A numerical criterion for very ample line bundles

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 K + L is very ample. The sufficient conditions are expressed in terms of lower bounds for the intersection numbers if 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 dimX > 3 and codimF > 2, the lower bounds for L p Y involve a numerical constant which depends on the geometry of X . By means of an iteration process, it is finally shown that 2K X + mL is very ample for m > I2n n . Our approach is mostly analytic and based on a combination of Hrmander's L 2 estimates for the operator ) , Lelong number theory and the Aubin-Calabi-Yau theorem.