arXiv subjects
Publications and source records attributed to Andreas Höring.
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 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.
It is well known that the Hodge conjecture with rational coefficients holds for degree 2n-2 classes on complex projective n-folds. In this paper we study the more precise question if on a rationally connected complex projective n-fold the integral Hodge classes of degree 2n-2 are generated over $\mathbb Z$ by classes of curves. We combine techniques from the theory of singularities of pairs on the one hand and infinitesimal variation of Hodge structures on the other hand to give an affirmative answer to this question for a large class of manifolds including Fano fourfolds. In the last case, one step in the proof is the following result of independent interest: There exist anticanonical divisors with isolated canonical singularities on a smooth Fano fourfold.
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.
Let X be a compact Kähler threefold that is not uniruled. We prove that X has a minimal model.
Given the Prym variety of an étale 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 $V^2$.
We classify non-algebraic compact Kähler threefolds admitting an endomorphism $f: X \to X$ of degree at least two.
Beauville asked if a compact Kähler 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.
Let X be a compact Kähler manifold whose universal covering is $\mathbb C^n$. A conjecture of Iitaka claims that X is a torus, up to finite étale cover. We prove this conjecture in various cases in dimension four. We also show that in the projective case Iitaka's conjecture is a consequence of the non-vanishing conjecture.
Generalising a classical theorem by Ueno, we prove structure results for manifolds with nef or semiample cotangent bundle.
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.
Let X be a compact Kähler manifold such that the anticanonical bundle $-K_X$ is nef. A classical conjecture claims that the Albanese map is submersive. We prove this conjecture if the general fibre is a weak Fano manifold. If X is projective we prove the conjecture also for fibres of dimension at most two.
Let X be a normal projective variety, and let A be an ample Cartier divisor on X. We prove that the twisted cotangent sheaf $Ω_X \otimes A$ is generically nef with respect to the polarisation A unless X is a projective space. As an application we prove a Kobayashi-Ochiai theorem for foliations: if $F \subsetneq T_X$ is a foliation of rank r such that $det F \equiv i_F A$, then we have $i_F \leq r$.
We present an inductive strategy to show the existence of rational curves on compact Kaehler manifolds which are not minimal models but have a pseudoeffective canonical bundle. The tool for this inductive strategy is a weak subadjunction formula for lc centers associated to certain big cohomology classes. This subadjunction formula is based, as in the projective case, on positivity arguments for relative adjoint classes.
Let X be a Fano variety of index k such that the non-klt locus Nklt(X) is not empty. We prove that Nklt(X) has dimension at least k-1 and equality holds if and only if Nklt(X) is a linear projective space P^{k-1}. In this case X has lc singularities and is a generalised cone with Nklt(X) as vertex. If X has lc singularities and Nklt(X) has dimension k we describe the non-klt locus and the global geometry of X. Moreover, we construct examples to show that all the classification results are effective.
Let X be a compact Kaehler threefold such that the base of the MRC-fibration has dimension two. We prove that X is bimeromorphic to a Mori fibre space. Together with our earlier result arXiv:1304.4013 this completes the MMP for compact Kaehler threefolds: let X be a non-projective compact Kaehler threefold. Then X has a minimal model or X is bimeromorphic to a Mori fibre space over a non-projective Kaehler surface.
Totally invariant divisors of endomorphisms of the projective space are expected to be always unions of linear spaces. Using logarithmic differentials we establish a lower bound for the degree of the non-normal locus of a totally invariant divisor. As a consequence we prove the linearity of totally invariant divisors for $\mathbb P^3$.
Let X be a simply connected projective manifold with nef anticanonical bundle. We prove that X is a product of a rationally connected manifold and a manifold with trivial canonical bundle. As an application we describe the MRC fibration of any projective manifold with nef anticanonical bundle.
We describe the recently established minimal model program for (non-algebraic) Kähler threefolds as well as the abundance theorem for these spaces.