We study the different notions of semipositivity for cohomology classes on 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 surfaces with a nef -divisor which is not semipositive.
Nous étudions les différentes notions de sémipositivité pour les classes de cohomologie 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.
Keywords: K3 surfaces, (1, 1) cohomology classes, smooth semipositive representatives
Mot clés : surfaces K3, classes de cohomologie (1, 1), représentants lisses semi-positifs
Filip, Simion 1; Tosatti, Valentino 2
@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{\textquoteright}institut Fourier}, volume = {68}, number = {7}, year = {2018}, doi = {10.5802/aif.3234}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3234/} }
TY - JOUR AU - Filip, Simion AU - Tosatti, Valentino TI - Smooth and Rough Positive Currents JO - Annales de l'Institut Fourier PY - 2018 SP - 2981 EP - 2999 VL - 68 IS - 7 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3234/ DO - 10.5802/aif.3234 LA - en ID - AIF_2018__68_7_2981_0 ER -
%0 Journal Article %A Filip, Simion %A Tosatti, Valentino %T Smooth and Rough Positive Currents %J Annales de l'Institut Fourier %D 2018 %P 2981-2999 %V 68 %N 7 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3234/ %R 10.5802/aif.3234 %G en %F AIF_2018__68_7_2981_0
Filip, Simion; Tosatti, Valentino. Smooth and Rough Positive Currents. Annales de l'Institut Fourier, Volume 68 (2018) no. 7, pp. 2981-2999. doi : 10.5802/aif.3234. https://aif.centre-mersenne.org/articles/10.5802/aif.3234/
[1] Some numerical criteria for contractability of curves on algebraic surfaces, Am. J. Math., Volume 84 (1962), pp. 485-496 | DOI | MR
[2] Compact complex surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, 4, Springer, 2004, xii+436 pages | DOI | MR
[3] Polynomial diffeomorphisms of . IV. The measure of maximal entropy and laminar currents, Invent. Math., Volume 112 (1993) no. 1, pp. 77-125 | DOI | MR
[4] The Dirichlet problem for a complex Monge–Ampère equation, Invent. Math., Volume 37 (1976) no. 1, pp. 1-44 | DOI | MR
[5] Fine topology, Šilov boundary, and , J. Funct. Anal., Volume 72 (1987) no. 2, pp. 225-251 | DOI | MR
[6] Existence of minimal models for varieties of log general type, J. Am. Math. Soc., Volume 23 (2010) no. 2, pp. 405-468 | DOI | MR
[7] Divisorial Zariski decompositions on compact complex manifolds, Ann. Sci. Éc. Norm. Supér., Volume 37 (2004) no. 1, pp. 45-76 | DOI | MR
[8] Monge–Ampère equations in big cohomology classes, Acta Math., Volume 205 (2010) no. 2, pp. 199-262 | DOI | MR
[9] Dynamique des automorphismes des surfaces complexes compactes, École Normale Superieure de Lyon (France) (1999) (Ph. D. Thesis)
[10] Dynamique des automorphismes des surfaces , Acta Math., Volume 187 (2001) no. 1, pp. 1-57 | DOI | MR
[11] Automorphisms of surfaces: Kummer rigidity and measure of maximal entropy (2015) (https://hal.archives-ouvertes.fr/hal-01071491)
[12] Kähler currents and null loci, Invent. Math., Volume 202 (2015) no. 3, pp. 1167-1198 | DOI | MR
[13] Complex Analytic and Differential Geometry (https://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/agbook.pdf)
[14] Mesures de Monge–Ampère et caractérisation géométrique des variétés algébriques affines, Mém. Soc. Math. Fr., Nouv. Sér. (1985) no. 19, 124 pages (124 p.) | DOI | MR | Zbl
[15] Regularization of closed positive currents and intersection theory, J. Algebr. Geom., Volume 1 (1992) no. 3, pp. 361-409 | MR
[16] Singular Hermitian metrics on positive line bundles, Complex algebraic varieties (Bayreuth, 1990) (Lecture Notes in Mathematics), Volume 1507, Springer, 1992, pp. 87-104 | DOI | MR
[17] Numerical characterization of the Kähler cone of a compact Kähler manifold, Ann. Math., Volume 159 (2004) no. 3, pp. 1247-1274 | DOI | MR
[18] Compact complex manifolds with numerically effective tangent bundles, J. Algebr. Geom., Volume 3 (1994) no. 2, pp. 295-345 | MR
[19] Regularity of dynamical Green’s functions, Trans. Am. Math. Soc., Volume 361 (2009) no. 9, pp. 4783-4805 | DOI | MR
[20] Pluripotential estimates on compact Hermitian manifolds, Advances in geometric analysis (Advanced Lectures in Mathematics (ALM)), Volume 21, International Press, 2012, pp. 69-86 | MR | Zbl
[21] Green currents for holomorphic automorphisms of compact Kähler manifolds, J. Am. Math. Soc., Volume 18 (2005) no. 2, pp. 291-312 | DOI | MR
[22] Rigidity of Julia sets for Hénon type maps, J. Mod. Dyn., Volume 8 (2014) no. 3-4, pp. 499-548 | DOI | MR
[23] Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, 153, Springer, 1969, xiv+676 pages | DOI | MR
[24] Kummer rigidity for K3 surface automorphisms via Ricci-flat metrics (2018) (https://arxiv.org/abs/1808.08673)
[25] 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) | DOI | MR | Zbl
[26] Kählerian normal complex surfaces, Tôhoku Math. J., Volume 35 (1983) no. 1, pp. 101-117 | DOI | MR
[27] Semipositive line bundles, J. Fac. Sci., Univ. Tokyo, Sect. I A, Volume 30 (1983) no. 2, pp. 353-378 | MR
[28] Über Modifikationen und exzeptionelle analytische Mengen, Math. Ann., Volume 146 (1962), pp. 331-368 | DOI | MR
[29] On the entropy of holomorphic maps, Enseign. Math., Volume 49 (2003) no. 3-4, pp. 217-235 | DOI | MR | Zbl
[30] Collapsing of abelian fibered Calabi-Yau manifolds, Duke Math. J., Volume 162 (2013) no. 3, pp. 517-551 | DOI | MR
[31] Remarks on the collapsing of torus fibered Calabi-Yau manifolds, Bull. Lond. Math. Soc., Volume 47 (2015) no. 6, pp. 1021-1027 | DOI | MR
[32] Adjoint -classes on threefolds (2018) (https://arxiv.org/abs/1807.08442)
[33] Minimal models for Kähler threefolds, Invent. Math., Volume 203 (2016) no. 1, pp. 217-264 | DOI | MR
[34] 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
[35] 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 | DOI | MR
[36] Le cône kählérien d’une surface, J. Math. Pures Appl., Volume 78 (1999) no. 3, pp. 249-263 | DOI | MR | Zbl
[37] Positivity in algebraic geometry. I. Classical setting: line bundles and linear series, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., 48, Springer, 2004, xviii+387 pages | DOI | MR
[38] Nef line bundles on Calabi-Yau threefolds, I (2018) (https://arxiv.org/abs/1601.01273)
[39] The metric entropy of diffeomorphisms. I. Characterization of measures satisfying Pesin’s entropy formula, Ann. Math., Volume 122 (1985) no. 3, pp. 509-539 | DOI | MR | Zbl
[40] The metric entropy of diffeomorphisms. II. Relations between entropy, exponents and dimension, Ann. Math., Volume 122 (1985) no. 3, pp. 540-574 | DOI | MR
[41] Dynamics on surfaces: Salem numbers and Siegel disks, J. Reine Angew. Math., Volume 545 (2002), pp. 201-233 | DOI | MR
[42] Singular Kählerian spaces, Manifolds (Tokyo, 1973), University of Tokyo Press, 1975, pp. 343-351 | MR
[43] 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 | Zbl
[44] Limits of Calabi-Yau metrics when the Kähler class degenerates, J. Eur. Math. Soc., Volume 11 (2009) no. 4, pp. 755-776 | DOI | MR
[45] Degenerations of Calabi-Yau metrics, Acta Phys. Polon. B Proc. Suppl., Volume 4 (2011) no. 3, pp. 495-505 | DOI
[46] Calabi-Yau manifolds and their degenerations, Ann. N.Y. Acad. Sci., Volume 1260 (2012) no. 1, pp. 8-13 | DOI
[47] KAWA lecture notes on the Kähler-Ricci flow, Ann. Fac. Sci. Toulouse, Math., Volume 27 (2018) no. 2, pp. 285-376 | DOI
[48] 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 | DOI
[49] The Chern-Ricci flow on complex surfaces, Compos. Math., Volume 149 (2013) no. 12, pp. 2101-2138 | DOI | MR
[50] The Kähler-Ricci flow, Ricci-flat metrics and collapsing limits, Am. J. Math., Volume 140 (2018) no. 3, pp. 653-698 | DOI | Zbl
[51] 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 | DOI | Zbl
[52] The Kähler cone on Calabi-Yau threefolds, Invent. Math., Volume 107 (1992) no. 3, pp. 561-583 | DOI | MR | Zbl
[53] Volume growth and entropy, Isr. J. Math., Volume 57 (1987) no. 3, pp. 285-300 | DOI | MR | Zbl
[54] The theorem of Riemann-Roch for high multiples of an effective divisor on an algebraic surface, Ann. Math., Volume 76 (1962), pp. 560-615 | DOI | MR
Cited by Sources: