Subvarieties of general type on a general projective hypersurface

Abstract

We study subvarieties of a general projective degree $d$ hypersurface $X_d\subset {\mathbf{P}}^n$. 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 $X_d\subset {\mathbf{P}}^n$, for $n\ge 6$ and $d\ge 2n-2$, is of general type.

Title changed, introduction completely rewritten, exposition improved. To appear in the Transactions of the A.M.S. 18 pages, Latex