2009/08/31 by Indranil Biswas, Biswas, Indranil, Jacques Hurtubise +1
Mathematics · #14F05 #14P99 #Algebraic Geometry (math.AG) #FOS: Mathematics #math.AG #msc:14F05 #msc:14P99
paper · pdf · doi:10.48550/arxiv.0909.0041
Final version; to appear in the Transactions of the AMS
arxiv created 2010/03/10 · arxiv updated 2010/03/11
Let XR be a geometrically irreducible smooth projective curve, defined over R, such that XR does not have any real points. Let X= XR×R C be the complex curve. We show that there is a universal real algebraic line bundle over XR x Picd(XR) if and only if χ(L) is odd for L in Picd(XR). There is a universal quaternionic algebraic line bundle over X x Picd(X) if and only if the degree d is odd. Take integers r and d such that r > 1, and d is coprime to r. Let MXR(r,d) (respectively, MX(r,d)) be the moduli space of stable vector bundles over XR (respectively, X) of rank r and degree d. We prove that there is a universal real algebraic vector bundle over XR x MXR(r,d) if and only if χ(E) is odd for E in MXR(r,d). There is a universal quaternionic vector bundle over X x MX(r,d) if and only if the degree d is odd. The cases where XR is geometrically reducible or XR has real points are also investigated.