1999/06/24 by Andrei Teleman, Teleman, Andrei
Mathematics · #53C07 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Differential Geometry (math.DG) #FOS: Mathematics #Finite Group Theory Research #Geometric and Algebraic Topology #math.AG #math.DG #msc:53C07
paper · pdf · doi:10.48550/arxiv.math/9906163
65 pages, LaTeX, to appear in Asian J. Math
arxiv created 1999/06/24 · openalex publication_date 1999/06/24 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We prove generic regularity and Uhlenbeck-type compactification theorems for the moduli spaces of PU(2)-monopoles. Generic regularity is NOT obtained in the usual way (by applying Sard theorem to a smooth parameterized moduli space), since the parameterized moduli space can be a priori singular. We explain why, using the standard order 0-perturbations, one cannot get the smoothness of the parameterized moduli space. (The same difficulty arises in the case of ASD-Spinc moduli spaces, so the definition of the Spinc-polynomial invariants should be revised.) We show that the singular locus of the parameterized moduli space is contained in a closed subspace which is (in a certain sense) of infinite codimension, and we apply Sard theorem to the complement of this subspace. Our approach to prove Uhlenbeck compactness follows closely the strategy developed in the instanton case by Donaldson and Kronheimer. Similar results, with different methods, were obtained by Feehan - Leness and Feehan.