Smooth and Rough Positive Currents  [ Courants Positifs Lisses et Peu Réguliers ]
Annales de l'Institut Fourier, Tome 68 (2018) no. 7, pp. 2981-2999.

Nous étudions les différentes notions de sémipositivité pour les classes de cohomologie (1,1) sur les surfaces K3. Nous montrons d’abord que chaque classe big et nef (et chaque classe nef et rationnelle) est semi-ample, et en particulier elle contient un représentant lisse semi-positif. En revanche, nous montrons qu’il existe des classes nef irrationnelles qui ne contiennent pas de courants positifs fermés lisses en dehors d’un sous-ensemble analytique, et nous répondons négativement à deux questions du deuxième auteur. En utilisant des résultats de Cantat et Dupont, nous construisons également des exemples de surfaces K3 projectives avec un R-diviseur nef mais non semi-positif.

We study the different notions of semipositivity for (1,1) cohomology classes on K3 surfaces. We first show that every big and nef class (and every nef and rational class) is semiample, and in particular it contains a smooth semipositive representative. By contrast, we show that there exist irrational nef classes with no closed positive current representative which is smooth outside a proper analytic subset. We use this to answer negatively two questions of the second-named author. Using a result of Cantat & Dupont, we also construct examples of projective K3 surfaces with a nef -divisor which is not semipositive.

Publié le : 2019-05-24
DOI : https://doi.org/10.5802/aif.3234
Classification : 14J28,  32Q25,  37F10,  14J50,  32J15
Mots clés : surfaces K3, classes de cohomologie (1,1), représentants lisses semi-positifs
@article{AIF_2018__68_7_2981_0,
     author = {Filip, Simion and Tosatti, Valentino},
     title = {Smooth and Rough Positive Currents},
     journal = {Annales de l'Institut Fourier},
     pages = {2981--2999},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {68},
     number = {7},
     year = {2018},
     doi = {10.5802/aif.3234},
     language = {en},
     url = {www.numdam.org/item/AIF_2018__68_7_2981_0/}
}
Filip, Simion; Tosatti, Valentino. Smooth and Rough Positive Currents. Annales de l'Institut Fourier, Tome 68 (2018) no. 7, pp. 2981-2999. doi : 10.5802/aif.3234. http://www.numdam.org/item/AIF_2018__68_7_2981_0/

[1] Artin, Michael Some numerical criteria for contractability of curves on algebraic surfaces, Am. J. Math., Volume 84 (1962), pp. 485-496 | Article | MR 0146182

[2] Barth, Wolf P.; Hulek, Klaus; Peters, Chris A. M.; Van de Ven, Antonius Compact complex surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, Volume 4, Springer, 2004, xii+436 pages | Article | MR 2030225

[3] Bedford, Eric; Lyubich, Mikhail; Smillie, John Polynomial diffeomorphisms of C 2 . IV. The measure of maximal entropy and laminar currents, Invent. Math., Volume 112 (1993) no. 1, pp. 77-125 | Article | MR 1207478

[4] Bedford, Eric; Taylor, Bert A. The Dirichlet problem for a complex Monge–Ampère equation, Invent. Math., Volume 37 (1976) no. 1, pp. 1-44 | Article | MR 0445006

[5] Bedford, Eric; Taylor, Bert A. Fine topology, Šilov boundary, and (dd c ) n , J. Funct. Anal., Volume 72 (1987) no. 2, pp. 225-251 | Article | MR 886812

[6] Birkar, Caucher; Cascini, Paolo; Hacon, Christopher D.; McKernan, James Existence of minimal models for varieties of log general type, J. Am. Math. Soc., Volume 23 (2010) no. 2, pp. 405-468 | Article | MR 2601039

[7] Boucksom, Sébastien Divisorial Zariski decompositions on compact complex manifolds, Ann. Sci. Éc. Norm. Supér., Volume 37 (2004) no. 1, pp. 45-76 | Article | MR 2050205

[8] Boucksom, Sébastien; Eyssidieux, Philippe; Guedj, Vincent; Zeriahi, Ahmed Monge–Ampère equations in big cohomology classes, Acta Math., Volume 205 (2010) no. 2, pp. 199-262 | Article | MR 2746347

[9] Cantat, Serge Dynamique des automorphismes des surfaces complexes compactes (1999) (Ph. D. Thesis)

[10] Cantat, Serge Dynamique des automorphismes des surfaces K3, Acta Math., Volume 187 (2001) no. 1, pp. 1-57 | Article | MR 1864630

[11] Cantat, Serge; Dupont, Christophe Automorphisms of surfaces: Kummer rigidity and measure of maximal entropy (2015) (https://hal.archives-ouvertes.fr/hal-01071491)

[12] Collins, Tristan C.; Tosatti, Valentino Kähler currents and null loci, Invent. Math., Volume 202 (2015) no. 3, pp. 1167-1198 | Article | MR 3425388

[13] Demailly, Jean-Pierre Complex Analytic and Differential Geometry (https://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/agbook.pdf)

[14] Demailly, Jean-Pierre Mesures de Monge–Ampère et caractérisation géométrique des variétés algébriques affines, Mém. Soc. Math. Fr., Nouv. Sér. (1985) no. 19, 124 pages (124 p.) | Article | MR 813252 | Zbl 0579.32012

[15] Demailly, Jean-Pierre Regularization of closed positive currents and intersection theory, J. Algebr. Geom., Volume 1 (1992) no. 3, pp. 361-409 | MR 1158622

[16] Demailly, Jean-Pierre Singular Hermitian metrics on positive line bundles, Complex algebraic varieties (Bayreuth, 1990) (Lecture Notes in Mathematics) Volume 1507, Springer, 1992, pp. 87-104 | Article | MR 1178721

[17] Demailly, Jean-Pierre; Paun, Mihai Numerical characterization of the Kähler cone of a compact Kähler manifold, Ann. Math., Volume 159 (2004) no. 3, pp. 1247-1274 | Article | MR 2113021

[18] Demailly, Jean-Pierre; Peternell, Thomas; Schneider, Michael Compact complex manifolds with numerically effective tangent bundles, J. Algebr. Geom., Volume 3 (1994) no. 2, pp. 295-345 | MR 1257325

[19] Diller, Jeffrey; Guedj, Vincent Regularity of dynamical Green’s functions, Trans. Am. Math. Soc., Volume 361 (2009) no. 9, pp. 4783-4805 | Article | MR 2506427

[20] Dinew, Sławomir; Kołodziej, Sławomir Pluripotential estimates on compact Hermitian manifolds, Advances in geometric analysis (Advanced Lectures in Mathematics (ALM)) Volume 21, International Press, 2012, pp. 69-86 | MR 3077248 | Zbl 1317.32066

[21] Dinh, Tien-Cuong; Sibony, Nessim Green currents for holomorphic automorphisms of compact Kähler manifolds, J. Am. Math. Soc., Volume 18 (2005) no. 2, pp. 291-312 | Article | MR 2137979

[22] Dinh, Tien-Cuong; Sibony, Nessim Rigidity of Julia sets for Hénon type maps, J. Mod. Dyn., Volume 8 (2014) no. 3-4, pp. 499-548 | Article | MR 3345839

[23] Federer, Herbert Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Volume 153, Springer, 1969, xiv+676 pages | Article | MR 0257325

[24] Filip, Simion; Tosatti, Valentino Kummer rigidity for K3 surface automorphisms via Ricci-flat metrics (2018) (https://arxiv.org/abs/1808.08673)

[25] Fornæss, John Erik; Sibony, Nessim Complex dynamics in higher dimensions, Complex potential theory (Montreal, 1993) (NATO ASI Series. Series C. Mathematical and Physical Sciences) Volume 439, Kluwer Academic Publishers, 1994, pp. 131-186 (Notes partially written by Estela A. Gavosto) | Article | MR 1332961 | Zbl 0811.32019

[26] Fujiki, Akira Kählerian normal complex surfaces, Tôhoku Math. J., Volume 35 (1983) no. 1, pp. 101-117 | Article | MR 695662

[27] Fujita, Takao Semipositive line bundles, J. Fac. Sci., Univ. Tokyo, Sect. I A, Volume 30 (1983) no. 2, pp. 353-378 | MR 722501

[28] Grauert, Hans Über Modifikationen und exzeptionelle analytische Mengen, Math. Ann., Volume 146 (1962), pp. 331-368 | Article | MR 0137127

[29] Gromov, Mikhaïl On the entropy of holomorphic maps, Enseign. Math., Volume 49 (2003) no. 3-4, pp. 217-235 | Article | MR 2026895 | Zbl 1080.37051

[30] Gross, Mark; Tosatti, Valentino; Zhang, Yuguang Collapsing of abelian fibered Calabi-Yau manifolds, Duke Math. J., Volume 162 (2013) no. 3, pp. 517-551 | Article | MR 3024092

[31] Hein, Hans-Joachim; Tosatti, Valentino Remarks on the collapsing of torus fibered Calabi-Yau manifolds, Bull. Lond. Math. Soc., Volume 47 (2015) no. 6, pp. 1021-1027 | Article | MR 3431582

[32] Höring, Andreas Adjoint (1,1)-classes on threefolds (2018) (https://arxiv.org/abs/1807.08442)

[33] Höring, Andreas; Peternell, Thomas Minimal models for Kähler threefolds, Invent. Math., Volume 203 (2016) no. 1, pp. 217-264 | Article | MR 3437871

[34] Kawamata, Yujiro; Matsuda, Katsumi; Matsuki, Kenji Introduction to the minimal model problem, Algebraic geometry (Sendai, 1985) (Advanced Studies in Pure Mathematics) Volume 10, North-Holland, 1987, pp. 283-360 http://faculty.ms.u-tokyo.ac.jp/~kawamata/kmm.pdf | MR 946243

[35] Koike, Takayuki On minimal singular metrics of certain class of line bundles whose section ring is not finitely generated, Ann. Inst. Fourier, Volume 65 (2015) no. 5, pp. 1953-1967 | Article | MR 3449202

[36] Lamari, Ahcène Le cône kählérien d’une surface, J. Math. Pures Appl., Volume 78 (1999) no. 3, pp. 249-263 | Article | MR 1687094 | Zbl 0941.32007

[37] Lazarsfeld, Robert Positivity in algebraic geometry. I. Classical setting: line bundles and linear series, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., Volume 48, Springer, 2004, xviii+387 pages | Article | MR 2095471

[38] Lazić, Vladimir; Oguiso, Keiji; Peternell, Thomas Nef line bundles on Calabi-Yau threefolds, I (2018) (https://arxiv.org/abs/1601.01273)

[39] Ledrappier, François; Young, Lai-Sang The metric entropy of diffeomorphisms. I. Characterization of measures satisfying Pesin’s entropy formula, Ann. Math., Volume 122 (1985) no. 3, pp. 509-539 | Article | MR 819556 | Zbl 0605.58028

[40] Ledrappier, François; Young, Lai-Sang The metric entropy of diffeomorphisms. II. Relations between entropy, exponents and dimension, Ann. Math., Volume 122 (1985) no. 3, pp. 540-574 | Article | MR 819557

[41] McMullen, Curtis T. Dynamics on K3 surfaces: Salem numbers and Siegel disks, J. Reine Angew. Math., Volume 545 (2002), pp. 201-233 | Article | MR 1896103

[42] Moishezon, Boris G. Singular Kählerian spaces, Manifolds (Tokyo, 1973), University of Tokyo Press, 1975, pp. 343-351 | MR 0379909

[43] Reid, Miles Chapters on algebraic surfaces, Complex algebraic geometry (Park City, 1993) (IAS/Park City Mathematics Series) Volume 3, American Mathematical Society, 1997, pp. 3-159 | MR 1442522 | Zbl 0910.14016

[44] Tosatti, Valentino Limits of Calabi-Yau metrics when the Kähler class degenerates, J. Eur. Math. Soc., Volume 11 (2009) no. 4, pp. 755-776 | Article | MR 2538503

[45] Tosatti, Valentino Degenerations of Calabi-Yau metrics, Acta Phys. Polon. B Proc. Suppl., Volume 4 (2011) no. 3, pp. 495-505 | Article

[46] Tosatti, Valentino Calabi-Yau manifolds and their degenerations, Ann. N.Y. Acad. Sci., Volume 1260 (2012) no. 1, pp. 8-13 | Article

[47] Tosatti, Valentino KAWA lecture notes on the Kähler-Ricci flow, Ann. Fac. Sci. Toulouse, Math., Volume 27 (2018) no. 2, pp. 285-376 | Article

[48] Tosatti, Valentino Nakamaye’s theorem on complex manifolds, Algebraic geometry (Salt Lake City, 2015). Part 1 (Proceedings of Symposia in Pure Mathematics) Volume 97.1, American Mathematical Society, 2018, pp. 633-655 | Article

[49] Tosatti, Valentino; Weinkove, Ben The Chern-Ricci flow on complex surfaces, Compos. Math., Volume 149 (2013) no. 12, pp. 2101-2138 | Article | MR 3143707

[50] Tosatti, Valentino; Weinkove, Ben; Yang, Xiaokui The Kähler-Ricci flow, Ricci-flat metrics and collapsing limits, Am. J. Math., Volume 140 (2018) no. 3, pp. 653-698 | Article | Zbl 06924834

[51] Tosatti, Valentino; Zhang, Yuguang Finite time collapsing of the Kähler-Ricci flow on threefolds, Ann. Sc. Norm. Super. Pisa, Cl. Sci., Volume 18 (2018) no. 1, pp. 105-118 | Article | Zbl 1391.53080

[52] Wilson, Pelham M. H. The Kähler cone on Calabi-Yau threefolds, Invent. Math., Volume 107 (1992) no. 3, pp. 561-583 | Article | MR 1150602 | Zbl 0766.14035

[53] Yomdin, Yosef Volume growth and entropy, Isr. J. Math., Volume 57 (1987) no. 3, pp. 285-300 | Article | MR 889979 | Zbl 0641.54036

[54] Zariski, Oscar The theorem of Riemann-Roch for high multiples of an effective divisor on an algebraic surface, Ann. Math., Volume 76 (1962), pp. 560-615 | Article | MR 0141668