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 d-th 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 constant which only depends on the dimension n of the manifold with the K\"ahlerian volume of its real locus RX and d^{n/2}. Actually, any closed affine real algebraic hypersurface appears with positive probability as part of such random real hypersurfaces in any ball of RX of radius O(d^{-1/2}).