Numerical and kodaira dimensions of cotangent bundles

TL;DR

This paper explores the conjectured equality of numerical and Kodaira dimensions for cotangent bundles of compact Kähler manifolds, providing evidence for special cases and connecting to classical results and recent conjectures.

Contribution

It extends the understanding of the relationship between numerical and Kodaira dimensions to cotangent bundles, proving the conjecture for certain classes of manifolds and relating it to the Beauville-Bogomolov decomposition.

Findings

Equality holds for rationally connected manifolds.

Equality holds for resolutions of varieties with klt singularities and trivial first Chern class.

Conjecture extends to 'special' manifolds, generalizing previous results.

Abstract

We conjecture the equality of the numerical and Kodaira dimensions $\nu_1^*(X)$ and $\kappa_1^*(X)$ for the cotangent bundle of compact K\"ahler 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 $\nu_1^*(X)=\kappa_1^*(X)=q'(X)-dim(X)$, where $q'(X)$ is the maximal irregularity of a finite \'etale 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 [43], the particular case of the preceding conjecture when $\kappa_1^*(X)=-dim(X)$ was introduced and studied in [29].