Clustering of critical points in Lefschetz fibrations and the symplectic Szpiro inequality. (English) Zbl 1033.53076
This paper proves a Szpiro type inequality for semistable symplectic Lefschetz fibrations of arbitrary base and fiber genus (a Lefschetz fibration is semistable if every fiber is a semistable complex curve). In the case of pencils (when the base curve has genus \(0\)), the result is that the number of critical points is bounded above by \(6h(D-1)\), where \(h\) is the genus of a smooth fiber and \(D\) is the number of singular fibers.
This result means that there are global obstructions to the clustering of critical points in a fiber. These obstructions give lower bounds for the commutator lengths of Dehn twists (more specifically, let \(t\) be a Dehn twist in a surface of genus \(h\) and write \(t^{k}\) as a product of \(g\) commutators in the mapping class group, then one has a lower bound for such \(g\)). The result generalizes [H. Endo and D. Kotschick, Invent. Math. 144, 169–175 (2001; Zbl 0987.57004)].
The authors also prove that the number of non-separating vanishing cycles in a Lefschetz pencil of genus \(h\) is not less than \((8h-3)/5\), generalising results of [T.-J. Li, Int. Math. Res. Notices 2000, 941–954 (2000; Zbl 0961.57022)] and [A. Stipsicz, Topology Appl. 117, 9–21 (2002; Zbl 1007.53059)].
This result means that there are global obstructions to the clustering of critical points in a fiber. These obstructions give lower bounds for the commutator lengths of Dehn twists (more specifically, let \(t\) be a Dehn twist in a surface of genus \(h\) and write \(t^{k}\) as a product of \(g\) commutators in the mapping class group, then one has a lower bound for such \(g\)). The result generalizes [H. Endo and D. Kotschick, Invent. Math. 144, 169–175 (2001; Zbl 0987.57004)].
The authors also prove that the number of non-separating vanishing cycles in a Lefschetz pencil of genus \(h\) is not less than \((8h-3)/5\), generalising results of [T.-J. Li, Int. Math. Res. Notices 2000, 941–954 (2000; Zbl 0961.57022)] and [A. Stipsicz, Topology Appl. 117, 9–21 (2002; Zbl 1007.53059)].
Reviewer: Vicente Muñoz (Madrid)
MSC:
| 53D35 | Global theory of symplectic and contact manifolds |
| 57R17 | Symplectic and contact topology in high or arbitrary dimension |
| 14H10 | Families, moduli of curves (algebraic) |
| 57R57 | Applications of global analysis to structures on manifolds |
References:
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