Given a reflexive sheaf on a mildly singular projective variety, we prove a flatness criterion under certain stability conditions. This implies the algebraicity of leaves for sufficiently stable… Expand

Beauville asked if a compact K\"ahler manifold with split tangent bundle has a universal covering that is a product of manifolds. We use Mori theory and elementary results about holomorphic… Expand

Let X be a projective manifold of dimension n. Beltrametti and Sommese conjectured that if A is an ample divisor such that $K_X+(n-1)A$ is nef, then $K_X+(n-1)A$ has non-zero global sections. We… Expand

Let $$X$$X be a normal variety such that $$K_X$$KX is $$\mathbb {Q}$$Q-Cartier, and let $$f:X \rightarrow X$$f:X→X be a finite surjective morphism of degree at least two. We establish a close… Expand

These are extended notes of the talks I gave at the workshop “Rencontre positivité” in Rennes. The aim of these talks was to illustrate the interaction between the geometry of a fibration and the… Expand

Using the minimal model program, it is proved that if the sectional genus is zero the Δ-genus is also zero, which leads to a birational classification of quasi-polarised varieties with sectionsal genus zero.Expand

Let X$X$ be a Fano manifold. While the properties of the anticanonical divisor −KX$-K_X$ and its multiples have been studied by many authors, the positivity of the tangent bundle TX$T_X$ is much more… Expand