= Higher cohomology vanishing from an ample hyperplane restriction
{title2=$H^{i\ge2}(X,L^m)=0\quad(m\gg0)$}
Let $H$ be a <very ample> <effective Cartier divisor> on a <projective scheme> $X$, and suppose $L|_H$ is <ample>. Then $H^i(X,L^m)=0$ for $i\ge2$ and sufficiently large $m$. By <uniform Serre vanishing for two ample twists>, cohomology of $L^m(tH)|_H$ vanishes uniformly for $t\ge0$. The restriction sequence identifies the higher groups of $L^m(tH)$ and $L^m((t+1)H)$ for $i\ge2$. For fixed $m$, large $t$ kills them by <Serre vanishing> for $H$; descend to $t=0$. No $H^1$ vanishing is claimed.
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