The families Seiberg-Witten invariants of smooth families of Kahler surfaces

Abstract

The Seiberg-Witten invariant has been an indispensable tool in understanding the topology and smooth structure of 4-manifolds, especially Kahler surfaces, where the mutually interacting symplectic and complex structures often allows for an explicit computation of the invariant. We concern ourselves with a natural generalisation of this setup to smooth families of 4-manifolds with fibres diffeomorphic to a single 4-manifold X, where one may define a generalisation of the Seiberg-Witten invariant known as the families Seiberg-Witten invariants. After introducing the necessary background into Seiberg-Witten theory, we provide an exposition on its generalisation to families of 4-manifolds and proceed to obtain a general formula for the invariants for smooth Kahler families with b1(X) = 0. Following this is a further explicit computation for three classes of example families, these being simple constructions of families of with fibres diffeomorphic to CP2, CP1 × CP1 and finally a family with fibres being the blowup of a Kahler surface at a point. We then conclude by looking at the consequences of the computations made, in particular investigating constraints on the cohomology of holomorphic line bundles over smooth Kahler families required for a non-vanishing diffeomorphism invariant, with a particular focus on when the base space of the family is S2, and further apply these considerations to the example families discussed prior.