A numerical criterion for very ample line bundles

Niveau: Supérieur, Licence, Bac+2
A numerical criterion for very ample line bundles Jean-Pierre Demailly Universite de Grenoble I, Institut Fourier, BP 74, Laboratoire associe au C.N.R.S. n˚ 188, F-38402 Saint-Martin d'Heres 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 dimX ≥ 3 and codimY ≥ 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 form ≥ 12nn. Our approach is mostly analytic and based on a combination of Hormander's L2 estimates for the operator ∂, Lelong number theory and the Aubin-Calabi-Yau theorem. Table of contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

• over subvarieties


• ample property


• l2 ·


• closed positive


• aubin-calabi-yau theorem


• positive currents


• bundle over


• nef line


• any vector bundle


• very ample