Smooth and Rough Positive Currents
Annales de l'Institut Fourier, Volume 68 (2018) no. 7, p. 2981-2999

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.

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.

Published online : 2019-05-24
DOI : https://doi.org/10.5802/aif.3234
Classification:  14J28,  32Q25,  37F10,  14J50,  32J15
Keywords: K3 surfaces, (1,1) cohomology classes, smooth semipositive representatives
@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},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {68},
     number = {7},
     year = {2018},
     pages = {2981-2999},
     doi = {10.5802/aif.3234},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2018__68_7_2981_0}
}
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/item/AIF_2018__68_7_2981_0/

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