Lower estimates for the expected Betti numbers of random real hypersurfaces

Abstract

We estimate from below the expected Betti numbers of real hypersurfaces taken at random in a smooth real projective n-dimensional manifold. These random hypersurfaces are chosen in the linear system of a large dth power of a real ample line bundle equipped with a Hermitian metric of positive curvature. As for the upper bounds that we recently established, these lower bounds read as a product of a factor which only depends on the dimension n of the manifold with the Kähler volume of its real locus R X and d n . Actually, any closed affine real algebraic hypersurface appears with positive probability as part of such random real hypersurfaces in any ball of R X of radius O ( 1 / d ) .