Birational rigidity of complete intersections

We prove that every smooth complete intersection X = X d 1 , … , d s ⊂ P ∑ i = 1 s d i defined by s hypersurfaces of degree d 1 , … , d s is birationally superrigid if 5 s + 1 ≤ 2 ( ∑ i = 1 s d i + 1 ) ∏ i = 1 s d i . 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.