Abundance for Kähler threefolds

Campana, Frédéric; Höring, Andreas; Peternell, Thomas

doi:10.24033/asens.2301

Abundance for Kähler threefolds
[Abondance pour les variétés kälhériennes de dimension 3]

Campana, Frédéric ; Höring, Andreas ; Peternell, Thomas

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 49 (2016) no. 4, pp. 971-1025.

Résumé
Abstract

Soit $X$ une variété kählérienne compacte à singularités terminales. Si $K_{X}$ est nef, nous montrons que $K_{X}$ est semi-ample, c'est-à-dire qu'un multiple $m K_{X}$ est engendré par ses sections globales.

Let $X$ be a compact Kähler threefold with terminal singularities such that $K_{X}$ is nef. We prove that $K_{X}$ is semiample, i.e., some multiple $m K_{X}$ is generated by global sections.

Publié le : 2016-11-12

MR Zbl

DOI : 10.24033/asens.2301

Classification : 32J27, 14E30, 14J30, 32J17, 32J25.
Keywords: log MMP, cone theorem, contraction theorem, rational curves, Zariski decomposition, Kähler manifolds, abundance.
Mot clés : log MMP, théorème du cône, théorème de contraction, courbes rationnelles, décomposition de Zariski, variétés kählériennes, abondance.

@article{ASENS_2016__49_4_971_0,
     author = {Campana, Fr\'ed\'eric and H\"oring, Andreas and Peternell, Thomas},
     title = {Abundance for {K\"ahler} threefolds},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {971--1025},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
     volume = {Ser. 4, 49},
     number = {4},
     year = {2016},
     doi = {10.24033/asens.2301},
     mrnumber = {3552019},
     zbl = {1386.32020},
     language = {en},
     url = {https://www.numdam.org/articles/10.24033/asens.2301/}
}

TY  - JOUR
AU  - Campana, Frédéric
AU  - Höring, Andreas
AU  - Peternell, Thomas
TI  - Abundance for Kähler threefolds
JO  - Annales scientifiques de l'École Normale Supérieure
PY  - 2016
SP  - 971
EP  - 1025
VL  - 49
IS  - 4
PB  - Société Mathématique de France. Tous droits réservés
UR  - https://www.numdam.org/articles/10.24033/asens.2301/
DO  - 10.24033/asens.2301
LA  - en
ID  - ASENS_2016__49_4_971_0
ER  -

%0 Journal Article
%A Campana, Frédéric
%A Höring, Andreas
%A Peternell, Thomas
%T Abundance for Kähler threefolds
%J Annales scientifiques de l'École Normale Supérieure
%D 2016
%P 971-1025
%V 49
%N 4
%I Société Mathématique de France. Tous droits réservés
%U https://www.numdam.org/articles/10.24033/asens.2301/
%R 10.24033/asens.2301
%G en
%F ASENS_2016__49_4_971_0

Campana, Frédéric; Höring, Andreas; Peternell, Thomas. Abundance for Kähler threefolds. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 49 (2016) no. 4, pp. 971-1025. doi : 10.24033/asens.2301. https://www.numdam.org/articles/10.24033/asens.2301/

Bibliographie
Cité par

Ancona, V. Faisceaux amples sur les espaces analytiques, Trans. Amer. Math. Soc., Volume 274 (1982), pp. 89-100 (ISSN: 0002-9947) | DOI | MR | Zbl

Ancona, V. Vanishing and nonvanishing theorems for numerically effective line bundles on complex spaces, Ann. Mat. Pura Appl., Volume 149 (1987), pp. 153-164 (ISSN: 0003-4622) | DOI | MR | Zbl

Araujo, C. The cone of pseudo-effective divisors of log varieties after Batyrev, Math. Z., Volume 264 (2010), pp. 179-193 (ISSN: 0025-5874) | DOI | MR | Zbl

Ancona, V.; Tan, V. V. On the blowing down problem in $𝐂$ -analytic geometry, J. reine angew. Math., Volume 350 (1984), pp. 178-182 (ISSN: 0075-4102) | MR | Zbl

Bauer, T.; Campana, F.; Eckl, T.; Kebekus, S.; Peternell, T.; Rams, S.; Szemberg, T.; Wotzlaw, L., Complex geometry (Göttingen, 2000), Springer, Berlin, 2002, pp. 27-36 | DOI | MR | Zbl

Birkar, C.; Cascini, P.; Hacon, C. D.; McKernan, J. Existence of minimal models for varieties of log general type, J. Amer. Math. Soc., Volume 23 (2010), pp. 405-468 (ISSN: 0894-0347) | DOI | MR | Zbl

Boucksom, S.; Demailly, J.-P.; Păun, M.; Peternell, T. The pseudo-effective cone of a compact Kähler manifold and varieties of negative Kodaira dimension, J. Algebraic Geom., Volume 22 (2013), pp. 201-248 (ISSN: 1056-3911) | DOI | MR | Zbl

Beauville, A. Variétés kählériennes dont la première classe de Chern est nulle, J. Differential Geom., Volume 18 (1983), pp. 755-782 http://projecteuclid.org/euclid.jdg/1214438181 (ISSN: 0022-040X) | MR | Zbl

(Boucksom, S.; Eyssidieux, P.; Guedj, V., eds.), Lecture Notes in Math., 2086, Springer, Cham, 2013, 333 pages (ISBN: 978-3-319-00818-9; 978-3-319-00819-6) | DOI | MR

Barth, W. P.; Hulek, K.; Peters, C. A. M.; Van de Ven, A., Ergebn. Math. Grenzg., 4, Springer, 2004, 436 pages (ISBN: 3-540-00832-2) | DOI | MR | Zbl

Boucksom, S. Cônes positifs des variétés complexes compactes (2002)

Boucksom, S. Divisorial Zariski decompositions on compact complex manifolds, Ann. Sci. École Norm. Sup., Volume 37 (2004), pp. 45-76 (ISSN: 0012-9593) | DOI | Numdam | MR | Zbl

Brunella, M. A positivity property for foliations on compact Kähler manifolds, Internat. J. Math., Volume 17 (2006), pp. 35-43 (ISSN: 0129-167X) | DOI | MR | Zbl

Bando, S.; Siu, Y.-T., Geometry and analysis on complex manifolds, World Sci. Publ., River Edge, NJ, 1994, pp. 39-50 | DOI | MR | Zbl

Beltrametti, M. C.; Sommese, A. J., de Gruyter Expositions in Mathematics, 16, Walter de Gruyter & Co., Berlin, 1995, 398 pages (ISBN: 3-11-014355-0) | DOI | MR | Zbl

Cao, J. A remark on compact Kähler manifolds with nef anticanonical bundles and its applications (preprint arXiv:1305.4397 )

Claudon, B.; Höring, A. Compact Kähler manifolds with compactifiable universal cover, with an appendix by Frédéric Campana, Bull. Soc. Math. France, Volume 141 (2013), pp. 355-375 (ISSN: 0037-9484) | DOI | MR | Zbl

(Corti, A., ed.), Oxford Lecture Series in Mathematics and its Applications, 35, Oxford Univ. Press, Oxford, 2007, 189 pages (ISBN: 978-0-19-857061-5) | DOI | MR

Campana, F.; Peternell, T. Complex threefolds with non-trivial holomorphic 2-forms, J. Algebraic Geom., Volume 9 (2000), pp. 223-264 (ISSN: 1056-3911) | MR | Zbl

Collins, T. C.; Tosatti, V. Kähler currents and null loci, Invent. math., Volume 202 (2015), pp. 1167-1198 | DOI | MR | Zbl

Debarre, O., Universitext, Springer, New York, 2001, 233 pages (ISBN: 0-387-95227-6) | DOI | MR | Zbl

Demailly, J.-P.; Hacon, C. D.; Păun, M. Extension theorems, non-vanishing and the existence of good minimal models, Acta Math., Volume 210 (2013), pp. 203-259 (ISSN: 0001-5962) | DOI | MR | Zbl

Demailly, J.-P.; Peternell, T. A Kawamata-Viehweg vanishing theorem on compact Kähler manifolds, J. Differential Geom., Volume 63 (2003), pp. 231-277 http://projecteuclid.org/euclid.jdg/1090426678 (ISSN: 0022-040X) | MR | Zbl

Demailly, J.-P.; Păun, M. Numerical characterization of the Kähler cone of a compact Kähler manifold, Ann. of Math., Volume 159 (2004), pp. 1247-1274 (ISSN: 0003-486X) | DOI | MR | Zbl

Enoki, I., Geometry and analysis on manifolds (Katata/Kyoto, 1987) (Lecture Notes in Math.), Volume 1339, Springer, Berlin, 1988, pp. 118-126 | DOI | MR | Zbl

Fujino, O., Flips for 3-folds and 4-folds (Oxford Lecture Ser. Math. Appl.), Volume 35, Oxford Univ. Press, Oxford, 2007, pp. 63-75 | DOI | MR | Zbl

Fujino, O. Non-vanishing theorem for log canonical pairs, J. Algebraic Geom., Volume 20 (2011), pp. 771-783 (ISSN: 1056-3911) | DOI | MR | Zbl

Fujino, O., Classification of algebraic varieties (EMS Ser. Congr. Rep.), Eur. Math. Soc., Zürich, 2011, pp. 305-315 | DOI | MR | Zbl

Fujita, T. On Kähler fiber spaces over curves, J. Math. Soc. Japan, Volume 30 (1978), pp. 779-794 (ISSN: 0025-5645) | DOI | MR | Zbl

Fujiki, A., Algebraic varieties and analytic varieties (Tokyo, 1981) (Adv. Stud. Pure Math.), Volume 1, North-Holland, Amsterdam, 1983, pp. 231-302 | DOI | MR | Zbl

Fujita, T., Algebraic geometry, Sendai, 1985 (Adv. Stud. Pure Math.), Volume 10, North-Holland, Amsterdam, 1987, pp. 167-178 | DOI | MR | Zbl

Fulton, W., Ergebn. Math. Grenzg., 2, Springer, Berlin, 1998, 470 pages (ISBN: 3-540-62046-X; 0-387-98549-2) | DOI | MR | Zbl

Grauert, H. Über Modifikationen und exzeptionelle analytische Mengen, Math. Ann., Volume 146 (1962), pp. 331-368 (ISSN: 0025-5831) | DOI | MR | Zbl

Grivaux, J. Chern classes in Deligne cohomology for coherent analytic sheaves, Math. Ann., Volume 347 (2010), pp. 249-284 (ISSN: 0025-5831) | DOI | MR | Zbl

Hanamura, M. On the birational automorphism groups of algebraic varieties, Compositio Math., Volume 63 (1987), pp. 123-142 (ISSN: 0010-437X) | Numdam | MR | Zbl

Hartshorne, R., Graduate Texts in Math., 52, Springer, New York-Heidelberg, 1977, 496 pages (ISBN: 0-387-90244-9) | MR | Zbl

Höring, A.; Peternell, T. Mori fibre spaces for Kähler threefolds, J. Math. Sci. Univ. Tokyo, Volume 22 (2015), pp. 219-246 (ISSN: 1340-5705) | MR | Zbl

Höring, A.; Peternell, T. Minimal models for Kähler threefolds, Invent. math., Volume 203 (2016), pp. 217-264 | DOI | MR | Zbl

Kollár, J. et al., Astérisque, 211, Soc. Math. France, 1992 (ISSN: 0303-1179) | MR

Kawamata, Y. Pluricanonical systems on minimal algebraic varieties, Invent. math., Volume 79 (1985), pp. 567-588 (ISSN: 0020-9910) | DOI | MR | Zbl

Kawamata, Y. Crepant blowing-up of 3-dimensional canonical singularities and its application to degenerations of surfaces, Ann. of Math., Volume 127 (1988), pp. 93-163 (ISSN: 0003-486X) | DOI | MR | Zbl

Kawamata, Y. On the length of an extremal rational curve, Invent. math., Volume 105 (1991), pp. 609-611 (ISSN: 0020-9910) | DOI | MR | Zbl

Kawamata, Y. Abundance theorem for minimal threefolds, Invent. math., Volume 108 (1992), pp. 229-246 (ISSN: 0020-9910) | DOI | MR | Zbl

Kawamata, Y. Termination of log flips for algebraic 3-folds, Internat. J. Math., Volume 3 (1992), pp. 653-659 (ISSN: 0129-167X) | DOI | MR | Zbl

Kollár, J.; Kovács, S. J. Log canonical singularities are Du Bois, J. Amer. Math. Soc., Volume 23 (2010), pp. 791-813 (ISSN: 0894-0347) | DOI | MR | Zbl

Kollár, J.; Mori, S., Cambridge Tracts in Mathematics, 134, Cambridge Univ. Press, Cambridge, 1998, 254 pages (ISBN: 0-521-63277-3) | DOI | MR | Zbl

Kobayashi, S., Publications of the Mathematical Society of Japan, 15, Princeton Univ. Press, Princeton, NJ; Iwanami Shoten, Tokyo, 1987, 305 pages (ISBN: 0-691-08467-X) | DOI | MR | Zbl

Kollàr, J. Kodaira's canonical bundle formula and adjunction, Flips for 3-folds and 4-folds (Oxford Lecture Ser. Math. Appl.), Volume 35, Oxford Univ. Press (2007), pp. 134-162 | DOI | MR | Zbl

Kollár, J., Cambridge Tracts in Mathematics, 200, Cambridge Univ. Press, Cambridge, 2013, 370 pages (ISBN: 978-1-107-03534-8) | DOI | MR | Zbl

Kollár, J. Flops, Nagoya Math. J., Volume 113 (1989), pp. 15-36 http://projecteuclid.org/euclid.nmj/1118781185 (ISSN: 0027-7630) | DOI | MR | Zbl

Kollár, J., Algebraic geometry—Santa Cruz 1995 (Proc. Sympos. Pure Math.), Volume 62, Amer. Math. Soc., Providence, RI, 1997, pp. 221-287 | DOI | MR | Zbl

Liu, W.; Rollenske, S. Geography of Gorenstein stable log surfaces, Trans. Amer. Math. Soc., Volume 368 (2016), pp. 2563-2588 (ISSN: 0002-9947) | DOI | MR | Zbl

Miyaoka, Y., Algebraic geometry, Sendai, 1985 (Adv. Stud. Pure Math.), Volume 10, North-Holland, Amsterdam, 1987, pp. 449-476 | DOI | MR | Zbl

Miyaoka, Y. Abundance conjecture for 3-folds: case $ν = 1$ , Compositio Math., Volume 68 (1988), pp. 203-220 (ISSN: 0010-437X) | Numdam | MR | Zbl

Mori, S. Flip theorem and the existence of minimal models for 3-folds, J. Amer. Math. Soc., Volume 1 (1988), pp. 117-253 (ISSN: 0894-0347) | DOI | MR | Zbl

Nakayama, N., Algebraic geometry, Sendai, 1985 (Adv. Stud. Pure Math.), Volume 10, North-Holland, Amsterdam, 1987, pp. 551-590 | DOI | MR | Zbl

Namikawa, Y. Projectivity criterion of Moishezon spaces and density of projective symplectic varieties, Internat. J. Math., Volume 13 (2002), pp. 125-135 (ISSN: 0129-167X) | DOI | MR | Zbl

Peternell, T. Towards a Mori theory on compact Kähler threefolds. III, Bull. Soc. Math. France, Volume 129 (2001), pp. 339-356 (ISSN: 0037-9484) | DOI | Numdam | MR | Zbl

Peternell, T. Towards a Mori theory on compact Kähler threefolds. II, Math. Ann., Volume 311 (1998), pp. 729-764 (ISSN: 0025-5831) | DOI | MR | Zbl

Păun, M. Sur l'effectivité numérique des images inverses de fibrés en droites, Math. Ann., Volume 310 (1998), pp. 411-421 (ISSN: 0025-5831) | DOI | MR | Zbl

Sakai, F. Weil divisors on normal surfaces, Duke Math. J., Volume 51 (1984), pp. 877-887 (ISSN: 0012-7094) | DOI | MR | Zbl

Shokurov, V. V. Three-dimensional log perestroikas, Izv. Ross. Akad. Nauk Ser. Mat., Volume 56 (1992), pp. 105-203 ; English translation: Russian Acad. Sci. Izv. Math. 40 (1993), 95–202 (ISSN: 0373-2436) | DOI | MR | Zbl

Toledo, D.; Tong, Y. L. L., The Lefschetz centennial conference, Part I (Mexico City, 1984) (Contemp. Math.), Volume 58, Amer. Math. Soc., Providence, RI, 1986, pp. 261-275 | DOI | MR | Zbl

Ueno, K. On compact analytic threefolds with nontrivial Albanese tori, Math. Ann., Volume 278 (1987), pp. 41-70 (ISSN: 0025-5831) | DOI | MR | Zbl

Varouchas, J. Kähler spaces and proper open morphisms, Math. Ann., Volume 283 (1989), pp. 13-52 (ISSN: 0025-5831) | DOI | MR | Zbl

Voisin, C., Cours spécialisés, 10, Soc. Math. France, Paris, 2002, 595 pages (ISBN: 2-85629-129-5) | DOI | MR | Zbl

Catanese, Fabrizio Manifolds with Trivial Chern Classes II: Manifolds Isogenous to a Torus Product, Coframed Manifolds and a Question by Baldassarri, International Mathematics Research Notices, Volume 2025 (2025) no. 5 | DOI:10.1093/imrn/rnaf041
Das, Omprokash; Hacon, Christopher; Păun, Mihai On the 4-dimensional minimal model program for Kähler varieties, Advances in Mathematics, Volume 443 (2024), p. 109615 | DOI:10.1016/j.aim.2024.109615
Das, Omprokash; Ou, Wenhao On the log abundance for compact Kähler threefolds, manuscripta mathematica, Volume 173 (2024) no. 1-2, p. 341 | DOI:10.1007/s00229-023-01467-6
Zhong, Guolei Existence of the equivariant minimal model program for compact Kähler threefolds with the action of an abelian group of maximal rank, Mathematische Nachrichten, Volume 296 (2023) no. 7, p. 3128 | DOI:10.1002/mana.202200127
Jian, Wangjian; Song, Jian Diameter estimates for long-time solutions of the Kähler–Ricci flow, Geometric and Functional Analysis, Volume 32 (2022) no. 6, p. 1335 | DOI:10.1007/s00039-022-00620-9
Wang, Juanyong On the Iitaka Conjecture C n,m for Kähler Fibre Spaces, Annales de la Faculté des sciences de Toulouse : Mathématiques, Volume 30 (2021) no. 4, p. 813 | DOI:10.5802/afst.1690
Höring, A. Adjoint -classes on threefolds, Izvestiya: Mathematics, Volume 85 (2021) no. 4, p. 823 | DOI:10.1070/im9084
FUJINO, OSAMU MINIMAL MODEL THEORY FOR LOG SURFACES IN FUJIKI’S CLASS, Nagoya Mathematical Journal, Volume 244 (2021), p. 256 | DOI:10.1017/nmj.2020.14
Neofytidis, Christoforos; Zhang, Weiyi Geometric structures, the Gromov order, Kodaira dimensions and simplicial volume, Pacific Journal of Mathematics, Volume 315 (2021) no. 1, p. 209 | DOI:10.2140/pjm.2021.315.209
Horing, Andreas Присоединенные $(1, 1)$ -классы на трехмерных многообразиях, Известия Российской академии наук. Серия математическая, Volume 85 (2021) no. 4, p. 215 | DOI:10.4213/im9084
Prokhorov, Yu. G.; Shramov, C. A. Finite groups of bimeromorphic selfmaps of uniruled Kähler threefolds, Izvestiya: Mathematics, Volume 84 (2020) no. 5, p. 978 | DOI:10.1070/im8983
Zhang, Yashan Infinite-Time Singularity Type of the Kähler–Ricci Flow, The Journal of Geometric Analysis, Volume 30 (2020) no. 2, p. 2092 | DOI:10.1007/s12220-017-9949-2
Prokhorov, Yuri Gennadievich; Shramov, Constantin Aleksandrovich Конечные группы бимероморфных автоморфизмов унилинейчатых трехмерных кэлеровых многообразий, Известия Российской академии наук. Серия математическая, Volume 84 (2020) no. 5, p. 169 | DOI:10.4213/im8983
Schreieder, Stefan; Tasin, Luca Kähler structures on spin 6-manifolds, Mathematische Annalen, Volume 373 (2019) no. 1-2, p. 397 | DOI:10.1007/s00208-017-1615-2
Tosatti, Valentino KAWA lecture notes on the Kähler–Ricci flow, Annales de la Faculté des sciences de Toulouse : Mathématiques, Volume 27 (2018) no. 2, p. 285 | DOI:10.5802/afst.1571
Eyssidieux, Phylippe; Guedj, Vincent; Zeriahi, Ahmed Convergence of Weak Kähler–Ricci Flows on Minimal Models of Positive Kodaira Dimension, Communications in Mathematical Physics, Volume 357 (2018) no. 3, p. 1179 | DOI:10.1007/s00220-018-3087-y
Diverio, Simone Quasi-Negative Holomorphic Sectional Curvature and Ampleness of the Canonical Class, Complex and Symplectic Geometry, Volume 21 (2017), p. 61 | DOI:10.1007/978-3-319-62914-8_5

Cité par 17 documents. Sources : Crossref