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

Abstract

We prove some results on effective very ampleness and projective normality for some varieties with trivial canonical bundle. In the first part we prove an effective projective normality result for an ample line bundle on regular smooth four-folds with trivial canonical bundle. More precisely we show that for a regular smooth four-fold with trivial canonical bundle, A⊗15 is projectively normal for A ample. In the second part we emphasize on the projective normality of multiples of ample and globally generated line bundles on certain classes of known examples (up to deformation) of projective hyperkähler varieties. As a corollary we show that excepting two extremal cases in dimensions 4 and 6, a general curve section of any ample and globally generated linear system on the above mentioned examples is non-hyperelliptic.