supporting facts to fujita conjecture

Question

I came across the Fujita conjecture which is perhaps very widely known. I want to know what are the supporting facts to the truth of the conjecture. http://en.wikipedia.org/wiki/Fujita_conjecture

Answer 1

There are a number of things known.

(1). As Libli answered, if $L$ is globally generated and ample (for example, very ample) then

• $K_X \otimes L^{\dim X + 1}$ is globally generated
• $K_X \otimes L^{\dim X+ 2}$ is very ample.

These follow from Castelnuovo-Mumford regularity and Kodaira vanishing (see Positivity I by Rob Lazarsfeld, the section on Castelnuovo-Mumford regularity). This actually holds in characteristic $p > 0$ as well, although you can't use Kodaira vanishing of course (see the work of Karen Smith).

(2). Of course it is easy to check it for curves (see Hartshorne Chapter IV).

From here on out we assume characteristic zero.

(3). Reider's theorem proves it for surfaces.

(4). Ein-Lazarsfeld did the base-point-freeness part for 3-folds. Global generation of pluricanonical and adjoint linear series on smooth projective threefolds.

(5). Kawamata did the base-point-freeness part for 4-folds. On Fujita's freeness conjecture for 3-folds and 4-folds.

Now, we also have evidence for weaker versions of Fujita's conjecture. For example, we could hope that $K_X \otimes L^{\phi(\dim X)}$ is globally generated when $\phi$ is some function we can control (likewise for very ampleness).

(6). The theorem of Angehrn and Siu does this Effective freeness and point separation for adjoint bundles. with a quadratic $\phi$. (Also see the paper of Demailly with $\phi(n) = 12n^n$, the title is A numerical criterion for very ample line bundles).

Anyway, a good reference for a lot of this are the books of Lazarsfeld (Positivity in Algebraic Geometry I and II).

There's a lot of other works too, so I'm sure I left some important things out.

Answer 2

The basepoint freeness part has been proved in dimension $5$ by Fei Ye and Zhixian Zhu, On Fujita's freeness conjecture in dimension $5$, Adv. Math., 2020 DOI:10.1016/j.aim.2020.107210, arXiv:1511.09154 [math.AG].

Answer 3

I think there is a result of Fujita which says that $K_X \otimes L^{dimX +2}$ is very ample if $L$ is very ample (a bordering case is given by the projective space, where you see that $K_{\mathbb{P}^n} \otimes \mathcal{O}(n + 1)$ is only globally generated)