Remarks on projective normality for certain Calabi-Yau and hyperk\\"ahler\n varieties

We prove some results on effective very ampleness and projective normality\nfor some varieties with trivial canonical bundle. In the first part we prove an\neffective projective normality result for an ample line bundle on regular\nsmooth four-folds with trivial canonical bundle. More precisely we show that\nfor a regular smooth fourfold with trivial canonical bundle, $A^{\\otimes 15}$\nis projectively normal for $A$ ample. In the second part we emphasize on the\nprojective normality of multiples of ample and globally generated line bundles\non certain classes of known examples (upto deformation) of projective\nhyperk\\"ahler varieties. As a corollary we show that excepting two extremal\ncases in dimensions $4$ and $6$, a general curve section of any ample and\nglobally generated linear system on the above mentioned examples is\nnon-hyperelliptic.\n