Real versus Complex Volumes on Real Algebraic Surfaces

Let X be a real algebraic surface. The comparison between the volume of D(R) and D(C) for ample divisors D brings us to define the concordance α(X), which is a number between 0 and 1. This number equals 1 when the Picard number ρ(XR) is 1, and for some surfaces with a “quite simple” nef cone, for example, Del Pezzo surfaces. For abelian surfaces, α(X) is 1/2 or 1, depending on the existence or not of positive entropy automorphisms on X. In the general case, the existence of such an automorphism gives an upper bound for α(X), namely the ratio of entropies htop(f|X(R))/htop(f|X(C)). Moreover, α(X) is equal to this ratio when the Picard number is 2. An interesting consequence of the inequality is the nondensity of Aut(XR) in Diff(X(R)) as soon as α(X)>0. Finally we show, thanks to this upper bound, that there exist K3 surfaces with arbitrary small concordance, considering a deformation of a singular surface of tridegree (2,2,2) in P¹×P¹×P¹.