  The Hirokado variety is a Calabi-Yau threefold in characteristic 3 that is
not liftable either to characteristic~0 or the ring $W_2$ of the second Witt
vectors. Although Deligne-Illusie-Raynaud type Kodaira vanishing cannot be
applied, we show that $H^1(X, L^{-1})=0$, for an ample line bundle such that
$L^3$ has a non-trivial global section, holds for this variety.
