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Annales scientifiques de l'ENS - Parutions - série 4, 49 (2016)

Parutions < série 4, 49

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE, série 4 49, fascicule 4 (2016)

Frédéric Campana, Andreas Höring, Thomas Peternell
Abundance for Kähler threefolds
Annales scientifiques de l'ENS 49, fascicule 4 (2016), 971-1025

Télécharger cet article : Fichier PDF

Résumé :
Abondance pour les variétés kälhériennes de dimension 3
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.

Mots-clefs : log MMP, théorème du cône, théorème de contraction, courbes rationnelles, décomposition de Zariski, variétés kählériennes, abondance.

Abstract:
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 mK_X is generated by global sections.

Keywords: log MMP, cone theorem, contraction theorem, rational curves, Zariski decomposition, Kähler manifolds, abundance.

Class. math. : 32J27, 14E30, 14J30, 32J17, 32J25.


ISSN : 0012-9593
Publié avec le concours de : Centre National de la Recherche Scientifique

Bibliographie:

1
Ancona, Vincenzo
Faisceaux amples sur les espaces analytiques
Trans. Amer. Math. Soc. 274 (1982) 89–100
Math Reviews MR670921 (84c:32029)
2
Ancona, Vincenzo
Vanishing and nonvanishing theorems for numerically effective line bundles on complex spaces
Ann. Mat. Pura Appl. 149 (1987) 153–164
Math Reviews MR932782 (89h:32058)
3
Araujo, Carolina
The cone of pseudo-effective divisors of log varieties after Batyrev
Math. Z. 264 (2010) 179–193
Math Reviews MR2564937 (2010k:14018)
4
Ancona, Vincenzo and Tan, Vo Van
On the blowing down problem in <b>C</b>-analytic geometry
J. reine angew. Math. 350 (1984) 178–182
Math Reviews MR743541 (86h:32025)
5
Bauer, Thomas and Campana, Frédéric and Eckl, Thomas and Kebekus, Stefan and Peternell, Thomas and Rams, Sławomir and Szemberg, Tomasz and Wotzlaw, Lorenz
A reduction map for nef line bundles
in Complex geometry (Göttingen, 2000)
(2002) 27–36
Math Reviews MR1922095 (2003h:14023)
6
Birkar, Caucher and Cascini, Paolo and Hacon, Christopher D. and McKernan, James
Existence of minimal models for varieties of log general type
J. Amer. Math. Soc. 23 (2010) 405–468
Math Reviews MR2601039 (2011f:14023)
7
Boucksom, Sébastien and Demailly, Jean-Pierre and Păun, Mihai and Peternell, Thomas
The pseudo-effective cone of a compact Kähler manifold and varieties of negative Kodaira dimension
J. Algebraic Geom. 22 (2013) 201–248
Math Reviews MR3019449
8
Beauville, Arnaud
Variétés kählériennes dont la première classe de Chern est nulle
J. Differential Geom. 18 (1983) 755–782
Math Reviews MR730926 (86c:32030)
9
An introduction to the Kähler-Ricci flow
Springer, Cham, 2013
Math Reviews MR3202578
10
Barth, Wolf P. and Hulek, Klaus and Peters, Chris A. M. and Van de Ven, Antonius
Compact complex surfaces
Springer, 2004
Math Reviews MR2030225 (2004m:14070)
11
Boucksom, Sébastien
Cônes positifs des variétés complexes compactes
thèse de doctorat, Université de Grenoble (2002)
12
Boucksom, Sébastien
Divisorial Zariski decompositions on compact complex manifolds
Ann. Sci. École Norm. Sup. 37 (2004) 45–76
Math Reviews MR2050205 (2005i:32018)
13
Brunella, Marco
A positivity property for foliations on compact Kähler manifolds
Internat. J. Math. 17 (2006) 35–43
Math Reviews MR2204838 (2006k:32044)
14
Bando, Shigetoshi and Siu, Yum-Tong
Stable sheaves and Einstein-Hermitian metrics
in Geometry and analysis on complex manifolds
(1994) 39–50
Math Reviews MR1463962 (98e:32053)
15
Beltrametti, Mauro C. and Sommese, Andrew J.
The adjunction theory of complex projective varieties
Walter de Gruyter Co., Berlin, 1995
Math Reviews MR1318687 (96f:14004)
16
17
Claudon, Benoît and Höring, Andreas
Compact Kähler manifolds with compactifiable universal cover, with an appendix by Frédéric Campana
Bull. Soc. Math. France 141 (2013) 355–375
Math Reviews MR3081560
18
Flips for 3-folds and 4-folds
Oxford Univ. Press, Oxford, 2007
Math Reviews MR2352762 (2008j:14031)
19
Campana, Frédéric and Peternell, Thomas
Complex threefolds with non-trivial holomorphic 2-forms
J. Algebraic Geom. 9 (2000) 223–264
Math Reviews MR1735771 (2002a:32016)
20
Collins, Tristan C. and Tosatti, Valentino
Kähler currents and null loci
Invent. math. 202 (2015) 1167–1198
21
Debarre, Olivier
Higher-dimensional algebraic geometry
Springer, New York, 2001
Math Reviews MR1841091 (2002g:14001)
22
Demailly, Jean-Pierre and Hacon, Christopher D. and Păun, Mihai
Extension theorems, non-vanishing and the existence of good minimal models
Acta Math. 210 (2013) 203–259
Math Reviews MR3070567
23
Demailly, Jean-Pierre and Peternell, Thomas
A Kawamata-Viehweg vanishing theorem on compact Kähler manifolds
J. Differential Geom. 63 (2003) 231–277
Math Reviews MR2015548 (2004m:32040)
24
Demailly, Jean-Pierre and Păun, Mihai
Numerical characterization of the Kähler cone of a compact Kähler manifold
Ann. of Math. 159 (2004) 1247–1274
Math Reviews MR2113021 (2005i:32020)
25
Enoki, Ichiro
Stability and negativity for tangent sheaves of minimal Kähler spaces
in Geometry and analysis on manifolds (Katata/Kyoto, 1987)
Lecture Notes in Math. 1339 (1988) 118–126
Math Reviews MR961477 (90a:32039)
26
Fujita, Takao
On Kähler fiber spaces over curves
J. Math. Soc. Japan 30 (1978) 779–794
Math Reviews MR513085 (82h:32024)
27
Fujiki, Akira
On the structure of compact complex manifolds in C
in Algebraic varieties and analytic varieties (Tokyo, 1981)
Adv. Stud. Pure Math. 1 (1983) 231–302
Math Reviews MR715653 (85g:32045b)
28
Fujita, Takao
On polarized manifolds whose adjoint bundles are not semipositive
in Algebraic geometry, Sendai, 1985
Adv. Stud. Pure Math. 10 (1987) 167–178
Math Reviews MR946238 (89d:14006)
29
Fujino, Osamu
Special termination and reduction to pl flips
in Flips for 3-folds and 4-folds
Oxford Lecture Ser. Math. Appl. 35 (2007) 63–75
Math Reviews MR2359342
30
Fujino, Osamu
Non-vanishing theorem for log canonical pairs
J. Algebraic Geom. 20 (2011) 771–783
Math Reviews MR2819675 (2012j:14021)
31
Fujino, Osamu
On Kawamata's theorem
in Classification of algebraic varieties
EMS Ser. Congr. Rep. (2011) 305–315
Math Reviews MR2779478 (2012c:14011)
32
Fulton, William
Intersection theory
Springer, Berlin, 1998
Math Reviews MR1644323 (99d:14003)
33
Grauert, Hans
Über Modifikationen und exzeptionelle analytische Mengen
Math. Ann. 146 (1962) 331–368
Math Reviews MR0137127 (25 \#583)
34
Grivaux, Julien
Chern classes in Deligne cohomology for coherent analytic sheaves
Math. Ann. 347 (2010) 249–284
Math Reviews MR2606937 (2011d:32029)
35
Hanamura, Masaki
On the birational automorphism groups of algebraic varieties
Compositio Math. 63 (1987) 123–142
Math Reviews MR906382 (88h:14018)
36
Hartshorne, Robin
Algebraic geometry
Springer, New York-Heidelberg, 1977
Math Reviews MR0463157 (57 \#3116)
37
Höring, Andreas and Peternell, Thomas
Minimal models for Kähler threefolds
Invent. math. 203 (2016) 217–264
38
Höring, Andreas and Peternell, Thomas
Mori fibre spaces for Kähler threefolds
J. Math. Sci. Univ. Tokyo 22 (2015) 219–246
Math Reviews MR3329195
39
Kawamata, Yujiro
Pluricanonical systems on minimal algebraic varieties
Invent. math. 79 (1985) 567–588
Math Reviews MR782236 (87h:14005)
40
Kawamata, Yujiro
Crepant blowing-up of 3-dimensional canonical singularities and its application to degenerations of surfaces
Ann. of Math. 127 (1988) 93–163
Math Reviews MR924674 (89d:14023)
41
Kawamata, Yujiro
On the length of an extremal rational curve
Invent. math. 105 (1991) 609–611
Math Reviews MR1117153 (92m:14026)
42
Kawamata, Yujiro
Abundance theorem for minimal threefolds
Invent. math. 108 (1992) 229–246
Math Reviews MR1161091 (93f:14012)
43
Kawamata, Yujiro
Termination of log flips for algebraic 3-folds
Internat. J. Math. 3 (1992) 653–659
Math Reviews MR1189678 (93j:14019)
44
Kollár, János and Kovács, Sándor J.
Log canonical singularities are Du Bois
J. Amer. Math. Soc. 23 (2010) 791–813
Math Reviews MR2629988 (2011m:14061)
45
Kollár, János and Mori, Shigefumi
Birational geometry of algebraic varieties
Cambridge Univ. Press, Cambridge, 1998
Math Reviews MR1658959 (2000b:14018)
46
Kobayashi, Shoshichi
Differential geometry of complex vector bundles
Princeton Univ. Press, Princeton, NJ; Iwanami Shoten, Tokyo, 1987
Math Reviews MR909698 (89e:53100)
47
Kollár, János
Flops
Nagoya Math. J. 113 (1989) 15–36
Math Reviews MR986434 (90e:14011)
48
Kollár, János
Singularities of pairs
in Algebraic geometry—Santa Cruz 1995
Proc. Sympos. Pure Math. 62 (1997) 221–287
Math Reviews MR1492525 (99m:14033)
49
Kollàr, János
Kodaira's canonical bundle formula and adjunction
in Flips for 3-folds and 4-folds
Oxford Lecture Ser. Math. Appl. 35 (2007) 134–162
50
Kollár, János
Singularities of the minimal model program
Cambridge Univ. Press, Cambridge, 2013
Math Reviews MR3057950
51
Kollár, János and others
Flips and abundance for algebraic threefolds
Soc. Math. France, 1992
Math Reviews MR1225842 (94f:14013)
52
Liu, Wenfei and Rollenske, Sönke
Geography of Gorenstein stable log surfaces
Trans. Amer. Math. Soc. 368 (2016) 2563–2588
Math Reviews MR3449249
53
Miyaoka, Yoichi
The Chern classes and Kodaira dimension of a minimal variety
in Algebraic geometry, Sendai, 1985
Adv. Stud. Pure Math. 10 (1987) 449–476
Math Reviews MR946247 (89k:14022)
54
Miyaoka, Yoichi
Abundance conjecture for 3-folds: case =1
Compositio Math. 68 (1988) 203–220
Math Reviews MR966580 (89m:14023)
55
Mori, Shigefumi
Flip theorem and the existence of minimal models for 3-folds
J. Amer. Math. Soc. 1 (1988) 117–253
Math Reviews MR924704 (89a:14048)
56
Nakayama, Noboru
The lower semicontinuity of the plurigenera of complex varieties
in Algebraic geometry, Sendai, 1985
Adv. Stud. Pure Math. 10 (1987) 551–590
Math Reviews MR946250 (89h:14028)
57
Namikawa, Yoshinori
Projectivity criterion of Moishezon spaces and density of projective symplectic varieties
Internat. J. Math. 13 (2002) 125–135
Math Reviews MR1891205 (2003b:32025)
58
Păun, Mihai
Sur l'effectivité numérique des images inverses de fibrés en droites
Math. Ann. 310 (1998) 411–421
Math Reviews MR1612321 (99c:32042)
59
Peternell, Thomas
Towards a Mori theory on compact Kähler threefolds. II
Math. Ann. 311 (1998) 729–764
Math Reviews MR1637984 (2001a:32027)
60
Peternell, Thomas
Towards a Mori theory on compact Kähler threefolds. III
Bull. Soc. Math. France 129 (2001) 339–356
Math Reviews MR1881199 (2003b:32024)
61
Sakai, Fumio
Weil divisors on normal surfaces
Duke Math. J. 51 (1984) 877–887
Math Reviews MR771385 (86m:14025)
62
Shokurov, V. V.
Three-dimensional log perestroikas
Izv. Ross. Akad. Nauk Ser. Mat. 56 (1992) 105–203 ; English translation: Russian Acad. Sci. Izv. Math. 40 (1993), 95–202
Math Reviews MR1162635 (93j:14012)
63
Toledo, Domingo and Tong, Yue Lin L.
Green's theory of Chern classes and the Riemann-Roch formula
in The Lefschetz centennial conference, Part I (Mexico City, 1984)
Contemp. Math. 58 (1986) 261–275
Math Reviews MR860421 (88b:32026)
64
Ueno, Kenji
On compact analytic threefolds with nontrivial Albanese tori
Math. Ann. 278 (1987) 41–70
Math Reviews MR909217 (88m:32054)
65
Varouchas, Jean
Kähler spaces and proper open morphisms
Math. Ann. 283 (1989) 13–52
Math Reviews MR973802 (89m:32021)
66
Voisin, Claire
Théorie de Hodge et géométrie algébrique complexe
Soc. Math. France, Paris, 2002
Math Reviews MR1988456 (2005c:32024a)