The generic complex hypersurfaces of degree d in complex projective space are all topologically identical. However, their geometry can differ significantly. I will explain that there exists a sequence of hypersurfaces (H_d)_d such that the relative volume of H_d where the Ricci curvature (of the restriction to H_d of the ambient metric) is negative, converges to 1. This is to be compared with a recent theorem of J.-P. Mohsen which asserts the existence of a sequence of complex projective submanifolds of codimension 2 with negative Ricci curvature. Our proof is probabilistic.
This is a joint work with Michele Ancona (LJAD, Nice).