Classes de cohomologie positives dans les variétés kählériennes compactes
Séminaire Bourbaki : volume 2004/2005, exposés 938-951, Astérisque, no. 307 (2006), Exposé no. 943, pp. 199-228.

Étant donnée une variété kählérienne compacte X, on étudie dans l’espace vectoriel réel de cohomologie de Dolbeault H 1,1 (X,𝐑)H 2 (X,𝐑) le cône convexe des classes de Kähler ainsi que celui, plus grand, des classes de courants positifs fermés de type (1,1). Lorsque X est projective, les traces de ces cônes sur l’espace de Néron-Severi NS (X) 𝐑 H 1,1 (X,𝐑) engendré par les classes entières sont respectivement le cône des classes de diviseurs amples et l’adhérence de celui des classes de diviseurs effectifs.

Let X be a compact Kähler manifold. In the real vector space H 1,1 (X,𝐑)H 2 (X,𝐑) of Dolbeault cohomology classes of type (1,1), we study the convex cone of Kähler classes and the larger cone of classes of positive closed currents of type (1,1). When X is projective, theses cones cut out, on the Néron-Severi subspace NS (X) 𝐑 H 1,1 (X,𝐑) generated by integral classes, the cone of classes of ample divisors and the closure of the cone of classes of effective divisors.

Classification : 32J27,  14M20,  14E30,  14C20,  14C17,  14C30,  32C30
Mots clés : variété kählérienne, variété hyperkählérienne, cône ample, cône nef, cône pseudo-effectif, classes grandes, cône de Kähler, courant, métrique singulière, décomposition de Zariski, volume d'un fibré en droites, variété uniréglée, courbe mobile
@incollection{SB_2004-2005__47__199_0,
     author = {Debarre, Olivier},
     title = {Classes de cohomologie positives dans les vari\'et\'es k\"ahl\'eriennes compactes},
     booktitle = {S\'eminaire Bourbaki : volume 2004/2005, expos\'es 938-951},
     author = {Collectif},
     series = {Ast\'erisque},
     note = {talk:943},
     pages = {199--228},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {307},
     year = {2006},
     zbl = {1125.32009},
     mrnumber = {2296419},
     language = {fr},
     url = {archive.numdam.org/item/SB_2004-2005__47__199_0/}
}
Debarre, Olivier. Classes de cohomologie positives dans les variétés kählériennes compactes, dans Séminaire Bourbaki : volume 2004/2005, exposés 938-951, Astérisque, no. 307 (2006), Exposé no. 943, pp. 199-228. http://archive.numdam.org/item/SB_2004-2005__47__199_0/

[1] T. Bauer, , A. Küronya & T. Szemberg - “Zariski chambers, volumes, and stable base loci”, J. Reine Angew. Math. 576 (2004), p. 209-233. | MR 2099205 | Zbl 1055.14007

[2] A. Beauville - “Variétés Kähleriennes dont la première classe de Chern est nulle”, J. Differential Geom. 18 (1983), no. 4, p. 755-782 (1984). | MR 730926 | Zbl 0537.53056

[3] P. Biran - “Symplectic packing in dimension 4, Geom. Funct. Anal. 7 (1997), no. 3, p. 420-437. | MR 1466333 | Zbl 0892.53022

[4] -, “From symplectic packing to algebraic geometry and back”, in European Congress of Mathematics, Vol. II (Barcelona, 2000), Progr. Math., vol. 202, Birkhäuser, Basel, 2001, p. 507-524. | MR 1909952 | Zbl 1047.53054

[5] S. Boucksom - “Le cône kählérien d'une variété hyperkählérienne”, C. R. Acad. Sci. Paris Sér. I Math. 333 (2001), no. 10, p. 935-938. | MR 1873811 | Zbl 1068.32014

[6] -, “Cônes positifs des variétés complexes compactes”, Thèse, Grenoble, 2002.

[7] -, “On the volume of a line bundle”, Internat. J. Math. 13 (2002), no. 10, p. 1043-1063. | MR 1945706 | Zbl 1101.14008

[8] -, “Divisorial Zariski decompositions on compact complex manifolds”, Ann. Sci. École Norm. Sup. (4) 37 (2004), no. 1, p. 45-76. | EuDML 82627 | Numdam | MR 2050205 | Zbl 1054.32010

[9] S. Boucksom, J.-P. Demailly, M. Păun & T. Peternell - “The pseudo-effective cone of a compact Kähler manifold and varieties of negative Kodaira dimension”, eprint math.AG/0405285. | Zbl 1267.32017

[10] N. Buchdahl - “On compact Kähler surfaces”, Ann. Inst. Fourier (Grenoble) 49 (1999), no. 1, p. vii, xi, 287-302. | EuDML 75337 | Numdam | MR 1688136 | Zbl 0926.32025

[11] F. Campana & M. Păun - “Une généralisation du théorème de Kobayashi-Ochiai”, eprint math.AG/0506366. | Zbl 1160.32020

[12] F. Campana & T. Peternell - “Algebraicity of the ample cone of projective varieties”, J. Reine Angew. Math. 407 (1990), p. 160-166. | EuDML 153233 | MR 1048532 | Zbl 0728.14004

[13] S. D. Cutkosky - “Zariski decomposition of divisors on algebraic varieties”, Duke Math. J. 53 (1986), no. 1, p. 149-156. | MR 835801 | Zbl 0604.14002

[14] S. D. Cutkosky & V. Srinivas - “On a problem of Zariski on dimensions of linear systems”, Ann. of Math. (2) 137 (1993), no. 3, p. 531-559. | MR 1217347 | Zbl 0822.14006

[15] J.-P. Demailly - “Champs magnétiques et inégalités de Morse pour la d '' -cohomologie”, C. R. Acad. Sci. Paris Sér. I Math. 301 (1985), no. 4, p. 119-122. | MR 799607 | Zbl 0595.58014

[16] -, “Singular Hermitian metrics on positive line bundles”, in Complex algebraic varieties (Bayreuth, 1990), Lecture Notes in Math., vol. 1507, Springer, Berlin, 1992, p. 87-104. | MR 1178721 | Zbl 0784.32024

[17] -, “Algebraic criteria for Kobayashi hyperbolic projective varieties and jet differentials”, in Algebraic geometry-Santa Cruz 1995, Proc. Sympos. Pure Math., vol. 62, Amer. Math. Soc., Providence, RI, 1997, p. 285-360. | MR 1492539 | Zbl 0919.32014

[18] -, “On the geometry of positive cones of projective and Kähler varieties”, in The Fano Conference, Univ. Torino, Turin, 2004, p. 395-422. | Zbl 1071.14013

[19] J.-P. Demailly & M. Păun - “Numerical characterization of the Kähler cone of a compact Kähler manifold”, Ann. of Math. (2) 159 (2004), no. 3, p. 1247-1274. | MR 2113021 | Zbl 1064.32019

[20] J.-P. Demailly, T. Peternell & M. a. Schneider - “Compact complex manifolds with numerically effective tangent bundles”, J. Algebraic Geom. 3 (1994), no. 2, p. 295-345. | MR 1257325 | Zbl 0827.14027

[21] L. Ein, R. Lazarsfeld, M. Mustaţă, M. Nakamaye & M. Popa - “Asymptotic invariants of base loci”, eprint math.AG/0308116, Ann. Inst. Fourier, à paraître. | Numdam | Zbl 1127.14010

[22] T. Fujita - “On Zariski problem”, Proc. Japan Acad. Ser. A Math. Sci. 55 (1979), no. 3, p. 106-110. | MR 531454 | Zbl 0444.14026

[23] J. E. Goodman - “Affine open subsets of algebraic varieties and ample divisors”, Ann. of Math. (2) 89 (1969), p. 160-183. | MR 242843 | Zbl 0159.50504

[24] D. Huybrechts - “Compact hyper-Kähler manifolds : basic results”, Invent. Math. 135 (1999), no. 1, p. 63-113. | MR 1664696 | Zbl 0953.53031

[25] -, “Erratum : “Compact hyper-Kähler manifolds : basic results” [Invent. Math. 135 (1999), no. 1, 63-113 ; MR1664696 (2000a :32039)]”, Invent. Math. 152 (2003), no. 1, p. 209-212. | Zbl 1029.53058

[26] -, “The Kähler cone of a compact hyperkähler manifold”, Math. Ann. 326 (2003), no. 3, p. 499-513. | MR 1992275 | Zbl 1023.14015

[27] Y. Kawamata - “The Zariski decomposition of log-canonical divisors”, in Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), Proc. Sympos. Pure Math., vol. 46, Amer. Math. Soc., Providence, RI, 1987, p. 425-433. | MR 927965 | Zbl 0656.14006

[28] K. Kodaira - “Holomorphic mappings of polydiscs into compact complex manifolds”, J. Differential Geometry 6 (1971/72), p. 33-46. | MR 301228 | Zbl 0227.32008

[29] A. Kouvidakis - “Divisors on symmetric products of curves”, Trans. Amer. Math. Soc. 337 (1993), no. 1, p. 117-128. | MR 1149124 | Zbl 0788.14019

[30] S. J. Kovács - “The cone of curves of a K3 surface”, Math. Ann. 300 (1994), no. 4, p. 681-691. | EuDML 165275 | MR 1314742 | Zbl 0813.14026

[31] A. Lamari - “Courants kählériens et surfaces compactes”, Ann. Inst. Fourier (Grenoble) 49 (1999), no. 1, p. vii, x, 263-285. | EuDML 75335 | Numdam | MR 1688140 | Zbl 0926.32026

[32] -, “Le cône kählérien d'une surface”, J. Math. Pures Appl. (9) 78 (1999), no. 3, p. 249-263. | MR 1687094 | Zbl 0941.32007

[33] R. Lazarsfeld - Positivity in algebraic geometry. I, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 48, Springer-Verlag, Berlin, 2004. | MR 2095471 | Zbl 1093.14500

[34] -, Positivity in algebraic geometry. II, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 49, Springer-Verlag, Berlin, 2004. | MR 2095472 | Zbl 0633.14016

[35] D. Mcduff & L. Polterovich - “Symplectic packings and algebraic geometry”, Invent. Math. 115 (1994), no. 3, p. 405-434. | EuDML 144175 | MR 1262938 | Zbl 0833.53028

[36] Y. Miyaoka & S. Mori - “A numerical criterion for uniruledness”, Ann. of Math. (2) 124 (1986), no. 1, p. 65-69. | MR 847952 | Zbl 0606.14030

[37] M. Nagata - “On the 14-th problem of Hilbert”, Amer. J. Math. 81 (1959), p. 766-772. | Article | MR 105409 | Zbl 0192.13801

[38] M. Nakamaye - “Stable base loci of linear series”, Math. Ann. 318 (2000), no. 4, p. 837-847. | MR 1802513 | Zbl 1063.14008

[39] -, “Base loci of linear series are numerically determined”, Trans. Amer. Math. Soc. 355 (2003), no. 2, p. 551-566 (electronic). | MR 1932713 | Zbl 1017.14017

[40] N. Nakayama - “A counterexample to the Zariski-decomposition conjecture”, Hodge Theory and Algebraic Geometry, Hokkaido Univ., Infinite Analysis Lecture Notes, vol. 19, Kyoto University, 1994, p. 96-101.

[41] -, “Zariski-decomposition and Abundance”, preprint, 1997.

[42] G. Pacienza - “On the nef cone of symmetric products of a generic curve”, Amer. J. Math. 125 (2003), no. 5, p. 1117-1135. | MR 2004430 | Zbl 1056.14042

[43] S. Payne - “Stable base loci, movable curves, and small modifications, for toric varieties”, eprint math.AG/0506622, Math. Zeit., à paraître. | Zbl 1097.14007

[44] P. R. Thie - “The Lelong number of a point of a complex analytic set”, Math. Ann. 172 (1967), p. 269-312. | Article | EuDML 161592 | MR 214812 | Zbl 0158.32804

[45] O. Zariski - “The theorem of Riemann-Roch for high multiples of an effective divisor on an algebraic surface”, Ann. of Math. (2) 76 (1962), p. 560-615. | MR 141668 | Zbl 0124.37001