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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 $ )

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    $\begingroup$ On n-dimensional projective space if $L$ is the hyperplane bundle then $L^n \otimes K_X =L^{-1}$. $\endgroup$
    – Pooter
    Oct 30 '17 at 15:16
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    $\begingroup$ @Pooter Maybe one could conjecture that $L^n \otimes K_X$ is globally generated unless $X$ is the projective space? Your question is equivalent to the characterization of extremal cases of Fujita conjecture. $\endgroup$
    – Chen Jiang
    Oct 30 '17 at 15:25
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    $\begingroup$ @ChenJiang: sure, I was just pointing out this is the best possible uniform statement. Another obvious comment: if $L$ is nef and big but not ample, then $L^k \otimes K_X$ may be negative on some curve for all $k$. $\endgroup$
    – Pooter
    Oct 30 '17 at 15:39
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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 $t=n+1$.

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