  Bott proved a strong vanishing theorem for sheaf cohomology on projective
space, namely that $H^j(X,\Omega^i_X\otimes L)=0$ for every $j>0$, $i\geq 0$,
and $L$ ample. This holds for toric varieties, but not for most other
varieties.
  We classify the smooth Fano 3-folds that satisfy Bott vanishing. There are
many more than expected.
  Along the way, we conjecture that for every projective birational morphism
$\pi\colon X\to Y$ of smooth varieties, and every line bundle $A$ on $X$ that
is ample over $Y$, the higher direct image sheaf $R^j\pi_*(\Omega^i_X\otimes
A)$ is zero for every $j>0$ and $i\geq 0$.
