\( \def\CC{{\mathbb C}} \def\RR{{\mathbb R}} \def\NN{{\mathbb N}} \def\ZZ{{\mathbb Z}} \def\TT{{\mathbb T}} \def\CF{{\operatorname{CF}^\bullet}} \def\HF{{\operatorname{HF}^\bullet}} \def\SH{{\operatorname{SH}^\bullet}} \def\ot{{\leftarrow}} \def\st{\;:\;} \def\Fuk{{\operatorname{Fuk}}} \def\emprod{m} \def\cone{\operatorname{Cone}} \def\Flux{\operatorname{Flux}} \def\li{i} \def\ev{\operatorname{ev}} \def\id{\operatorname{id}} \def\grad{\operatorname{grad}} \def\ind{\operatorname{ind}} \def\weight{\operatorname{wt}} \def\Sym{\operatorname{Sym}} \def\HeF{\widehat{CHF}^\bullet} \def\HHeF{\widehat{HHF}^\bullet} \def\Spinc{\operatorname{Spin}^c} \def\min{\operatorname{min}} \def\div{\operatorname{div}} \def\SH{{\operatorname{SH}^\bullet}} \def\CF{{\operatorname{CF}^\bullet}} \def\Tw{{\operatorname{Tw}}} \def\Log{{\operatorname{Log}}} \def\TropB{{\operatorname{TropB}}} \def\wt{{\operatorname{wt}}} \def\Span{{\operatorname{span}}} \def\Crit{\operatorname{Crit}} \def\CritVal{\operatorname{CritVal}} \def\FS{\operatorname{FS}} \def\Sing{\operatorname{Sing}} \def\Coh{\operatorname{Coh}} \def\Vect{\operatorname{Vect}} \def\into{\hookrightarrow} \def\tensor{\otimes} \def\CP{\mathbb{CP}} \def\eps{\varepsilon} \) 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*}