Universal Vector Bundle over the Reals
نویسنده
چکیده
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 × Pic d(XR) if and only if χ(L) is odd for L ∈ Picd(XR). There is a universal quaternionic algebraic line bundle over X × Pic(X) if and only if the degree d is odd. Take integers r and d such that r ≥ 2, 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 ×MXR(r, d) if and only if χ(E) is odd for E ∈ MXR(r, d). There is a universal quaternionic vector bundle over 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.
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