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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.

##### MSC:
 14C20 Divisors, linear systems, invertible sheaves 14F17 Vanishing theorems in algebraic geometry 32L15 Bundle convexity
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##### References:
 [1] Andreotti, A; Grauert, H., Théorème de finitude pour la cohomologie des espaces complexes, Bull. Soc. Math. France, 90, 193-259, (1962) · Zbl 0106.05501 [2] Barlet, D., Functions of several complex variables, IV (Sem. François Norguet, 1977-1979) (French), 807, Convexité au voisinage d’un cycle, (French), 102-121, (1980), Springer, Berlin · Zbl 0434.32012 [3] Boucksom, S., Divisorial Zariski decompositions on compact complex manifolds, Ann. Sci. École Norm. Sup. (4), 37, 1, 45-76, (2004) · Zbl 1054.32010 [4] Boucksom, S.; Demailly, J. P.; Paun, M.; T., Peternell, The pseudo-effective cone of a compact Kähler manifold and varieties of negative Kodaira dimension · Zbl 1267.32017 [5] Demailly, J. P., Champs magnétiques et inégalités de Morse pour la $$d^{′ ′ }$$-cohomologie, Ann. Inst. Fourier (Grenoble), 35, 4, 189-229, (1985) · Zbl 0565.58017 [6] Demailly, J. P., Cohomology of q-convex spaces in top degrees, Math. Z, 204, 2, 283-295, (1990) · Zbl 0682.32017 [7] Demailly, J. P., Holomorphic Morse inequalities and asymptotic cohomology groups: a tribute to bernhard Riemann, Milan J. Math., 78, 1, 265-277, (2010) · Zbl 1205.32017 [8] Demailly, J. P., A converse to the andreotti-grauert theorem, Ann. Fac. Sci. Toulouse Math. (6), 20, (2011) · Zbl 1228.32020 [9] Demailly, J. P.; Peternell, T.; Schneider, M., Holomorphic line bundles with partially vanishing cohomology, Proceedings of the Hirzebruch 65 Conference on Algebraic Geometry (Ramat Gan, 1993), 9, 165-198, (1996), Bar-Ilan Univ · Zbl 0859.14005 [10] Ein, L.; Lazarsfeld, R.; Musţǎ, M.; Nakamaye, M.; Popa, M., Asymptotic invariants of base loci, Ann. Inst. Fourier (Grenoble), 56, 6, 1701-1734, (2006) · Zbl 1127.14010 [11] Fujita, T., Semipositive line bundles, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 30, 2, 353-378, (1983) · Zbl 0561.32012 [12] Fuse, H.; Ohsawa, T., On a curvature property of effective divisors and its application to sheaf cohomology, Publ. Res. Inst. Math. Sci., 45, 4, 1033-1039, (2009) · Zbl 1190.32009 [13] Küronya, A., Positivity on subvarieties and vanishing of higher cohomology · Zbl 1291.14018 [14] Lazarsfeld, R., Positivity in Algebraic Geometry I, 48, (2004), Springer Verlag, Berlin · Zbl 1093.14500 [15] Matsumura, S., Restricted volumes and divisorial Zariski decompositions · Zbl 1277.14006 [16] Matsumura, S., On the curvature of holomorphic line bundles with partially vanishin cohomology, RIMS, Kôkyûroku, Potential theory and fiber spaces, 1783, 155-169, (2012) [17] Ohsawa, T., Completeness of noncompact analytic spaces, Publ. Res. Inst. Math. Sci., 20, 3, 683-692, (1984) · Zbl 0568.32008 [18] Richberg, R., Stetige streng pseudokonvexe funktionen, Math. Ann., 175, 257-286, (1968) · Zbl 0153.15401 [19] Siu, Y. T., Every Stein subvariety admits a Stein neighborhood, Invent. Math., 38, 1, 89-100, (197677) · Zbl 0343.32014 [20] Sommese, A. J., Submanifolds of abelian varieties, Math. Ann., 233, 3, 229-256, (1978) · Zbl 0381.14007 [21] Totaro, B., Line bundles with partially vanishing cohomology · Zbl 1277.14007 [22] Yau, S. T., On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation, I, Comm. Pure Appl. Math., 31, 3, 339-411, (1978) · Zbl 0369.53059 [23] Zariski, O., The theorem of Riemann-Roch for high multiples of an effective divisor on an algebraic surface, Ann. of Math. (2), 76, 560-615, (1962) · Zbl 0124.37001
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