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

Shin-ichi Matsumura[1]

  • [1] Department of Mathematics and Computer Science, Kagoshima University, 1-21-35 Koorimoto, Kagoshima 890-0065, Japan.

Annales de l’institut Fourier (2013)

  • Volume: 63, Issue: 6, page 2199-2221
  • ISSN: 0373-0956

Abstract

top
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.

How to cite

top

Matsumura, Shin-ichi. "Asymptotic cohomology vanishing and a converse to the Andreotti-Grauert theorem on surfaces." Annales de l’institut Fourier 63.6 (2013): 2199-2221. <http://eudml.org/doc/275657>.

@article{Matsumura2013,
abstract = {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.},
affiliation = {Department of Mathematics and Computer Science, Kagoshima University, 1-21-35 Koorimoto, Kagoshima 890-0065, Japan.},
author = {Matsumura, Shin-ichi},
journal = {Annales de l’institut Fourier},
keywords = {Asymptotic cohomology groups; partial cohomology vanishing; $q$-positivity; hermitian metrics; Chern curvatures; asymptotic cohomology groups; -positivity; Hermitian metrics},
language = {eng},
number = {6},
pages = {2199-2221},
publisher = {Association des Annales de l’institut Fourier},
title = {Asymptotic cohomology vanishing and a converse to the Andreotti-Grauert theorem on surfaces},
url = {http://eudml.org/doc/275657},
volume = {63},
year = {2013},
}

TY - JOUR
AU - Matsumura, Shin-ichi
TI - Asymptotic cohomology vanishing and a converse to the Andreotti-Grauert theorem on surfaces
JO - Annales de l’institut Fourier
PY - 2013
PB - Association des Annales de l’institut Fourier
VL - 63
IS - 6
SP - 2199
EP - 2221
AB - 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.
LA - eng
KW - Asymptotic cohomology groups; partial cohomology vanishing; $q$-positivity; hermitian metrics; Chern curvatures; asymptotic cohomology groups; -positivity; Hermitian metrics
UR - http://eudml.org/doc/275657
ER -

References

top
  1. A Andreotti, H. Grauert, Théorème de finitude pour la cohomologie des espaces complexes, Bull. Soc. Math. France 90 (1962), 193-259 Zbl0106.05501MR150342
  2. D. Barlet, Convexité au voisinage d’un cycle, (french), Functions of several complex variables, IV (Sem. François Norguet, 1977–1979) (French) 807 (1980), 102-121, Springer, Berlin Zbl0434.32012MR592787
  3. S. Boucksom, Divisorial Zariski decompositions on compact complex manifolds, Ann. Sci. École Norm. Sup. (4) 37 (2004), 45-76 Zbl1054.32010MR2050205
  4. S. Boucksom, J. P. Demailly, M. Paun, Peternell T., The pseudo-effective cone of a compact Kähler manifold and varieties of negative Kodaira dimension Zbl1267.32017
  5. J. P. Demailly, Champs magnétiques et inégalités de Morse pour la d ' ' -cohomologie, Ann. Inst. Fourier (Grenoble) 35 (1985), 189-229 Zbl0565.58017MR812325
  6. J. P. Demailly, Cohomology of q-convex spaces in top degrees, Math. Z 204 (1990), 283-295 Zbl0682.32017MR1055992
  7. J. P. Demailly, Holomorphic Morse inequalities and asymptotic cohomology groups: a tribute to Bernhard Riemann, Milan J. Math. 78 (2010), 265-277 Zbl1205.32017MR2684780
  8. J. P. Demailly, A converse to the Andreotti-Grauert theorem, Ann. Fac. Sci. Toulouse Math. (6) 20 (2011) Zbl1228.32020MR2858170
  9. J. P. Demailly, T. Peternell, M. Schneider, Holomorphic line bundles with partially vanishing cohomology, Proceedings of the Hirzebruch 65 Conference on Algebraic Geometry (Ramat Gan, 1993) 9 (1996), 165-198, Bar-Ilan Univ Zbl0859.14005MR1360502
  10. L. Ein, R. Lazarsfeld, M. Musţǎ, M. Nakamaye, M. Popa, Asymptotic invariants of base loci, Ann. Inst. Fourier (Grenoble) 56 (2006), 1701-1734 Zbl1127.14010MR2282673
  11. T. Fujita, Semipositive line bundles, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 30 (1983), 353-378 Zbl0561.32012MR722501
  12. H. Fuse, T. Ohsawa, On a curvature property of effective divisors and its application to sheaf cohomology, Publ. Res. Inst. Math. Sci. 45 (2009), 1033-1039 Zbl1190.32009MR2597128
  13. A. Küronya, Positivity on subvarieties and vanishing of higher cohomology Zbl1291.14018
  14. R. Lazarsfeld, Positivity in Algebraic Geometry I, 48 (2004), Springer Verlag, Berlin Zbl1093.14500MR2095471
  15. S. Matsumura, Restricted volumes and divisorial Zariski decompositions Zbl1277.14006
  16. S. Matsumura, On the curvature of holomorphic line bundles with partially vanishin cohomology, RIMS, Kôkyûroku, Potential theory and fiber spaces (2012), 155-169 
  17. T. Ohsawa, Completeness of noncompact analytic spaces, Publ. Res. Inst. Math. Sci. 20 (1984), 683-692 Zbl0568.32008MR759689
  18. R. Richberg, Stetige streng pseudokonvexe Funktionen, Math. Ann. 175 (1968), 257-286 Zbl0153.15401MR222334
  19. Y. T. Siu, Every Stein subvariety admits a Stein neighborhood, Invent. Math. 38 (1976/77), 89-100 Zbl0343.32014MR435447
  20. A. J. Sommese, Submanifolds of Abelian varieties, Math. Ann. 233 (1978), 229-256 Zbl0381.14007MR466647
  21. B. Totaro, Line bundles with partially vanishing cohomology Zbl1277.14007
  22. S. T. Yau, On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation, I, Comm. Pure Appl. Math. 31 (1978), 339-411 Zbl0369.53059MR480350
  23. O. Zariski, The theorem of Riemann-Roch for high multiples of an effective divisor on an algebraic surface, Ann. of Math. (2) 76 (1962), 560-615 Zbl0124.37001MR141668

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.