A numerical criterion for very ample line bundles. (English) Zbl 0783.32013

Demailly, Jean-Pierre

J. Differ. Geom. 37, No. 2, 323-374 (1993).

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 \(K_ X+L\) is very ample. The sufficient conditions are expressed in terms of lower bounds for the intersection numbers \(L^ p\cdot Y\) 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 [Ann. Math., II. Ser. 127, No. 2, 309-316 (1988; Zbl 0663.14010)]. When \(\dim X\geq 3\) and \(\text{codim} Y\geq 2\), the lower bounds for \(L^ p\cdot Y\) 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 \(m\geq 12n^ n\). Our approach is mostly analytic and based on a combination of Hörmander’s \(L^ 2\) estimates for the operator \(\overline\partial\), Lelong number theory and the Aubin-Calabi-Yau theorem.

Reviewer: J.-P.Demailly (Saint-Martin d’Hères)

MSC:

32J15	Compact complex surfaces
32L10	Sheaves and cohomology of sections of holomorphic vector bundles, general results
32C30	Integration on analytic sets and spaces, currents

Keywords:

very ample line bundle; plurisubharmonic function; closed positive current; Monge-Ampère equation; intersection theory; numerical criterion; Lelong number; Aubin-Calabi-Yau theorem

Citations:

Zbl 0663.14010