Numerical and Kodaira dimensions of cotangent bundles

Campana, Frédéric

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

We conjecture the equality of the numerical and Kodaira dimensions ${\nu }_1^{*}\left(X\right)$ and ${\kappa }_1^{*}\left(X\right)$ 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 ${\nu }_1^{*}\left(X\right)={\kappa }_1^{*}\left(X\right)=q^{\prime }\left(X\right)-dim\left(X\right)$, where $q^{\prime }\left(X\right)$ 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 ${\kappa }_1^{*}$ was already introduced and studied by Fumio Sakai in [46], the particular case of the preceding conjecture when ${\kappa }_1^{*}\left(X\right)=-dim\left(X\right)$ was introduced and studied in [31].

Nous conjecturons l’égalité entre les dimensions numérique et de Kodaira ${\nu }_1^{*}\left(X\right)$ et ${\kappa }_1^{*}\left(X\right)$ 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 ${\nu }_1^{*}\left(X\right)={\kappa }_1^{*}\left(X\right)=q^{\prime }\left(X\right)-dim\left(X\right)$, où $q^{\prime }\left(X\right)$ 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 ${\kappa }_1^{*}$ a été introduit et étudié auparavant par Fumio Sakai dans [46], et l’égalité ${\nu }_1^{*}\left(X\right)$ and ${\kappa }_1^{*}\left(X\right)$ conjecturée et étudiée lorsque ${\kappa }_1^{*}\left(X\right)=-dim\left(X\right)$ dans [31].

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