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

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.

### 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

Zbl 0663.14010
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