Rational curves on hypersurfaces of a projective variety

In this paper, we extend our result in [3] to hypersurfaces of any smooth projective variety $Y$. Precisely we let $X_0$ be a generic hypersurface of $Y$ and $c_0:{\mathbf{P}}^1\to X_0$ be a generic birational morphism to its image, i.e. $c_0\in Ho{m}_{bir}({\mathbf{P}}^1,X_0)$ is generic, such that (1) $dim(X_0)\ge 3$, (2) $H^1(N_{c_0\mathrm{/}Y})=0$. Then \begin{equation} H^1(N_{c_0/X_0})=0. \end{equation} As an application we prove that the Clemens’ conjecture holds for Calabi-Yau complete intersections of dimension 3.