On the projective normality and normal presentation on higher dimensional varieties with nef canonical bundle

Abstract

In this article we prove new results on projective normality and normal presentation of adjunction bundle associated to an ample and globally generated line bundle on higher dimensional smooth projective varieties with nef canonical bundle. As one of the consequences of the main theorem, we give bounds on very ampleness and projective normality of pluricanonical linear systems on varieties of general type in dimensions three, four and five. These improve known such results.Ein and Lazarsfeld proved that for a very ample line bundle L on a smooth projective variety X, K X + (n + 1 + p)A satisfies the property N p . (see [10]).Another very interesting and related conjecture is the conjecture by Fujita. The precise statement is the following:Fujita's Conjecture: On a smooth projective variety of dimension n, K X + (n + 1)A is globally generated and K X + (n + 2)A is very ample where A is an arbitrary ample line bundle.Fujita's conjecture has been proved for surfaces by Reider (cf. [37]) using Bogomolov's instability theorem (see [2]) on rank two vector bundles. Fujita's freeness conjecture has been proved by Ein and Lazarsfeld (see [9]) for n = 3, by Kawamata (see [24]) for n = 3, 4 and by Fei Ye and Zhixian Zhu (see [39]) for n = 5.Since the first and the last terms are zero by Kodaira vanishing, hence H i (M L ⊗ (K + (l − i)B)) = 0 for all 2 ≤ i ≤ n.

Remarks on projective normality for certain Calabi–Yau and hyperkähler varieties

“…Then lB satisfies the property N 0 for all l ≥ n. Moreover, if X is Calabi-Yau, then lB satisfies the property N 1 for all l ≥ n.Proof. Follows immediately from Theorem 2.3 and Theorem 3.4 of[23].…”

“…Proof. Follows immediately from Theorem 2.3 and Theorem 3.4 of [23]. Now we want to find out what multiple of an ample line bundle is very ample on a four dimensional variety with trivial canonical bundle.…”

Syzygies of adjoint linear series on projective varieties

“…Relaxing the positivity assumption on A from very ample to ample and basepoint free poses significant challenges and our methods are very different from those of [EL93]. Results concerning N 0 and N 1 were obtained in [MR19] for varieties with K X nef with methods similar to [Pur05], and optimal bounds for property N p have been proved for abelian varieties [Par00] and Calabi-Yau varieties [GP98,Niu19].…”