Divisors on real algebraic varieties without real points

Abstract

The image of the Picard group of a nonsingular projective variety X over R into the Galois-invariant part of the Picard group of X⊗C is described in terms of cohomological invariants of the analytic manifold X(C) with its antiholomorphic involution. This description gives for example a link between the level of the function field of X and the topological level of (Zariski-open subsets of) X(C); it easily follows that a nontrivial principal homogeneous space over a real abelian variety has a function field of level 2.