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

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