A numerical criterion for very ample line bundles

Abstract

Let X be a projective algebraic manifold of dimension n and let L be an ample line bundle over X. We give a numerical criterion ensuring that the adjoint bundle Kx + L is very ample. The sufficient conditions are expressed in terms of lower bounds for the intersection numbers Lp • Y over subvarieties Y of X. In the case of surfaces, our criterion gives universal bounds and is only slightly weaker than I. Reider’s criterion. When dim X ≥ 3 and codim Y ≥ 2, the lower bounds for Lp • Y involve a numerical constant which depends on the geometry of X. By means of an iteration process, it is finally shown that 2Kx + mL is very ample for m ≥ 12nn. Our approach is mostly analytic and based on a combination of Hörmander’s L2 estimates for the operator ∂, Lelong number theory and the Aubin-Calabi-Yau theorem. © 1993, International Press of Boston, Inc. All Rights Reserved.