Birational rigidity of complete intersections

We prove that every smooth complete intersection X defined by s hypersurfaces of degree d_1, … , d_s in a projective space of dimension d_1 + … + d_s is birationally superrigid if 5s +1 is at most 2(d_1 + … + d_s + 1)/sqrt{d_1…d_s}. In particular, X is non-rational and Bir(X)=Aut(X). We also prove birational superrigidity of singular complete intersections with similar numerical condition. These extend the results proved by Tommaso de Fernex.