Surfaces elliptiques r\'eelles et in\'egalit\'e de Ragsdale-Viro

Abstract

On a real regular elliptic surface without multiple fiber, the Betti number $h_1$ and the Hodge number $h^{1,1}$ are related by $h_1\leq h^{1,1}$. We prove that it's always possible to deform such algebraic surface to obtain $h_1=h^{1,1}$. Furthermore, we can impose that each homology class can be represented by a real algebraic curve. We use a real version of the modular construction of elliptic surfaces.