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 fourfold 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 (upto 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. Definition 0.2. A compact Kähler manifold M of dimension n ≥ 3 is called Calabi-Yau if it has trivial canonical bundle and the hodge numbers h p,0 (M) vanish for all 0 < p < n.With this definition, Calabi-Yau manifolds are neccessarily projective. However the following definition of hyperkähler manifolds does not imply projectivity in general. Definition 0.3. A compact Kähler manifold M is called hyperkähler if it is simply connected and its space of global holomorphic two forms is spanned by a symplectic form.The decomposition theorem of Bogomolov (see [2]) says, any complex manifold with trivial first Chern class admits a finiteétale cover isomorphic to a product of complex tori, Calabi-Yau manifolds and hyperkähler manifolds. Thus, these spaces can be thought as the "building blocks" for manifolds with trivial first Chern class.The theorem of Saint-Donat that we stated (see Theorem 0.1) deals with ample line bundles.