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Asymptotic cohomology vanishing and a converse to the Andreotti-Grauert theorem on surfaces. (Annulation de la cohomologie asymptotique et réciproque du théorème de Andreotti-Grauert sur les surfaces.) (English. French summary) Zbl 1298.14012
The question investigated in this article is the relation between partial positivity for line bundles and vanishing of cohomology groups. A line bundle \(L\) is said to be \(q\)-positive if it carries a hermitian metric such that its curvature form has at least \((n-q)\) positive eigenvalues at any point of the ambient space (\(n\) being the dimension of the latter). The Andreotti-Grauert Theorem [A. Andreotti and H. Grauert, Bull. Soc. Math. Fr. 90, 193–259 (1962; Zbl 0106.05501)] states that such a \(q\)-positive line bundle (on a compact manifold) is also cohomologically \(q\)-ample, i.e. if \(\mathcal{F}\) is any coherent sheaf on \(X\), \(H^i(X,\mathcal{F}\otimes L^m)=0\) for any \(i>q\) and for any \(m\) large enough. This paper is concerned with the converse statement and several results are established in this direction. For instance, the converse is true on projective surfaces. In the semi-ample case, both of these properties are shown to be equivalent (to each other and) to the following condition: the Iitaka fibration of \(L\) has fibres of dimension at most \(q\) (in particular \(\kappa(L)\geq n-q\)). The third part of the article contains some interesting lemmas on local positivity in the setting of germs of analytic sets.

14C20 Divisors, linear systems, invertible sheaves
14F17 Vanishing theorems in algebraic geometry
32L15 Bundle convexity
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