Smooth and Rough Positive Currents

Filip, Simion; Tosatti, Valentino

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 $ℝ$-divisor which is not semipositive.

Classification : 14J28, 32Q25, 37F10, 14J50, 32J15
Keywords: K3 surfaces, (1, 1) cohomology classes, smooth semipositive representatives
Mot clés : surfaces K3, classes de cohomologie (1, 1), représentants lisses semi-positifs

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