Effective nonvanishing, effective global generation.

*(English)*Zbl 0934.14002From the introduction: Let \(g:X\to S\) be a surjective morphism of proper varieties, where \(X\) is nonsingular and complete, \(M\) be a nef and \(g\)-big line bundle on \(X\), \(L\) be a nef and big line bundle on \(S\) and \(N=K_X+ M+mg^*L\) be a line bundle varying with the positive integer \(m\). J. Kollár [Math. Ann. 296, No. 4, 595-605 (1993; Zbl 0818.14002)]; theorem 3.2 proved, under the necessary assumption that the first direct image sheaf \(g_*N\neq 0\), that \(h^0(X,N)= h^0(S,g_*N)>0\) and the sections of \(g_*N\) generate this sheaf at a general point of \(S\) for every \(m>(1/2) (\dim S^2+\dim S)\) (this is what makes the result “effective”). The purpose of this note is to observe that more precise statements are possible if one considers the local Seshadri constants of \(L\) on \(S\). The main result is the effective nonvanishing theorem 2.2, a “multiple-points higher-jets” version of theorem 3.2 of J. Kollár’s paper cited above.

As a first application, some generalizations to the case of nef vector bundles of the results concerning line bundles by L. Ein, O. Küchle and R. Lazarsfeld in J. Differ. Geom. 42, No. 2, 193-219 (1995; Zbl 0866.14004) and by J. Kollár in Invent. Math. 113, No. 1, 177-215 (1993; Zbl 0819.14006) are given: effective construction of rational and birational maps, and nonvanishing on varieties with big enough algebraic fundamental group.

As another application, it is shown that the global generation results for line bundles of Anghern-Siu, Demailly, Tsuji and Siu generalize to vector bundles of the form \(K_X^{\otimes a}\otimes E\otimes\text{det} E\otimes L^{\otimes m}\), where \(a\) and \(m\) are appropriate positive integers, \(E\) is a nef vector bundle and \(L\) is an ample line bundle. Explicit upper bounds on \(m\) are given and they depend only on the dimension of the variety, and not also on the Chern classes of the variety and the bundles in question.

The paper by M. A. A. de Cataldo in J. Reine Angew. Math. 502, 53-122 (1998; Zbl 0902.32012) provides upper bounds as above for vector bundles \(E\) subject to curvature conditions. The methods employed there are analytic. A global generation result for nef vector bundles is proved in the final section by the use of algebraic Nadel ideals.

As a first application, some generalizations to the case of nef vector bundles of the results concerning line bundles by L. Ein, O. Küchle and R. Lazarsfeld in J. Differ. Geom. 42, No. 2, 193-219 (1995; Zbl 0866.14004) and by J. Kollár in Invent. Math. 113, No. 1, 177-215 (1993; Zbl 0819.14006) are given: effective construction of rational and birational maps, and nonvanishing on varieties with big enough algebraic fundamental group.

As another application, it is shown that the global generation results for line bundles of Anghern-Siu, Demailly, Tsuji and Siu generalize to vector bundles of the form \(K_X^{\otimes a}\otimes E\otimes\text{det} E\otimes L^{\otimes m}\), where \(a\) and \(m\) are appropriate positive integers, \(E\) is a nef vector bundle and \(L\) is an ample line bundle. Explicit upper bounds on \(m\) are given and they depend only on the dimension of the variety, and not also on the Chern classes of the variety and the bundles in question.

The paper by M. A. A. de Cataldo in J. Reine Angew. Math. 502, 53-122 (1998; Zbl 0902.32012) provides upper bounds as above for vector bundles \(E\) subject to curvature conditions. The methods employed there are analytic. A global generation result for nef vector bundles is proved in the final section by the use of algebraic Nadel ideals.

##### MSC:

14C20 | Divisors, linear systems, invertible sheaves |

14J60 | Vector bundles on surfaces and higher-dimensional varieties, and their moduli |

14Q20 | Effectivity, complexity and computational aspects of algebraic geometry |

14E05 | Rational and birational maps |

14F17 | Vanishing theorems in algebraic geometry |

##### Keywords:

algebraic fundamental group; local Seshadri constant; maps to Grassmannians; nef vector bundles; effective global generation; jets; effective nonvanishing##### References:

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