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

Symplectic snippets

symplectic lefschetz fibration

definition 0.0.1 [Sei08]

Let \((X, \omega , J)\) be a symplectic manifold equipped with compatible almost complex structure. A symplectic Lefschetz fibration is map \(\pi: X\to \CC\) satisfying the following properties:
  • \(\pi\) is \(J\)-holomorphic, in the sense that \(J\pi_*=\pi_*\jmath\), where \(\jmath\) is the standard complex structure on \(\CC\);
  • The map \(\pi\) has finitely many critical points;
  • The set of critical values \(\{\pi(x)\;|\;z\in \Crit(\pi)\}\) are disjoint and;
  • In a neighborhood of each critical point, there exists holomorphic coordinates \((z_1, \ldots, z_n)\) for \(X\) so that \(\pi=\sum_{i=1}^n z_i^2\).
If the map \(\pi\) is not proper (i.e. the fibers are allowed to be non-compact,) then we impose the additional requirement:
  • There exists a compact set \(X_0\subset X\) so that \(\pi:X_0\to \CC\) is a proper fibration and;
  • The fibration \(\pi:X\setminus X_0\to \CC\) is a trivial symplectic fibration, with split complex and symplectic structure.

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References

[Sei08]Paul Seidel. Fukaya categories and Picard-Lefschetz theory, volume 10. European Mathematical Society, 2008.