Clustering of critical points in Lefschetz fibrations and the symplectic Szpiro inequality

Abstract

We prove upper bounds for the number of critical points in semistable symplectic Lefschetz fibrations. We also obtain a new lower bound for the number of nonseparting vanishing cycles in Lefschetz pencils, and reprove the known lower bounds for the commutator lengths of Dehn twists.

minor corrections; to appear in Trans. Amer. Math. Soc