Subvarieties of general type on a general projective hypersurface

Abstract

We study subvarieties of a general projective degree d d hypersurface X d ⊂ P n X_d\subset \mathbf {P}^n . Our main theorem, which improves previous results of L. Ein and C. Voisin, implies in particular the following sharp corollary: any subvariety of a general hypersurface X d ⊂ P n X_{d}\subset {\mathbf P}^n , for n ≥ 6 n\geq 6 and d ≥ 2 n − 2 d\geq 2n-2 , is of general type.