Fujita Conjecture

Fujita conjectured in 1985 that adjoint bundles become positive at a uniform, dimension-linear rate: for a smooth projective complex $n$-fold $X$ with ample line bundle $L$, the bundle $K_X\otimes L^{\otimes(n+1)}$ should be globally generated and $K_X\otimes L^{\otimes(n+2)}$ very ample [Fujita1987Polarized]. The proposed thresholds are exactly those attained on projective space with $L=\mathcal O(1)$, where they cannot be improved, so the conjecture asserts that $\mathbb P^n$ is the extremal case.

The conjecture is known in low dimensions and for special classes of $X$ and $L$. In general, effective results give weaker, dimension-dependent bounds: Angehrn and Siu proved global generation of $K_X\otimes L^{\otimes m}$ once $m$ grows roughly quadratically in $n$ [AngehrnSiu1995], and Lazarsfeld's book surveys the effective freeness and very-ampleness theorems in this area [Lazarsfeld2004Positivity].

Both halves of the conjecture remain open for arbitrary $X$ and $L$ in higher dimensions; closing the gap between the known dimension-dependent bounds and the conjectured linear ones is the central difficulty.