Manifolds with nef cotangent bundle

Generalising a classical theorem by Ueno, we prove structure results for manifolds with nef or semiample cotangent bundle.Comment: 9 pages, changed metadat

M-regularity of the Fano surface

Let $(A,\Theta)$ be a principally polarised abelian variety, and let Y be a subvariety. Pareschi and Popa conjectured that Y has minimal cohomology class if and only if the structure sheaf of Y satisfies a property that they call M-regularity. Let now X be a smooth cubic threefold. By a classical result due to Clemens and Griffiths, its intermediate Jacobian J(X) is a principally polarised abelian variety; furthermore the Fano surface of lines on X can be embedded in J(X) and has minimal cohomology class. In this short note we show that its structure sheaf is M-regular.Comment: 5 pages, changed metadat

Uniruled varieties with split tangent bundle

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 foliations to study this problem for projective uniruled varieties. In particular we obtain an affirmative answer for rationally connected varieties in any dimension and uniruled varieties in dimension 4.Comment: changed metadat

Geometry of Brill-Noether loci on Prym varieties

Given the Prym variety of an \'etale double cover one can define analogues of the classical Brill-Noether loci on Jacobians of curves. Recent work by Lahoz and Naranjo shows that the Brill-Noether locus $V^2$ completely determines the covering. In this paper we describe the singular locus and the irreducible components of $V^2$.Comment: 19 pages, changed metadat

Adjoint (1,1)(1,1)-classes on threefolds

We answer a question of Filip and Tosatti concerning a basepoint-free theorem for transcendental $(1,1)$-classes on threefolds.Comment: 8 page

The structure of uniruled manifolds with split tangent bundle

In this paper we show that a uniruled manifold with a split tangent bundle admits almost holomorphic fibrations that are related to the splitting. We analyse these fibrations in detail in several special cases, this yields new results about the integrability of the direct factors and the universal covering of the manifold.Comment: 14 page

Non-algebraic compact K\"ahler threefolds admitting endomorphisms

We classify non-algebraic compact K\"ahler threefolds admitting an endomorphism $f: X \to X$ of degree at least two.Comment: 39 pages, changed metadat

Algebraic integrability of foliations with numerically trivial canonical bundle

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 foliations with numerically trivial canonical bundle such that the second Chern class does not vanish. Combined with the recent work of Druel and Greb-Guenancia-Kebekus this establishes the Beauville-Bogomolov decomposition for minimal models with trivial canonical class.Comment: 20 pages, removed an assumption from Thm.1.

Quasi-lines and their degenerations

In this paper we study the structure of manifolds that contain a quasi-line and give some evidence towards the fact that the irreducible components of degenerations of the quasi-line should determine the Mori cone. We show that the minimality with respect to a quasi-line yields strong restrictions on fibre space structures of the manifold.Comment: 20 pages, changed metadat

Singularities of varieties admitting an endomorphism

Let X be a normal variety such that $K_X$ is Q-Cartier, and let $f: X \rightarrow X$ be a finite surjective morphism of degree at least two. We establish a close relation between the irreducible components of the locus of singularities that are not log-canonical and the dynamics of the endomorphism f. As a consequence we prove that if X is projective and f polarised, then X has at most log-canonical singularities.Comment: 15 pages, changed metadat