Stéphane Druel

Cohomological characterizations of projective spaces and hyperquadrics

Projective spaces and hyperquadrics are the simplest projective algebraic varieties, and they can be characterized in many ways. The aim of this paper is to provide a new characterization of them in terms of positivity properties of the tangent bundle (Theorem 1.1). The first result in this direction was Mori’s proof of the Hartshorne conjecture in [Mor79] (see also Siu and Yau [SY80]), that characterizes projective spaces as the only manifolds having ample tangent bundle. Then, in [Wah83], Wahl characterized projective spaces as the only manifolds whose tangent bundles contain ample invertible subsheaves. Interpolating Mori’s and Wahl’s results, Andreatta and Wiśniewski gave the following characterization:

A decomposition theorem for singular spaces with trivial canonical class of dimension at most five

In this paper we partly extend the Beauville–Bogomolov decomposition theorem to the singular setting. We show that any complex projective variety of dimension at most five with canonical singularities and numerically trivial canonical class admits a finite cover, étale in codimension one, that decomposes as a product of an Abelian variety, and singular analogues of irreducible Calabi–Yau and irreducible holomorphic symplectic varieties.