Semipositive line bundles.

*(English)*Zbl 0561.32012From the author’s introduction: In this paper we want to study semipositive line bundles. As we explained in Proc. Jap. Acad., Ser. A 56, 393-396 (1980; Zbl 0477.14010), there are several different ways to define the notion of semipositivity. The strongest one is the semiampleness, i.e, a line bundle L is semiample if \(Bs| mL| =\emptyset\) for some positive integer m. The weakest one is the numerical semipositivity, i.e. a line bundle L on space S is numerically semipositive id \(LC>0\) for any curve C in S. Semiample line bundles have many nice properties. For example, the graded algebra \(G(S,L)=\otimes_{t>0}H^ 0(S,tL)\) is finitely generated. So, it would be important to find a good sufficient condition for a line bundle to be semiample. Our theorem improves upon Zariski’s famous result. Although our criterion is still too strong, it seems difficult to obtain a better one.

On complex analytic manifolds there is another notion (called geometrical semipositivity) based on the real differential (1,1)-form representing the Chern class \(c_ 1(L)\). The significance of this notion lies in a vanishing theorem, which is slightly stronger than Kodaira’s original one. In the paper cited above we introduced a couple of other notions, but now the cohomological semipositivity turned out to be equivalent to the numerical semipositivity. This is a consequence of a strengthened version of Serre’s vanishing theorem. As applications we obtain the results, which were known in the cited paper only in case of characteristic zero. - In the final section we give a generalization of Ramanujam’s vanishing theorem in positive characteristic cases.

On complex analytic manifolds there is another notion (called geometrical semipositivity) based on the real differential (1,1)-form representing the Chern class \(c_ 1(L)\). The significance of this notion lies in a vanishing theorem, which is slightly stronger than Kodaira’s original one. In the paper cited above we introduced a couple of other notions, but now the cohomological semipositivity turned out to be equivalent to the numerical semipositivity. This is a consequence of a strengthened version of Serre’s vanishing theorem. As applications we obtain the results, which were known in the cited paper only in case of characteristic zero. - In the final section we give a generalization of Ramanujam’s vanishing theorem in positive characteristic cases.

Reviewer: H.Röhrl

##### MSC:

32L05 | Holomorphic bundles and generalizations |

32L20 | Vanishing theorems |

14F05 | Sheaves, derived categories of sheaves, etc. (MSC2010) |