The families Seiberg-Witten invariants of smooth families of K\"ahler surfaces

TL;DR

This paper extends Seiberg-Witten invariants to smooth families of K"ahler surfaces, providing explicit formulas for cases with trivial first Betti number and applying them to various complex surface families.

Contribution

It introduces a generalization of Seiberg-Witten invariants for families of 4-manifolds with K"ahler structures and computes these invariants explicitly in specific cases.

Findings

Computed invariants for families with $b_1=0$ using characteristic classes.

Derived explicit formulas for certain K"ahler families.

Applied formulas to examples including projective bundles and blowups.

Abstract

We consider a generalisation of the Seiberg-Witten invariant to the families Seiberg-Witten invariants of a smooth family of 4-manifolds with fibres diffeomorphic to a 4-manifold $X$. Of particular interest is the special case when the family has a smoothly varying K\"ahler structure. We obtain a general computation of the invariants when $b_1=0$ in terms of characteristic classes of some vector bundles corresponding to the cohomology groups of holomorphic line bundles of the family. Finally, we apply the formula to some examples of K\"ahler families where some more further explicit computations can be made. We consider a family of $\mathbb{CP}^2$'s obtained from the projectivisation of a rank 3 complex vector bundle, a family of $\mathbb{CP}^1\times\mathbb{CP}^1$'s obtained as the fibre product of the projectivisation of two rank 2 complex vector bundles and a family with fibres being the blowup of a K\"ahler surface at a point known as the universal blowup family.