Andreas Höring

On a conjecture of Beltrametti and Sommese

Let X be a projective manifold of dimension n. Beltrametti and Sommese conjectured that if A is an ample divisor such that

Positivity of direct image sheaves a geometric point of view

K_X+(n-1)A

M-regularity of the Fano surface

is nef, then

Mori Contractions of Maximal Length

K_X+(n-1)A

MINIMAL CLASSES ON THE INTERMEDIATE JACOBIAN OF A GENERIC CUBIC THREEFOLD

has non-zero global sections. We prove a weak version of this conjecture in arbitrary dimension. In dimension three, we prove the stronger non-vanishing conjecture of Ambro, Ionescu and Kawamata and give an application to Seshadri constants.

Algebraic varieties with quasi-projective universal cover

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 positivity of direct image sheaves. We give numerous examples and an introduction to the KollárViehweg method for proving direct image theorems. As an application, we study the geometry of projective manifolds with nef anticanonical bundle.

The structure of uniruled manifolds with split tangent bundle

Let

Totally invariant divisors of endomorphisms of projective spaces

(A,\Theta)

ALGEBRAIC FOLIATIONS DEFINED BY QUASI-LINES

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.

Anticanonical Divisors and Curve Classes on Fano Manifolds

We prove a relative version of the theorem of Cho, Miyaoka and Shepherd-Barron: a Mori fibre space of maximal length is birational to a projective bundle.