2022/08/18 by Joshua Celeste, Celeste, Joshua
Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Differential Geometry (math.DG) #FOS: Mathematics #Geometric and Algebraic Topology #Geometry and complex manifolds
paper · pdf · doi:10.48550/arxiv.2208.08700
openalex publication_date 2022/08/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Kähler structure. We obtain a general computation of the invariants when b1=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 Kähler families where some more further explicit computations can be made. We consider a family of \mathbbCP2's obtained from the projectivisation of a rank 3 complex vector bundle, a family of \mathbbCP1×\mathbbCP1'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 Kähler surface at a point known as the universal blowup family.