Numerical and kodaira dimensions of cotangent bundles

Abstract

We conjecture the equality of the numerical and Kodaira dimensions ${\nu }_1^{\ast }(X)$ and ${\kappa }_1^{\ast }(X)$ 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 ${\nu }_1^{\ast }(X)={\kappa }_1^{\ast }(X)=q^{\prime }(X)-dim(X)$, where $q^{\prime }(X)$ 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 ${\kappa }_1^{\ast }$ was already introduced and studied by Fumio Sakai in [43], the particular case of the preceding conjecture when ${\kappa }_1^{\ast }(X)=-dim(X)$ was introduced and studied in [29].