[Modifications de faisceaux analytiques cohérents sans torsion]
Nous étudions la transformation de faisceaux analytiques cohérents à travers des modifications propres. Plus précisément, nous étudions les images directes de faisceaux images-inverses, et les préimages sans torsion de faisceaux images directes. Sous certaines conditions, il est démontré que les faisceaux cohérents sans torsion peuvent être réalisés comme images directes de faisceaux localement libres à travers des modifications. Ainsi, il est possible d’étudier les faisceaux cohérents modulo torsion en ramenant le problème à l’étude de fibrés vectoriels sur des variétés. Nous appliquons ceci aux faisceaux d’idéaux réduits et au faisceau canonique de -formes holomorphes de Grauert–Riemenschneider.
We study the transformation of torsion-free coherent analytic sheaves under proper modifications. More precisely, we study direct images of inverse image sheaves, and torsion-free preimages of direct image sheaves. Under some conditions, it is shown that torsion-free coherent sheaves can be realized as the direct image of locally free sheaves under modifications. Thus, it is possible to study coherent sheaves modulo torsion by reducing the problem to study vector bundles on manifolds. We apply this to reduced ideal sheaves and to the Grauert–Riemenschneider canonical sheaf of holomorphic -forms.
Révisé le :
Accepté le :
Publié le :
Keywords: modifications, coherent analytic sheaves, torsion-free preimage
Mot clés : modifications, faisceaux analytiques cohérents, préimages sans torsion
Ruppenthal, Jean 1 ; Sera, Martin 1
CC-BY-ND 4.0
@article{AIF_2017__67_1_237_0,
author = {Ruppenthal, Jean and Sera, Martin},
title = {Modifications of torsion-free coherent analytic sheaves},
journal = {Annales de l'Institut Fourier},
pages = {237--265},
publisher = {Association des Annales de l{\textquoteright}institut Fourier},
volume = {67},
number = {1},
year = {2017},
doi = {10.5802/aif.3080},
language = {en},
url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3080/}
}
TY - JOUR AU - Ruppenthal, Jean AU - Sera, Martin TI - Modifications of torsion-free coherent analytic sheaves JO - Annales de l'Institut Fourier PY - 2017 SP - 237 EP - 265 VL - 67 IS - 1 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3080/ DO - 10.5802/aif.3080 LA - en ID - AIF_2017__67_1_237_0 ER -
%0 Journal Article %A Ruppenthal, Jean %A Sera, Martin %T Modifications of torsion-free coherent analytic sheaves %J Annales de l'Institut Fourier %D 2017 %P 237-265 %V 67 %N 1 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3080/ %R 10.5802/aif.3080 %G en %F AIF_2017__67_1_237_0
Ruppenthal, Jean; Sera, Martin. Modifications of torsion-free coherent analytic sheaves. Annales de l'Institut Fourier, Tome 67 (2017) no. 1, pp. 237-265. doi : 10.5802/aif.3080. https://aif.centre-mersenne.org/articles/10.5802/aif.3080/
[1] Complex analytic cones, Math. Ann., Volume 273 (1986) no. 4, pp. 601-627 | DOI
[2] A closure operation on complex analytic cones and torsion, Ann. Fac. Sci. Toulouse Math. (6), Volume 7 (1998) no. 1, pp. 5-33
[3] Graded algebras of powers of ideals generated by -sequences, J. Algebra, Volume 25 (1973), pp. 90-99
[4] Complex Analytic and Differential Geometry (2012) (Institut Fourier, Université de Grenoble I. OpenContent AG-Book, http://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/agbook.pdf)
[5] Commutative algebra, Graduate Texts in Mathematics, 150, Springer-Verlag, New York, 1995, xvi+785 pages (With a view toward algebraic geometry) | DOI
[6] Eine Charakterisierung von holomorphen Vektorraumbündeln, Bayer. Akad. Wiss. Math.-Natur. Kl. S.-B., Volume 1966 (1967), pp. 101-107
[7] Lineare Faserräume und kohärente Modulgarben über komplexen Räumen, Arch. Math. (Basel), Volume 18 (1967), pp. 609-617
[8] Complex analytic geometry, Lecture Notes in Mathematics, Vol. 538, Springer-Verlag, Berlin, 1976, vii+201 pages
[9] Über Modifikationen und Exzeptionelle Analytische Mengen, Math. Ann., Volume 146 (1962), pp. 331-368
[10] Coherent analytic sheaves, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 265, Springer-Verlag, Berlin, 1984, xviii+249 pages | DOI
[11] Verschwindungssätze für analytische Kohomologiegruppen auf komplexen Räumen, Invent. Math., Volume 11 (1970), pp. 263-292
[12] Techniques de construction en géométrie analytique. V: Fibres vectoriels, fibres projectifs, fibres en drapeaux., Sem. H. Cartan 13 (1960/61) (1962) no. 12, 15 pages
[13] On the symmetric and Rees algebra of an ideal generated by a -sequence, J. Algebra, Volume 62 (1980) no. 2, pp. 268-275 | DOI
[14] Vector bundles on complex projective spaces, Modern Birkhäuser Classics, Birkhäuser/Springer Basel AG, Basel, 2011, viii+239 pages (Corrected reprint of the 1988 edition, With an appendix by S. I. Gelfand)
[15] Modifications, Several complex variables, VII (Encyclopaedia Math. Sci.), Volume 74, Springer, Berlin, 1994, pp. 285-317 | DOI
[16] Differential calculus, holomorphic maps and linear structures on complex spaces, Several complex variables, VII (Encyclopaedia Math. Sci.), Volume 74, Springer, Berlin, 1994, pp. 97-144 | DOI
[17] Moišezon spaces and positive coherent sheaves, Proc. Amer. Math. Soc., Volume 71 (1978) no. 2, pp. 237-240
[18] On monoidal transformations of coherent sheaves, Proc. Amer. Math. Soc., Volume 74 (1979) no. 2, pp. 389-390
[19] Characterizing Moišezon spaces by almost positive coherent analytic sheaves, Math. Z., Volume 123 (1971), pp. 263-284
[20] Picard variety of an isolated singular point, Rice Univ. Studies, Volume 54 (1968) no. 4, pp. 63-73
[21] -theory for the -operator on compact complex spaces, Duke Math. J., Volume 163 (2014) no. 15, pp. 2887-2934 | DOI
[22] A generalization of Takegoshi’s relative vanishing theorem, J. Geom. Anal., Volume 26 (2016) no. 3, pp. 1891-1912
[23] Relative vanishing theorems in analytic spaces, Duke Math. J., Volume 52 (1985) no. 1, pp. 273-279
Cité par Sources :
