Asymptotiques de nombres de Betti d'hypersurfaces projectives r\'eelles

Abstract

We are interested in the maximal values of the Betti numbers b_i({\mathbb R}X_m^n) for fixed i,m,n; where {\mathbb R}X_m^n is the real part of a real nonsingular hypersurface of degree m in the complex projective space {\mathbb C}P^n, and in the maximal values of the Betti numbers b_i({\mathbb R}Y_{2k}^n) for fixed i,k,n; where {\mathbb R}Y_{2k}^n is the real part of a double covering Y_{2k}^n of{\mathbb C}P^n ramified over some real nonsingular hypersurface of degree 2k. We show the existence of limits lim_{m -> +\infty} [Max b_i({\mathbb R}X_m^n)]/m^n=h_{i,n} and lim_{k -> +\infty} [Max b_i({\mathbb R}Y_{2k}^n)]/k^n =d_{i,n}. We construct real nonsingular hypersurfaces as small perturbations of double hypersurfaces using the Viro method. This construction enables us to obtain recursive lower bounds for the h_{0,n} and d_{0,n}, and inequalities h_{0,3} >= d_{0,2}/6+1/12, h_{1,3} >= d_{1,2}/6+1/6. As applications, we show the existence, for any integer n >= 5, of real algebraic hypersurfaces in {\mathbb C}P^n which are not T-hypersurfaces, and we prove inequalities 35/96 <= h_{0,3} <= 5/12, 35/48 <= h_{1,3} <= 5/6.