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
CC-BY-ND 4.0
@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/
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