# Variants of Fujita conjecture

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 $L^{\otimes n+2}\otimes K_X$ )

• On n-dimensional projective space if $L$ is the hyperplane bundle then $L^n \otimes K_X =L^{-1}$. Oct 30 '17 at 15:16
• @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. Oct 30 '17 at 15:25
• @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 $k$. Oct 30 '17 at 15:39

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 $t=n+1$.