Let $X$ be a smooth complex projective manifold of dimension $n$, and $L$ a big and nef line bundle (or ample, if you prefer). Is there a Fujita type conjecture for the adjoint line bundle
$
L^{\otimes n}\otimes K\_X
$
?


(Fujita conjecture is for
$
L^{\otimes n+
1}\otimes K\_X
$
and
$
L^{\otimes n+2}\otimes K\_X
$
)



Variants of Fujita conjecture
Recently I got a copy of Fujita's original paper [T. Fujita, On Polarized Manifolds Whose Adjoint Bundles Are Not Semipositive] from a friend, and it is surprising that his original conjecture is a little more general than the Fujita's freeness conjecture but less well-known to people. So I feel it might be interesting to post it under this question.


The conjecture is: Let $X$ be a projective smooth variety (or a projective variety with at worst Gorenstein rational singularities) of dimension $n$ and $L$ be an ample line bundle on $X$. Let $t$ be a positive integer such that $K\_X+tL$ is nef. Then $m(K\_X+tL)$ is base point free for all $m>n+1-t$.


Note that in the paper Fujita showed that $K\_X+(n+1)L$ is always nef and hence Fujita's freeness conjecture is a special case of this conjecture when $t=n+1$.


