To sing, to brag, to blaze, to grumble . . . Abstract.We prove that for a compact Kähler threefold with canonical singularities and vanishing first Chern class, the projective fibres are dense in the semiuniversal deformation space. This implies that every Kähler threefold of Kodaira dimension zero admits small projective deformations after a suitable bimeromorphic modification. As a corollary, we see that the fundamental group of any Kähler threefold is a quotient of an extension of fundamental groups of projective manifolds, up to subgroups of finite index.In the course of the proof, we show that for a canonical threefold with c 1 = 0, the Albanese map decomposes as a product after a finiteétale base change. This generalizes a result of Kawamata, valid in all dimensions, to the Kähler case. Furthermore we generalize a Hodge-theoretic criterion for algebraic approximability, due to Green and Voisin, to quotients of a manifold by a finite group.

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Abstract. Let (X, D) be a projective log canonical pair. We show that for any natural number p, the sheaf Ω p X (log⌊D⌋) * * of reflexive logarithmic pforms does not contain a Weil divisorial subsheaf whose Kodaira-Iitaka dimension exceeds p. This generalizes a classical theorem of Bogomolov and Sommese.In fact, we prove a more general version of this result which also deals with the orbifoldes géométriques introduced by Campana. The main ingredients to the proof are the extension theorem of Greb-Kebekus-Kovács-Peternell, a new version of the Negativity lemma, the minimal model program, and a residue map for symmetric differentials on dlt pairs.We also give an example showing that the statement cannot be generalized to spaces with Du Bois singularities. As an application, we give a KodairaAkizuki-Nakano-type vanishing result for log canonical pairs which holds for reflexive as well as for Kähler differentials.

Deswegen zurück zum echten Leben. In dem, wenn uns Ruhe umgibt, eine Erinnerung hochkommen kann. Und die kündigt sich leise an, wiederholt sich ein paar Mal und man fragt sich: ist das jetzt wirklich so gewesen oder doch anders? Aber es war schmerzhaft, diese Erinnerung. Und die kommt immer näher, und wird immer erlebbarer, und dann ist es plötzlich so, als wäre es ganz aktuell, als würde es wieder durch einen durchgehen im Hier und Jetzt.

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.