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A converse to the Andreotti-Grauert theorem. (English. French summary) Zbl 1228.32020
Summary: The goal of this paper is to show that there are strong relations between certain Monge-Ampère integrals appearing in holomorphic Morse inequalities, and asymptotic cohomology estimates for tensor powers of holomorphic line bundles. Especially, we prove that these relations hold without restriction for projective surfaces, and in the special case of the volume, i.e. of asymptotic \(0\)-cohomology, for all projective manifolds. These results can be seen as a partial converse to the Andreotti-Grauert vanishing theorem.

32J25 Transcendental methods of algebraic geometry (complex-analytic aspects)
32L10 Sheaves and cohomology of sections of holomorphic vector bundles, general results
14C20 Divisors, linear systems, invertible sheaves
14F99 (Co)homology theory in algebraic geometry
Full Text: DOI EuDML
[1] Andreotti (A.), Grauert (H.).— Théorèmes de finitude pour la cohomologie des espaces complexes, Bull. Soc. Math. France 90, p. 193-259 (1962). · Zbl 0106.05501
[2] Angelini (F.).— An algebraic version of Demailly’s asymptotic Morse inequalities; arXiv: alg-geom/9503005, Proc. Amer. Math. Soc. 124 p. 3265-3269 (1996). · Zbl 0860.14019
[3] Boucksom (S.).— On the volume of a line bundle, Internat. J. Math. 13, p. 1043-1063 (2002). · Zbl 1101.14008
[4] Boucksom (S.), Demailly (J.-P.), Păun (M.), Peternell (Th.).— The pseudo-effective cone of a compact Kähler manifold and varieties of negative Kodaira dimension, arXiv: math.AG/0405285, see also Proceedings of the ICM 2006 in Madrid.
[5] Demailly (J.-P.).— Estimations \(L^2\) pour l’opérateur \(\overline{∂ }\) d’un fibré vectoriel holomorphe semi-positif au dessus d’une variété kählérienne complète, Ann. Sci. École Norm. Sup. 15, p. 457-511 (1982). · Zbl 0507.32021
[6] Demailly (J.-P.).— Champs magnétiques et inégalités de Morse pour la \(d^{′′}\)-cohomologie, Ann. Inst. Fourier (Grenoble) 35, p. 189-229 (1985). · Zbl 0565.58017
[7] Demailly (J.-P.).— Holomorphic Morse inequalities, Lectures given at the AMS Summer Institute on Complex Analysis held in Santa Cruz, July 1989, Proceedings of Symposia in Pure Mathematics, Vol. 52, Part 2, p. 93-114 (1991). · Zbl 0755.32008
[8] Demailly (J.-P.).— Regularization of closed positive currents and Intersection Theory, J. Alg. Geom. 1, p. 361-409 (1992). · Zbl 0777.32016
[9] Demailly (J.-P.).— Holomorphic Morse inequalities and asymptotic cohomology groups: a tribute to Bernhard Riemann, Milan Journal of Mathematics 78, p. 265-277 (2010). · Zbl 1205.32017
[10] Demailly (J.-P.), Ein (L.) and Lazarsfeld (R.).— A subadditivity property of multiplier ideals, Michigan Math. J. 48, p. 137-156 (2000). · Zbl 1077.14516
[11] Demailly (J.-P.), Păun (M.).— Numerical characterization of the Kähler cone of a compact Kähler manifold, arXiv: math.AG/\(0105176\,\); Annals of Math. 159, p. 1247-1274 (2004). · Zbl 1064.32019
[12] de Fernex (T.), Küronya (A.), Lazarsfeld (R.).— Higher cohomology of divisors on a projective variety, Math. Ann. 337, p. 443-455 (2007). · Zbl 1127.14012
[13] Fujita (T.).— Approximating Zariski decomposition of big line bundles, Kodai Math. J. 17, p. 1-3 (1994). · Zbl 0814.14006
[14] Hironaka (H.).— Resolution of singularities of an algebraic variety over a field of characteristic zero, Ann. of Math. 79, p. 109-326 (1964). · Zbl 0122.38603
[15] Küronya (A.).— Asymptotic cohomological functions on projective varieties, Amer. J. Math. 128, p. 1475-1519 (2006). · Zbl 1114.14005
[16] Lazarsfeld (R.).— Positivity in Algebraic Geometry I.-II, Ergebnisse der Mathematik und ihrer Grenzgebiete, Vols. 48-49, Springer Verlag, Berlin, 2004. · Zbl 1093.14500
[17] Totaro (B.).— Line bundles with partially vanishing cohomology, July 2010, arXiv: math.AG/1007.3955.
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