Related papers: Uniruled varieties with split tangent bundle

Consider a projective manifold X and suppose that some wedge power of the cotangent bundle contains a subsheaf whose determinant bundle has maximal Kodaira dimension. Then we prove that X is of general type. More generally we compute the…

Let $M=P(E)$ be the complex manifold underlying the total space of the projectivization of a holomorphic vector bundle $E \to \Sigma$ over a compact complex curve $\Sigma$ of genus $\ge 2$. Building on ideas of Fujiki, we prove that $M$…

We introduce and study the notion of "surface decomposable" variety, and discuss the possibility that any projective hyper-K\"ahler manifold is surface decomposable, which would produce new evidence for Beauville's weak splitting…

Any smooth projective variety contains many complete intersection subvarieties with ample cotangent bundles, of each dimension up to half its own dimension.

Theorem (uniformization). Let X be a compact Kahler manifold of dimension n with large, residually finite and nonamenable fundamental group. Then its universal covering is a bounded domain in the n-dimensional affine space.

Let $X$ be a projective variety with log terminal singularities and vanishing augmented irregularity. In this paper we prove that if $X$ admits a relatively minimal genus one fibration then it does contain a subvariety of codimension one…

Fixed a polarised variety $X$, we can ask if it admits Ulrich bundles and, in case, what is their minimal possible rank. In this thesis, after recalling general properties of Ulrich sheaves, we show that any finite covering of…

The question of existence of Ulrich bundles on nonsingular projective varieties is posed here in weaker terms: either to find a K-theoretic solution, or to find one in the derived category of the variety. We observe that if any motivic…

In this note we study two features of submanifolds (subvarieties) with ample normal bundles in a compact K\"ahler manifold X. First, we study how algebraic X can be, i.e. we investigate the algebraic dimension. Second, we study curves with…

We confirm Beauville's conjecture that claims that if the p-th exterior power of the tangent bundle of a smooth projective variety contains the p-th power of an ample line bundle, then the variety is either the projective space or the…

This survey paper discusses some of the recent progress in the study of rational curves on algebraic varieties. It was written for the survey volume of the priority programme "Global Methods in Complex Geometry", supported by the DFG. To…

A toric variety is called fibered if it can be represented as a total space of fibre bundle over toric base and with toric fiber. Fibered toric varieties form a special case of toric variety bundles. In this note we first give an…

We study varieties $X \subset P^r$ such that is $N_X^*(k)$ is an Ulrich vector bundle for some integer $k$. We first prove that such an $X$ must be a curve. Then we give several examples of curves with $N_X^*(k)$ an Ulrich vector bundle.

In this paper, with the aim of establishing a structure theorem for a compact K\"ahler manifold $X$ with semi-positive holomorphic sectional curvature, we study a morphism $\phi: X \to Y$ to a compact K\"ahler manifold $Y$ with…

Classification theory and the study of projective varieties which are covered by rational curves of minimal degrees naturally leads to the study of families of singular rational curves. Since families of arbitrarily singular curves are hard…

We give an almost complete classification of Ulrich bundles $\mathcal E$ with $c_2(\mathcal E)^2=0$ on a variety $X$ of dimension $n \ge 4$. Moreover, we show that there are strong constraints on the geometry of $X$ and we study…

We study the Euler class of smooth orientable infinite-type surface bundles with a section. For many such surfaces, we show that this cohomology class is nontrivial, and that the behavior of its powers depends on the genus and the type of…

These are lecture notes for a course to be held. They provide a full discussion of certain analytic aspects of the uniformisation theory of (singular) holomorphic foliations by curves on compact Kaehler manifolds, with emphasis on their…

Given a projective variety $X$ and a very ample line bundle $\mathcal{L}$ on $X$, we classify for which $X$ and $\mathcal{L}$ the twisted syzygies and twisted dual syzygies bundles are Ulrich with respect to the polarizations…

We provide supplements and open problems related to structure theorems for maximal rationally connected fibrations of certain positively curved projective varieties, including smooth projective varieties with semi-positive holomorphic…