\(
\def\CC{{\mathbb C}}
\def\RR{{\mathbb R}}
\def\NN{{\mathbb N}}
\def\ZZ{{\mathbb Z}}
\def\TT{{\mathbb T}}
\def\CF{{\operatorname{CF}^\bullet}}
\def\HF{{\operatorname{HF}^\bullet}}
\def\SH{{\operatorname{SH}^\bullet}}
\def\ot{{\leftarrow}}
\def\st{\;:\;}
\def\Fuk{{\operatorname{Fuk}}}
\def\emprod{m}
\def\cone{\operatorname{Cone}}
\def\Flux{\operatorname{Flux}}
\def\li{i}
\def\ev{\operatorname{ev}}
\def\id{\operatorname{id}}
\def\grad{\operatorname{grad}}
\def\ind{\operatorname{ind}}
\def\weight{\operatorname{wt}}
\def\Sym{\operatorname{Sym}}
\def\HeF{\widehat{CHF}^\bullet}
\def\HHeF{\widehat{HHF}^\bullet}
\def\Spinc{\operatorname{Spin}^c}
\def\min{\operatorname{min}}
\def\div{\operatorname{div}}
\def\SH{{\operatorname{SH}^\bullet}}
\def\CF{{\operatorname{CF}^\bullet}}
\def\Tw{{\operatorname{Tw}}}
\def\Log{{\operatorname{Log}}}
\def\TropB{{\operatorname{TropB}}}
\def\wt{{\operatorname{wt}}}
\def\Span{{\operatorname{span}}}
\def\Crit{\operatorname{Crit}}
\def\CritVal{\operatorname{CritVal}}
\def\FS{\operatorname{FS}}
\def\Sing{\operatorname{Sing}}
\def\Coh{\operatorname{Coh}}
\def\Vect{\operatorname{Vect}}
\def\into{\hookrightarrow}
\def\tensor{\otimes}
\def\CP{\mathbb{CP}}
\def\eps{\varepsilon}
\)
SympSnip:
We first show that \(\ker(\pi_*)\) is a symplectic subspace.
Let \(0\neq v\in \ker(\pi_*)\) be any tangent vector.
Since \(\pi_*\) is \(J\)-holomorphic, \(Jv\in \ker(\pi_*)\).
We conclude that \(\omega(v, Jv)=g(v, v)\neq 0\), and so \(\omega\) is non-degenerate on the subspace \(\ker(\pi_*)=T_zX\).
It remains to show that the form is closed.
Let \(i:X_z\into X\) be and inclusion of a fiber of the Lefschetz fibration.
Then \(di^*\omega= i^*d\omega =0\), so the \(\omega|_{X_z}\) is a symplectic form on the fiber.
We can construct a connection by picking a horizontal complement to the kernel of the projection.
As \(\ker(\pi_*)\) is a symplectic subspace, \((\ker(\pi_*))^{\omega\bot}\), its symplectic complement, is a complementary subspace in the sense that \(\ker(\pi_*)\oplus (\ker(\pi_*))^{\omega\bot}=TX\).
This is a choice of horizontal complement, defining a connection on this fiber bundle.
In fact, the splitting of the tangent bundle locally splits the symplectic form.
proposition 0.0.1
In the local splitting \(T_xX=\ker(\pi_*)\oplus(\ker(\pi_*))^{\omega^\bot}\), the symplectic form \(\omega_X\) can be written as
\[\omega_X=\omega|_{X_z}\oplus f\omega_\CC.\]
for some smooth function \(f:X\to \RR\).
Pick \(p\in X\) a point, and tangent vectors \((v_1, w_1), (v_2, w_2)\in T_pX=\ker(\pi_*)\oplus(\ker(\pi_*))^{\omega^\bot}.\)
Because this these two spaces are symplectic orthogonal, \(\omega_X(v_1, w_2)=0\) and \(\omega_X(v_2, w_1)=0\).
Since \(v_1, v_2\) are tangent vectors to \(X_p\), and symplectic forms on \(\CC\) are all scalar multiples, there exists a function \(f: X\to \RR\) yielding the decomposition
\begin{align*}
\omega_X((v_1, w_1), (v_2, w_2))=&\omega_X((v_1, 0), (v_2, 0))+ \omega_X((0, w_1), (0, w_2)).
=&\omega_{X_p}(v_1, v_2)+f(p) \pi^*\omega_\CC(w_1, w_2)
\end{align*}