Lagrangian matching invariants for fibred four-manifolds: I

Abstract

In a pair of papers, we construct invariants for smooth four-manifolds equipped with 'broken fibrations' - the singular Lefschetz fibrations of Auroux, Donaldson and Katzarkov - generalising the Donaldson-Smith invariants for Lefschetz fibrations. The 'Lagrangian matching invariants' are designed to be comparable with the Seiberg-Witten invariants of the underlying four-manifold; formal properties and first computations support the conjecture that equality holds. They fit into a field theory which assigns Floer homology groups to three-manifolds fibred over S1. The invariants are derived from moduli spaces of pseudo-holomorphic sections of relative Hubert schemes of points on the fibres, subject to Lagrangian boundary conditions. Part I - the present paper - is devoted to the symplectic geometry of these Lagrangians.