Iterated fibre sums of algebraic Lefschetz fibrations

TL;DR

This paper studies the construction of algebraic Lefschetz fibrations through fibre sums, computes Seiberg-Witten invariants for these sums, and derives obstructions to certain boundary diffeomorphisms, extending known results from elliptic surfaces.

Contribution

It generalizes fibre sum constructions to algebraic Lefschetz fibrations and calculates their Seiberg-Witten invariants, providing new obstructions for boundary diffeomorphisms.

Findings

Partial calculation of Seiberg-Witten invariants for fibre sums

Obstructions for extending boundary diffeomorphisms

Extension of known elliptic surface results to more general fibrations

Abstract

Let M denote the total space of a Lefschetz fibration, obtained by blowing up a Lefschetz pencil on an algebraic surface. We consider the n-fold fibre sum M(n), generalizing the construction of the elliptic surfaces E(n). For a Lefschetz pencil on a simply-connected minimal surface of general type we partially calculate the Seiberg-Witten invariants of the fibre sum M(n) using a formula of Morgan-Szabo-Taubes. As an application we derive an obstruction for self-diffeomorphisms of the boundary of the tubular neighbourhood of a general fibre in M(n) to extend over the complement of the neighbourhood. Similar obstructions are known in the case of elliptic surfaces.