= Vanishing for divisor twists on a complex torus
{title2=$H^i(M,L\otimes\mathcal O(aV))=0\quad(i>0,\ a\geq0)$}
Let $L$ be positive and $V$ a closed smooth hypersurface in a <complex torus>. The <semipositive canonical bundle of a submanifold of a complex torus> makes $P_a=(L\otimes\mathcal O(aV))|_V$ positive. Since $K_V=\mathcal O(V)|_V$, the restriction of the $a$-th ambient twist is $K_V\otimes P_{a-1}$ when $a\geq1$. Apply the <Kodaira vanishing theorem> on $V$, and use $0\to L((a-1)V)\to L(aV)\to L(aV)|_V\to0$ and its long exact sequence to induct from the positive bundle $L$ on the torus. No unproved ambient positivity of $\mathcal O(V)$ is needed.
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