Real versus Complex Volumes on Real Algebraic Surfaces

TL;DR

This paper introduces the concept of concordance to compare real and complex volumes on real algebraic surfaces, revealing relationships with automorphisms, entropy, and deformation properties.

Contribution

It defines the concordance for real algebraic surfaces and explores its connections with automorphisms, entropy, and deformation, providing new bounds and examples.

Findings

Concordance equals 1 for surfaces with Picard number 1.

Automorphisms with positive entropy bound the concordance from above.

Existence of K3 surfaces with arbitrarily small concordance.

Abstract

Let X be a real algebraic surface. The comparison between the volume of real and complex loci of ample divisors D brings us to define the concordance, which is a number between 0 and 1. This number equals 1 when the Picard number is 1, and for some surfaces with a "quite simple" nef cone, e.g. Del Pezzo surfaces. For abelian surfaces, it 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 the concordance, namely the ratio of entropies in real and complex loci of X. Moreover the concordance is equal to this ratio when the Picard number is 2. An interesting consequence of the inequality is the non-density of the automorphisms in the group of diffeomorphisms of the real surface X(R), as soon as the concordance is positive. 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 the product of three projective lines.