Kähler spaces with zero first Chern class: Bochner principle, Albanese map and fundamental groups

Benoît Claudon (1) , Patrick Graf (2) , Henri Guenancia (3) , Philipp Naumann (2)

1 IRMAR - Institut de Recherche Mathématique de Rennes
2 Mathematisches Institut [Bayreuth]
3 IMT - Institut de Mathématiques de Toulouse UMR5219

Let $X$ be a compact Kähler space with klt singularities and vanishing first Chern class. We prove the Bochner principle for holomorphic tensors on the smooth locus of $X$: any such tensor is parallel with respect to the singular Ricci-flat metrics. As a consequence, after a finite quasi-étale cover $X$ splits off a complex torus of the maximum possible dimension. We then proceed to decompose the tangent sheaf of $X$ according to its holonomy representation. In particular, we classify those $X$ which have strongly stable tangent sheaf: up to quasi-étale covers, these are either irreducible Calabi--Yau or irreducible holomorphic symplectic. As an application of these results, we show that if $X$ has dimension four, then it satisfies Campana's Abelianity Conjecture.

Domaines: Géométrie algébrique [math.AG]

Benoît Claudon, Patrick Graf, Henri Guenancia, Philipp Naumann. Kähler spaces with zero first Chern class: Bochner principle, Albanese map and fundamental groups. Journal für die reine und angewandte Mathematik, 2022, 786, pp.245-275. ⟨10.1515/crelle-2022-0001⟩. ⟨hal-02936352v2⟩