Numerical and Kodaira dimensions of cotangent bundles

Frédéric Campana

doi:10.5802/afst.1780

Frédéric Campana ¹

¹ Université Lorraine, Institut Elie Cartan, Nancy, France

Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 33 (2024) no. 2, pp. 523-573.

Abstract
Abstract

We conjecture the equality of the numerical and Kodaira dimensions $ν_{1}^{*} (X)$ and $κ_{1}^{*} (X)$ for the cotangent bundle of compact Kähler manifolds $X$ , generalising the classical case of the canonical bundle. We show or reduce it to the classical case of the canonical bundle for some peculiar manifolds: among them, the rationally connected ones, or resolutions of varieties with klt singularities and trivial first Chern class, in which case we show that $ν_{1}^{*} (X) = κ_{1}^{*} (X) = q^{'} (X) - dim (X)$ , where $q^{'} (X)$ is the maximal irregularity of a finite étale cover of $X$ . The proof rests on the Beauville–Bogomolov decomposition, and a direct computation for smooth models of quotients $A / G$ of complex tori by finite groups. We conjecture that these equalities hold true, much more generally, when $X$ is “special”. The invariant $κ_{1}^{*}$ was already introduced and studied by Fumio Sakai in [46], the particular case of the preceding conjecture when $κ_{1}^{*} (X) = - dim (X)$ was introduced and studied in [31].

Nous conjecturons l’égalité entre les dimensions numérique et de Kodaira $ν_{1}^{*} (X)$ et $κ_{1}^{*} (X)$ pour le fibré cotangent des variétés Kählériennes compactes $X$ , généralisant le cas classique du fibré canonique. Nous la démontrons ou la réduisons au cas classique du fibré canonique pour certaines classes de variétés, parmi lesquelles : les variétés rationnellement connexes, ainsi que les modèles lisses des variétés à singularités klt et première classe de Chern triviale, pour lesquelles nous montrons que $ν_{1}^{*} (X) = κ_{1}^{*} (X) = q^{'} (X) - dim (X)$ , où $q^{'} (X)$ est l’irrégularité maximale des revêtements étales de $X$ . La preuve repose sur la décomposition de Bogomolov–Beauville, et un calcul direct pour les modèles lisses des quotients $A / G$ de tores complexes par un groupe fini. Nous conjecturons que ces égalités restent vraies, bien plus généralement, lorsque $X$ est « spéciale ». L’invariant $κ_{1}^{*}$ a été introduit et étudié auparavant par Fumio Sakai dans [46], et l’égalité $ν_{1}^{*} (X)$ and $κ_{1}^{*} (X)$ conjecturée et étudiée lorsque $κ_{1}^{*} (X) = - dim (X)$ dans [31].

Received: 2022-03-09
Accepted: 2022-11-09
Published online: 2024-09-23

DOI: 10.5802/afst.1780

Author's affiliations:

Frédéric Campana ¹

¹ Université Lorraine, Institut Elie Cartan, Nancy, France

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {Fr\'ed\'eric Campana},
     title = {Numerical and {Kodaira} dimensions of cotangent bundles},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {523--573},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
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Frédéric Campana. Numerical and Kodaira dimensions of cotangent bundles. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 33 (2024) no. 2, pp. 523-573. doi : 10.5802/afst.1780. https://afst.centre-mersenne.org/articles/10.5802/afst.1780/

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