  We construct and study relations between Chern classes and Galois cohomology
classes in the Gal(C/R)-equivariant cohomology of real algebraic varieties with
no real points. We give applications to the topology of their sets of complex
points, and to sums of squares problems. In particular, we show that -1 is a
sum of 2 squares in the function field of any smooth projective real algebraic
surface with no real points and with vanishing geometric genus, as well as
higher-dimensional generalizations of this result.
