Take the 2-minute tour ×  Given a Fano threefold $X$, its index
$ind(X)$ is the largest integer $r$ such that there exists a
divisor $H$ such that $rH \cong -K_X$. Let $\mathcal{L}$ be the
associated (ample) line bundle and define the twisted forms
$\Omega^q(k) = \Omega^q \otimes \mathcal{L}^k$. When do the
twisted cohomology groups $H^p(X, \Omega^q(k))$ vanish? The
Kodaira-Nakano vanishing theorem states that the cohomology
groups vanish for $p+q > 3.$ For simple examples such as
$\mathbb{CP}^3$ and the quadric hypersurface, many more of the
twisted cohomology groups vanish. What results are known? Can
anything stronger be said if $X$ has a Kahler-Einstein metric?
Added: There are few scattered results in the literature. For
example:  The only two threefolds with non-vanishing $H^0(X,
\Omega^1_X (1))$ are the Mukai–Umemura threefold $V_{22}$ and
$V_{18}$ [math.AG/0310390].  share|improve this question add
comment  1 Answer  up vote 4 down vote accepted  The Hodge
groups $H^1(X,\Omega^2_X)$ and $H^2(X,\Omega