arXiv subjects
Publications and source records attributed to Andreas Höring.
Let $f\colon X\to X$ be a polarised endomorphism of a complex projective manifold, and let $D\subset X$ be a reduced totally invariant hypersurface. We prove that every irreducible component of the singular locus of $D$ has codimension one in $D$. In particular, if $D$ is normal, then it is smooth. As an application we prove the linearity of totally invariant divisors for endomorphisms of $\mathbb P^4$.
Let $X$ be a mildly singular Fano variety such that the tangent sheaf is a direct sum. We show that the direct factors are algebraically integrable, so the infinitesimal decomposition induces a product structure on a quasi-étale cover of $X$.
Let $X$ be a hyperkähler manifold, and let $A$ be a nef and big divisor on $X$. We show that the fixed part of the linear system $|A|$ is reduced and as a consequence $|2A|$ is mobile. If $X$ has dimension four we also show that if the fixed part of $|A|$ is not empty, the mobile part induces a (rational) Lagrangian fibration.
We give a classification of smooth Fano fourfolds such that the base scheme of the anticanonical system is a smooth surface. As a consequence we show that there are exactly 22 deformation families of such manifolds and they are all obtained by the same geometric construction. These 22 families are closely related to the list of smooth Fano threefolds that admit a $\mathbb P^1$-bundle structure.
Nonvanishing theorems play a central role in birational geometry, since they derive geometric consequences from numerical information and constitute a crucial step towards abundance and semiampleness problems. General nonvanishing statements remain rare, especially in the Kähler setting. We present two types of nonvanishing results for compact Kähler varieties. First, on non-uniruled varieties with nonzero Euler-Poincaré characteristic, we prove nonvanishing for adjoint bundles of numerical dimension one on Kähler klt pairs, as well as nonvanishing for nef line bundles of numerical dimension one on $K$-trivial varieties. Second, on hyperkähler manifolds we study line bundles $\mathcal L$ which are nef but not big, and establish a dichotomy: either nonvanishing holds for $\mathcal L$, or any closed positive current in the cohomology class of $\mathcal L$ has maximal Lelong components with a rather restricted geometry. We obtain much stronger abundance-type results in dimension $4$.
The cotangent bundle $T^*X$ of a smooth intersection $X$ of two quadrics admits a Lagrangian fibration determined by the intrinsic geometry of $X$. We show that this fibration is actually the Hitchin morphism if we endow $X$ with a structure of moduli space of twisted Spin-bundles. This generalises the classical result for threefolds, in which case it recovers the Hitchin fibration for the moduli space of rank two bundles with fixed determinant of odd degree on a curve of genus two.
We show that a divisor in a rational homogenous variety with split normal sequence is the preimage of a hyperplane section in either the projective space or a quadric.
A famous theorem of Shokurov states that a general anticanonical divisor of a smooth Fano threefold is a smooth K3 surface. This is quite surprising since there are several examples where the base locus of the anticanonical system has codimension two. In this paper we show that for four-dimensional Fano manifolds the behaviour is completely opposite: if the base locus is a normal surface, hence has codimension two, all the anticanonical divisors are singular.
We study the relative cone conjecture for families of $K$-trivial varieties with vanishing irregularity. As an application we prove that the relative movable and the relative nef cone conjectures hold for fibrations in projective IHS manifolds of the 4 known deformation types.
Given a Q-Cartier divisor $S \subset X$ admitting a fibration $S \rightarrow B$ onto a curve we give sufficient conditions for the existence of a bimeromorphic contraction contracting S onto B. As a corollary we recover a contraction result for compact Kähler threefolds.
We study degenerations of complex projective spaces $\mathbb P^n$ into normal projective klt varieties $X$. If the tangent sheaf of $X$ is semi-stable, we show that $X$ itself is a projective space. If $X$ is a threefold with canonical singularities, we show that there are only three varieties which satisfy all the conditions.
In this short note we prove two projectivity criteria for fibrations between mildly singular compact Kähler spaces. They are the relative versions of the celebrated criteria of Kodaira and Moishezon. As an application we obtain that the MRC fibration always has a model that is a projective morphism.
K3 surfaces have been studied from many points of view, but the positivity of the cotangent bundle is not well understood. In this paper we explore the surprisingly rich geometry of the projectivised cotangent bundle of a very general polarised K3 surface $S$ of degree two. In particular, we describe the geometry of a surface $D_S \subset \mathbb{P}(Ω_S)$ that plays a similar role to the surface of bitangents for a quartic in $\mathbb{P}^3$.
In this text we signal a serious gap in the proof of the main theorem of our paper and explain which parts of the statement remain valid. In fact, the main theorem remains valid unless possibly the variety does not admit positive-dimensional subvarieties through a very general point and is not bimeromorphic to a quotient of a torus. This latter case would be ruled out by a Chern class inequality which holds in the algebraic case but is still unknown in the Kähler setting.
Let A be an elliptic curve, and let $V_A$ be the Serre vector bundle on A. A famous example of Demailly-Peternell-Schneider shows that the tautological class of $V_A$ contains a unique closed positive current. In this survey we start by generalising this statement to arbitrary compact Kähler manifolds. We then give an application to abelian fibrations $X \rightarrow Y$ where the total space X has pseudoeffective cotangent bundle and raise some questions about nonvanishing properties of these bundles.
Let $M$ be a smooth Fano threefold such that a canonical extension of the tangent bundle is an affine manifold. We show that $M$ is rational homogeneous.
Let $M \subset X$ be a submanifold of a rational homogeneous space $X$ such that the normal sequence splits. We prove that $M$ is also rational homogeneous.
Demailly proved that on a smooth compact Kähler manifold the distribution defined by a holomorphic $p$-form with values in an anti-pseudoeffective line bundle is always integrable. We generalise his result to compact Kähler spaces with klt singularities.