with a non-degenerate 2-form}.


\definition
{\bf \blue An almost K\"ahler manifold}
is a manifold  $(M, \omega, I)$
equipped with a symplectic form $\omega$ and
an almost complex structure $I$
compatible with $\omega$.

\theorem
Let $(M, \omega)$ be a manifold equipped
with a non-degenerate skew-symmetric 2-form.
Then {\bf \red the space of almost complex structures
compatible with $\omega$ is contractible.}


\newpage

{\bf \blue Space of almost complex structures compatible with $\omega$}

{\bf \green THEOREM 1:}
Let $(M, \omega)$ be a manifold equipped
with a non-degenerate skew-symmetric 2-form.
Then {\bf \red the space $C$ of almost complex structures
compatible with $\omega$ is contractible.}


\pstep
We identify $C$ with the space of Riemannian metrics
$g$ such that $g^{-1}\omega$ is an almost complex
structure compatible with the $\omega$. Since 
the space $R$ of Riemannian metrics is convex,
it is contractible, hence {\bf \purple to prove that $C$
is conractible it would suffice to show that
$C$ is a deformational retract of $R$.}

{\bf \green Step 2:} 
Let $A:= g^{-1}\omega$, that is, $g(Ax,y)=\omega(x,y)$.
{\bf \purple The matrix $A$ us skew-symmetric:}
$g(Ax,y)= \omega(x,y)= - \omega(y, x)=-g(Ay,x)=-g(x,Ay)$.
A skew-symmetric matrix can be written in some
orthonormal basis as
 \[ \small \omega=\begin{pmatrix}
A=\begin{matrix}0 & \alpha_1\\ -\alpha_1 & 0\end{matrix} & & 0 \\
 & \ddots & \\
0 & & \begin{matrix}0 & \alpha_n \\ -\alpha_n & 0\end{matrix}
\end{pmatrix} 
\]
Since $g(A^2x,x)=-g(Ax,Ax)$, 
{\bf \purple the matrix $A^^2$ is symmetric and negative definite.}
{\bf \red Then $B_t:=e^{-\frac{t}2\log (-A^2)}$
is correctly defined, symmetric, and continuously depends
on $t$ and $A$.}

\newpage

{\bf \blue Space of almost complex structures compatible with $\omega$ (2)}

{\bf \green Step 3:}
The operator $AB_1$ is written in the same coordinates
as 
\[ \begin{pmatrix}
\omega=\begin{matrix}0 & 1\\ -1 & 0\end{matrix} & & 0 \\
 & \ddots & \\
0 & & \begin{matrix}0 & 1 \\ -1 & 0\end{matrix}
\end{pmatrix},
\]
that is, defines an almost complex structure on $M$.
Since $\omega(AB_1x, y)= g(B_1x, y)$, and $B_1$
is symmetric, {\bf \purple this almost complex
structure is compatible with $\omega$.}

{\bf \green Step 4:} The map
$g,t \stackrel{\Psi_t}\arrow g(B_tx, y)$ for $t=1$ 
gives the metric $g(B_1\cdot, \cdot)\in C$,
and for $t=0$ gives $g$. {\bf\purple Therefore,
the map $\Psi_t$ retracts $C$ to $R$.}
\endproof



\newpage




{\bf \blue Conformal structure}

\definition
Let $h, h'$ be Riemannian structures on $M$.
These Riemannian structures are called {\bf \blue
  conformally equivalent} if $h' = fh$, where 
$f$ is a positive smooth function. 

\definition
{\bf \blue Conformal structure} on $M$ is a class
of conformal equivalence of Riemannian metrics.

\exercise
Let $I$ be an almost complex structure on a 2-dimensional 
Riemannian manifold, and $h, h'$ two Hermitian metrics.
{\bf \red Prove that $h$ and $h'$ are conformally equivalent}. 
Prove that any metric conformally equivalent to 
a Hermitian metric is Hermitian.

\remark {\bf \purple The last statement is clear from the definition,
and true in any dimension. }

\newpage

{\bf \blue Stereographic projection}



\centerline{\epsfig{file=1280px-Stereographic_projection_in_3D.png,width=0.4\linewidth} \ 
\ \epsfig{file=Stereographic_projection_SW.JPG,width=0.4\linewidth}}


{\bf \blue Stereographic projection} is a light projection from the south pole
to a plane tangent to the north pole.

{\bf \purple stereographic projection is conformal} {\bf \red (prove it!)}

\newpage

{\bf \blue Stereographic projection (2)}

\ \ 

\centerline{\epsfig{file=1024px-Stereographic_with_Tissot_Indicatrices.png,width=0.5\linewidth}}

{\em \green The stereographic projection with Tissot's indicatrix of deformation. }


\newpage

{\bf \blue Cylindrical projection}

\ \ 

\centerline{\epsfig{file=cylindrical-proj.png,width=\linewidth}}

{\em \green Cylindrical projection is not conformal. However, it is volume-preserving. }


\newpage

{\bf \blue Conformal structures and almost complex structures}

\remark
The following theorem 
implies that almost complex structures on 
a 2-dimensional oriented manifold are equivalent
to conformal structures.

\theorem
Let $M$ be a 2-dimensional oriented manifold.
Given a complex structure $I$, let $\nu$ be
the conformal class of its Hermitian metric
(it is unique as shown above). 
{\bf \red Then $\nu$ determines
$I$ uniquely.}

\proof
Choose a Riemannian structure $h$ compatible
with the conformal structure $\nu$.
Since $M$ is oriented, the group $SO(2)=U(1)$
acts in its tangent bundle in a natural way:
$\rho:\; U(1) \arrow GL(TM)$.
Rescaling $h$ does not change this action,
hence it is determined by $\nu$.
Now, define $I$ as $\rho(\1)$; then $I^2= \rho(-1)=-\Id$.
Since $U(1)$ acts by isometries, this
almost complex structure is compatible with $h$ and
with $\nu$. \endproof

\definition
{\bf \blue A Riemann surface} is a complex manifold of dimension 1,
or (equivalently) an oriented 2-manifold equipped with
a conformal structure.

\exercise Prove that
{\bf \purple a smooth map 
from one Riemann surface to another is holomorphic
if and only if it preserves the conformal structure
everywhere.}


\newpage

{\bf \blue Calibrations}

\newcommand{\comass}{\operatorname{\sf comass}}

\definition
(Harvey-Lawson, 1982)\\
 Let $W\subset V$ be a $p$-dimensional subspace in a Euclidean space,
and $\Vol(W)$ denote the Riemannian volume form of 
$W \subset V$, defined up to a sign. For any $p$-form $\eta
\in \Lambda^p V$, let {\bf\blue comass} 
$\comass(\eta)$ be  the maximum of $\frac{\eta(v_1, v_2, ...,
v_p)}{|v_1||v_2|...|v_p|}$, for all $p$-tuples $(v_1, ...,
v_p)$ of vectors in $V$ and {\bf\blue face} be the set of  
planes $W\subset V$ where $\frac\eta {\Vol(W)}=\comass(\eta)$.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\definition\label{_precalibra_Definition_}
A {\bf\blue precalibration} on a Riemannian 
manifold is a differential form with comass 
$\leq 1$ everywhere.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\definition
A {\bf\blue calibration} is a precalibration which is closed.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\definition
Let $\eta$ be a $k$-dimensional precalibration on a
Riemannian manifold, and $Z\subset M$  a $k$-dimensional
subvariety (we always assume that the Hausdorff dimension
of the set of singular points of $Z$ is $\leq k-2$,
because in this case a compactly supported differential
form can be integrated over $Z$).  We say that $Z$ is
{\bf\blue calibrated by $\eta$} if at any 
smooth point $z\in Z$, the space $T_zZ$ is a face of 
the precalibration $\eta$.


\newpage 

{\small \sf
\begin{center} 
\begin{tabular}{cc}
\epsfig{file=Blaine_Lawson-1972-Berkeley.jpg,width=0.4\linewidth}
& \epsfig{file=Reese_Harvey_1968_Berkeley.jpg,width=0.4\linewidth}\\
H. Blaine Lawson, Jr.,  &
F. Reese Harvey,  \\ Berkeley, 1972 & Berkeley, 1968
\end{tabular}

Source: George M. Bergman, Berkeley
\end{center}
}

\newpage 

{\bf \blue Calibrations and minimal submanifolds}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\remark Clearly, for any precalibration $\eta$,
one has \[
\Vol(Z) \geq \int_Z\eta,\ \ \ \ \ (*)
\] where
$\Vol(Z)$ denotes the Riemannian volume of a compact $Z$, 
and the equality happens iff $Z$
 is calibrated by $\eta$. If, in
addition, $\eta$ is closed, the number
$\int_Z\eta$ is a cohomological invariant.
Then, {\bf \purple the inequality (*) implies that $Z$ 
minimizes the Riemannian volume in its 
homology class.}

\definition
A subvariety $Z$ is called {\bf \blue minimal}
if for any sufficiently small deformation 
$Z'$ of $Z$ in class $C^1$, one has 
$\Vol(Z')\geq \Vol(Z)$.

\remark
{\bf \red Calibrated subvarieties are obviously minimal.}


\definition
An almost complex Hermitian manifold $(M, I, h)$
with closed Hermitian form $\omega:= h(I(\cdot), \cdot)$
is called {\bf \blue almost K\"ahler}.

\example
{\bf \blue (Wirtenger's inequality):} \\
Let $(M, I, \omega)$ be an almost K\"ahler manifold.
{\bf \purple Then $\frac{\omega^d}{d! 2^d}$
is a calibration which calibrates 
$d$-dimensional almost complex subvarieties.}
In patricular, {\bf \red almost complex submanifolds
in almost K\"ahler manifolds are minimal}.



\newpage 

{\bf \blue Pseudoholomorphic curves}

\definition
Let $(M,J)$ be an almost complex manifold,
$(\Sigma, I)$ a Riemann surface,
and $\phi:\; \Sigma \arrow M$ an $I$-holomorphic map, 
that is, a smooth map with $D\phi(Ix)= J(D\phi(x))$.
Then $\phi(\Sigma)$ is called {\bf \blue
a pseudo-holomorphic curve}, or
{\bf \blue a $J$-holomorphic curve}.

\theorem
{\bf \blue (Wirtenger's inequality):} \\
Let $(M, I, \omega)$ be an almost K\"ahler manifold.
{\bf \purple Then $\frac{1}{2}\omega$ is a calibration
which calibrates pseudo-holomorphic curves.}

\proof
Let $g_S$ be the Riemannian volume form on $S$,
and $x, y\in T_sS$ be orthogonal vectors of length 1.
Then $g_S(x, y)=1$ and $\omega(x, y)= g(x, Iy)\leq 1$,
and the equality is realized if and only if $x=Iy$,
by Cauchy-Bunyakovsky-Schwarz inequality.
\endproof

\corollary {\bf \red Pseudoholomorphic curves are
minimal.}

\newpage

{\bf \blue Symplectic capacity and the pseudoholomorphic curves}

{\bf \green THEOREM 2:}
Let  $M=\C P^1 \times T^{2n}$ be the product of 
$\C P^1$ and a torus, equipped with the standard
symplectic structure, and $J$ a compatible
almost complex structure. {\bf \red Then
for any $x\in M$ there exists a pseudo-holomorphic
curve $S$ homologous to $\C P^1 \times \{m \}$
and passing through $x$.}

This theorem implies Gromov's non-squeezing theorem.

\theorem {\bf \blue (Gromov)}
{\bf \red Symplectic capacity of a symplectic cylinder
$\Cyl_1$ is equal to $\pi$.}





\end{document} 

When it is singular, we need to show that
the contribution of the singular points vanishes.
{\bf \purple This would follow if we prove that
$\lim\limits{\epsilon \to 0} \int_{S_\epsilon} \omega=0$,}
where $S_\epsilon$ is an intersection of $S$ and
an $\epsilon$-ball with the center in a singular point.

{\bf\green Step 2:}
Since $\int_{S_{\epsilon+\delta}} - \omega \int_{S_\epsilon} \omega
= \int_{S_{\epsilon+\delta}\backslash S_\epsilon}\omega >0$,
the function $\epsilon \arrow \int_{S_\epsilon}$ is monotonous.
Since $\int_{S_{\epsilon}}\omega= \sum_k 
\int_{S_{{\frac 1 {2^k}\epsilon}\backslash S_{{\frac 1 {2^{k+1}}\epsilon}}\omega$,
to prove that it converges to 0
it remains to show that $\int_{S_{\epsilon}}\omega$ is finite.


\newpage

{\bf \blue K\"ahler manifolds}


{\bf\green DEFINITION:} An Riemannian metric $g$ on
an almost complex manifiold $M$ is called 
{\bf \blue Hermitian} if $g(Ix, Iy)= g(x,y)$.
In this case, $g(x, Iy)= g(Ix, I^2y) = - g(y, Ix)$,
hence $\omega(x,y):= g(x, Iy)$ is skew-symmetric.

{\bf\green DEFINITION:} The differential 
form $\omega\in \Lambda^{1,1}(M)$ is called
{\bf \blue the Hermitian form} of $(M,I,g)$.

{\bf\green THEOREM:} Let $(M,I,g)$ be an almost complex Hermitian
manifold. {\bf \purple Then the following conditions are equivalent.}

(i) The complex structure $I$ is integrable, and 
the Hermitian form $\omega$ is closed.

(ii) One has $\nabla(I)=0$, where $\nabla$ is the Levi-Civita connection 
\[ \nabla:\; \End(TM) \arrow \End(TM)\otimes \Lambda^1(M).\]

{\bf\green DEFINITION:} A complex Hermitian manifold $M$
is called {\bf \blue K\"ahler} if either of these conditions hold.
The cohomology class $[\omega]\in H^2(M)$ of a form $\omega$ 
is called {\bf \blue the K\"ahler class} of $M$. The set
of all K\"ahler classes is called {\bf \blue the K\"ahler cone}.

