Asymptotic cohomology vanishing and a converse to the Andreotti-Grauert theorem on surfaces

Matsumura, Shin-ichi

Annales de l'Institut Fourier, Tome 63 (2013) no. 6, pp. 2199-2221.

Dans cet article, nous étudions les relations entre la positivité de la courbure et le comportement asymptotique de la cohomologie de degré supérieur des puissances tensorielles d’un fibré en droites holomorphe. Le théorème d’annulation d’Andreotti-Grauert affirme que la positivité partielle de la courbure implique l’annulation asymptotique de la cohomologie de certains degrés supérieurs. Nous étudions la réciproque de ce théorème dans plusieurs situations. Par exemple, nous considérons le cas d’un fibré en droite semi-ample ou gros. De plus, nous montrons que la réciproque du théorème d’Andreotti-Grauert est vraie sur les surfaces projectives sans aucune hypothèse sur le fibré en droites.

In this paper, we study relations between positivity of the curvature and the asymptotic behavior of the higher cohomology group for tensor powers of a holomorphic line bundle. The Andreotti-Grauert vanishing theorem asserts that partial positivity of the curvature implies asymptotic vanishing of certain higher cohomology groups. We investigate the converse implication of this theorem under various situations. For example, we consider the case where a line bundle is semi-ample or big. Moreover, we show the converse implication holds on a projective surface without any assumptions on a line bundle.

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