Kähler spaces with zero first Chern class: Bochner principle, fundamental groups, and the Kodaira problem

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 ten-sors 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 Calabi-Yau or irreducible holomorphic symplectic. Several applications of these results are given, the strongest ones of which apply in dimension four. In this case, we prove Campana's Abelianity Conjecture for X. If in addition X has only algebraic singularities then a cover of X admits small projective deformations and the Beauville-Bogomolov Decomposition Theorem holds for X.

Domaines: Géométrie algébrique [math.AG]

Benoît Claudon, Patrick Graf, Henri Guenancia, Philipp Naumann. Kähler spaces with zero first Chern class: Bochner principle, fundamental groups, and the Kodaira problem. Journal für die reine und angewandte Mathematik, 2022, ⟨10.1515/crelle-2022-0001⟩. ⟨hal-02936352v1⟩