Related papers: On anti-hyperbolicity for hyperk\"ahler varieties

First we show that any group of automorphisms of null-entropy of a projective hyperk\"ahler manifold $M$ is almost abelian of rank at most $\rho(M) - 2$. We then characterize automorphisms of a K3 surface with null-entropy and those with…

An n-dimensional complex manifold M is said to be (holomorphically) dominable by $\CC^n$ if there is a map $F:\CC^n \ra M$ which is holomorphic such that the Jacobian determinant $\det(DF)$ is not identically zero. Such a map F is called a…

Let M be a (bounded or not) domain of C^n which is complete with respect to a K\"ahler metric, or more generally, a complete K\"ahler manifold with trivial canonical bundle. Let f be a linearly nondegenerate meromorphic map from M to the…

The aim of these notes is to acquaint the reader with important objects in complex algebraic geometry: K3 surfaces and their higher-dimensional analogs, hyperk\"ahler manifolds. These manifolds are interesting from several points of view:…

Using the Kodaira dimension and the fundamental group of X, we succeed in classifying algebraic surfaces which are dominable by C^2 except for certain cases in which X is an algebraic surface of Kodaira dimension zero and the case when X is…

Let $M$ be a complete K\"{a}hler manifold, whose universal covering is biholomorphic to a ball $\mathbb B^m(R_0)$ in $\mathbb C^m$ ($0<R_0\le +\infty$). In this article, we will show that if three meromorphic mappings $f^1,f^2,f^3$ of $M$…

Hypersurfaces are studied and classified under multiple additional assumptions in any Riemannian homogeneous space $(\mathbb{C}P^3, g_a)$, including nearly K\"ahler $\mathbb{C}P^3$. Notably, all extrinsically homogeneous hypersurfaces are…

Let $M$ be a complete K\"{a}hler manifold, whose universal covering is biholomorphic to a ball $\mathbb B^m(R_0)$ in $\mathbb C^m$ ($0<R_0\le +\infty$). Our first aim in this paper is to study the algebraic dependence problem of…

By a theorem of Wahl, the canonically embedded curves which are hyperplane section of K3 surfaces are distinguished by the non-surjectivity of their Wahl map. In this paper we address the problem of distinguishing hyperplane sections of…

We introduce the notion of a Hyper-K\"{a}hler manifold $X$ induced by a Hodge structure of K3-type. We explore this notion for the known deformation types of hyper-K\"{a}hler manifolds studying those that are induced by a K3 or abelian…

The present paper provides several results on automorphisms of hyperk\"ahler (or irreducible holomorphic symplectic) manifolds. In particular it focuses on the symplectic case and contains a classification of prime order symplectic…

We study deformations of complex hyperbolic surfaces which furnish the simplest examples of: (i) negatively curved K\"ahler manifolds and (ii) negatively curved Riemannian manifolds not having {\it constant} curvature. Although such complex…

We introduce two notions of hyperbolicity for not necessarily K\"ahler $n$-dimensional compact complex manifolds $X$. The first, called {\it balanced hyperbolicity}, generalises Gromov's K\"ahler hyperbolicity by means of Gauduchon's…

We first prove an isomorphism between the moduli space of smooth cubic threefolds and the moduli space of hyperkaehler fourfolds of K3^{[2]}-type with a non-symplectic automorphism of order three, whose invariant lattice has rank one and is…

We determine the Hodge endomorphism algebras of non-projective complex K3 surfaces (and more generally, hyperk\"ahler manifolds). We show that they are either totally real fields or number fields generated by Salem numbers. This is unlike…

Let $M$ be a complete K\"{a}hler manifold, whose universal covering is biholomorphic to a ball $\mathbb B^m(R_0)$ in $\mathbb C^m$ ($0<R_0\le +\infty$). In this article, we will show that if three meromorphic mappings $f^1,f^2,f^3$ of $M$…

In this paper we define the notion of a hyperk\"ahler manifold (potentially) of Jacobian type. If we view hyperk\"ahler manifolds as "abelian varieties", then those of Jacobian type should be viewed as "Jacobian varieties". Under a minor…

We show that the non-measure hyperbolicity of K3 surfaces -- which M. Green and P. Griffiths verified for certain cases in 1980 -- holds for all K3 surfaces. As a byproduct, we prove the non-measure hyperbolicity of any Hilbert schemes of…

The paper generalizes some of the well-known results for K3 surfaces to higher-dimensional irreducible symplectic (or, equivalently, compact irreducible hyperkaehler) manifolds. In particular, we discuss the projectivity of such manifolds…

A well known consequence of the Wirtinger inequality is that in a Kaehler surface a holomorphic curve is an area minimizer in its homology class. In light of this result it is natural, given a Kaehler surface, to investigate the relation…