Projectively flat klt varieties
Journal de l’École polytechnique — Mathématiques, Volume 8 (2021), pp. 1005-1036.

In the context of uniformisation problems, we study projective varieties with klt singularities whose cotangent sheaf admits a projectively flat structure over the smooth locus. Generalising work of Jahnke-Radloff, we show that torus quotients are the only klt varieties with semistable cotangent sheaf and extremal Chern classes. An analogous result for varieties with nef normalised cotangent sheaves follows.

Dans le cadre des problèmes d’uniformisation, nous étudions les variétés projectives avec singularités klt dont le faisceau cotangent admet une structure projective plate sur le lieu lisse. En généralisant le travail de Jahnke-Radloff, nous montrons que les quotients des tores sont les seules variétés klt avec un faisceau cotangent semi-stable et des classes de Chern extrémales. Un résultat analogue pour les variétés avec un faisceau cotangent normalisé nef s’ensuit.

Received:
Accepted:
Published online:
DOI: 10.5802/jep.164
Classification: 32Q30,  32Q26,  14E20,  14E30,  53B10
Keywords: Bogomolov-Gieseker inequality, Abelian variety, klt singularities, Miyaoka-Yau inequality, stability, projective flatness, uniformisation
Daniel Greb 1; Stefan Kebekus 2, 3; Thomas Peternell 4

1 Essener Seminar für Algebraische Geometrie und Arithmetik, Fakultät für Mathematik, Universität Duisburg–Essen 45117 Essen, Germany
2 & Freiburg Institute for Advanced Studies (FRIAS) Freiburg im Breisgau, Germany
3 Mathematisches Institut, Albert-Ludwigs-Universität Freiburg Ernst-Zermelo-Straße 1, 79104 Freiburg im Breisgau, Germany
4 Mathematisches Institut, Universität Bayreuth 95440 Bayreuth, Germany
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Daniel Greb; Stefan Kebekus; Thomas Peternell. Projectively flat klt varieties. Journal de l’École polytechnique — Mathématiques, Volume 8 (2021), pp. 1005-1036. doi : 10.5802/jep.164. https://jep.centre-mersenne.org/articles/10.5802/jep.164/

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