2022/03/07 by F. Campana, Campana, Frederic Bruno
Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Complex Variables (math.CV) #FOS: Mathematics #Geometric and Algebraic Topology #Geometry and complex manifolds
paper · pdf · doi:10.48550/arxiv.2203.03273
openalex publication_date 2022/03/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We conjecture the equality of the numerical and Kodaira dimensions ν1^*(X) and κ1^*(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 ν1^*(X)=κ1^*(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 κ1^* was already introduced and studied by Fumio Sakai in [43], the particular case of the preceding conjecture when κ1^*(X)=-dim(X) was introduced and studied in [29].