\(
\def\CC{{\mathbb C}}
\def\RR{{\mathbb R}}
\def\NN{{\mathbb N}}
\def\ZZ{{\mathbb Z}}
\def\TT{{\mathbb T}}
\def\CF{{\operatorname{CF}^\bullet}}
\def\HF{{\operatorname{HF}^\bullet}}
\def\SH{{\operatorname{SH}^\bullet}}
\def\ot{{\leftarrow}}
\def\st{\;:\;}
\def\Fuk{{\operatorname{Fuk}}}
\def\emprod{m}
\def\cone{\operatorname{Cone}}
\def\Flux{\operatorname{Flux}}
\def\li{i}
\def\ev{\operatorname{ev}}
\def\id{\operatorname{id}}
\def\grad{\operatorname{grad}}
\def\ind{\operatorname{ind}}
\def\weight{\operatorname{wt}}
\def\Sym{\operatorname{Sym}}
\def\HeF{\widehat{CHF}^\bullet}
\def\HHeF{\widehat{HHF}^\bullet}
\def\Spinc{\operatorname{Spin}^c}
\def\min{\operatorname{min}}
\def\div{\operatorname{div}}
\def\SH{{\operatorname{SH}^\bullet}}
\def\CF{{\operatorname{CF}^\bullet}}
\def\Tw{{\operatorname{Tw}}}
\def\Log{{\operatorname{Log}}}
\def\TropB{{\operatorname{TropB}}}
\def\wt{{\operatorname{wt}}}
\def\Span{{\operatorname{span}}}
\def\Crit{\operatorname{Crit}}
\def\CritVal{\operatorname{CritVal}}
\def\FS{\operatorname{FS}}
\def\Sing{\operatorname{Sing}}
\def\Coh{\operatorname{Coh}}
\def\Vect{\operatorname{Vect}}
\def\into{\hookrightarrow}
\def\tensor{\otimes}
\def\CP{\mathbb{CP}}
\def\eps{\varepsilon}
\)
SympSnip:
example 0.0.1
Recall our running example \(\pi:\CC^2\to \CC\) from Missing Label (exm:LocalModelOfASingularity)!.
We will prove that the symplectic parallel transport map preserves a class of Lagrangian submanifolds of the fiber.
Consider the function \(H(z_1, z_2)= \frac{1}{2}\left(|z_1|^2-|z_2|^2\right)=\frac{1}{2}\left( x_1^2+y_1^2-x_2^2-y_2^2\right)\).
The exterior derivative of this function, in local coordinates, is given by
\[dH= x_1dx_1 +y_1dy_1 -x_2dx_2- y_2dy_2.\]
We prove that \(H\) is invariant under the action of symplectic parallel transport along the fibration \(\pi:\CC^2\to \CC\).
In this example, we can explicitly compute that \(H\) is invariant under vectors contained in \(\ker(d\pi)^{\omega_\bot}\).
The kernel of \(d\pi=z_2dz_1+z_1dz_2\) at a point \((z_1, z_2)\) is the complex subspace generated by the vector \begin{align*}
\ker_{(z_1, z_2)}(d\pi)=&\Span_\CC(\langle z_1, -z_2\rangle)\\
=&\Span_\RR(\langle x_1, y_1, -x_2, -y_2\rangle, \langle -y_1, x_1, y_2, -x_2\rangle ).
\end{align*}
In this setting, the symplectic complement is described by the orthogonal complement, and so
\begin{align*}
(\ker_{(z_1, z_2)}(d\pi))^{\omega\bot}=&\Span_\CC(\langle \bar z_2, \bar z_1\rangle)\\
=&\Span_\RR(\langle x_2, -y_2, x_1, -y_1\rangle, \langle y_2, x_2, y_1, x_1\rangle ).
\end{align*}
One then checks that \(dH\) vanishes on this by computing \(dH(v)=0\) for \(v\in (\ker_{(z_1, z_2)}(d\pi))^{\omega\bot}\)
\begin{align*}
\langle x_1, y_1, -x_2,- y_2\rangle\cdot \langle x_2, -y_2, x_1, -y_1\rangle=&0\\
\langle x_1, y_1, -x_2,-y_2\rangle\cdot \langle y_2, x_2, y_1, x_1 \rangle=&0
\end{align*}
This means that the level sets of \(H\) are preserved under parallel transport.
We use to this to describe some Lagrangian submanifolds of \(\CC^2\).
If we take a level set of \(H\) and restrict to a fiber above the point \(re^{\jmath c}\), the level set \(H^{-1}(\lambda)\cap \pi^{-1}(re^{\jmath c})\) can be explicitly parameterized by \(S^1:=\theta\mapsto re^{\jmath c}\cdot(s e^{\jmath\theta}, s^{-1} e^{-\jmath\theta})\), where \(s\) is determined by \(r^2(s^2-s^{-2})=\lambda\).
Simply because every curve is a Lagrangian submanifold of a \(\CC^*\), the level set of \(H\) restricted to a fiber of \(\pi\) is a Lagrangian submanifold.
We can now apply Missing Label (prp:ParallelTransportOfLagrangianSubmanifold)! to obtain some new Lagrangian submanifolds of \(\CC^2\) from parallel transport of these level sets.
Let \(\gamma:[0, 1]\to \CC\setminus 0\) be a closed curve, and \(\lambda\in \RR\) some value.
Define the Lagrangian \(L_{\gamma, \lambda}\) to be the parallel transport of the \(\lambda\)-level set along the curve \(\gamma\).
This already gives several interesting examples of Lagrangian submanifolds inside of \(\CC^2\).
These Lagrangian submanifolds can also be characterized in the following way:
\[L_{\gamma, \lambda}:=\{(z_1, z_2)\;|\; H(z_1, z_2)=\lambda, \pi(z_1, z_2)\in \Im(\gamma)\}.\]
A good example of one of these Lagrangians is the product torus. Let \(\gamma_r=re^{\jmath\theta}\).
Let \(s\) be the real value so that \(r^2(s^2-s^{-2})=\lambda\). Then the Lagrangian \(L_{\gamma_r, \lambda}\) is explicitly parameterized by:
\[L_{\gamma_r, \lambda}=\{r(se^{\jmath\theta_1}, se^{\jmath\theta_2})\}\]
This agrees with the definition of the product torus from (product torus).