  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. They fit into a field theory which assigns Floer homology groups
to fibred 3-manifolds. The invariants are derived from moduli spaces of
pseudo-holomorphic sections of relative Hilbert schemes of points on the
fibres, subject to Lagrangian boundary conditions. Part I is devoted to the
symplectic geometry of these Lagrangians.
