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 \(\pi=\sum_{i=1}^n z_i^2\).
- 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.
References
[Sei08] | Paul Seidel. Fukaya categories and Picard-Lefschetz theory, volume 10. European Mathematical Society, 2008. |