Searcharxiv⌕ Search

arXiv subjects

Keiji Oguiso

Publications and source records attributed to Keiji Oguiso.

At least 73 records · Page 4Linked to original sources
arXiv2010

Local structure of principally polarized stable Lagrangian fibrations

A holomorphic Lagrangian fibration is stable if the characteristic cycles of the singular fibers are of type $I_m, 1 \leq m <\infty,$ or $A_{\infty}$. We will give a complete description of the local structure of a stable Lagrangian fibration when it is principally polarized. In particular, we give an explicit form of the period map of such a fibration and conversely, for a period map of the described type, we construct a principally polarized stable Lagrangian fibration with the given period map. This enables us to give a number of examples exhibiting interesting behavior of the characteristic cycles.

math.AG↗
arXiv2010

Bounded derived categories of very simple manifolds

An unrepresentable cohomological functor of finite type of the bounded derived category of coherent sheaves of a compact complex manifold of dimension greater than one with no proper closed subvariety is given explicitly in categorical terms. This is a partial generalization of an impressive result due to Bondal and Van den Bergh.

math.AG↗
arXiv2010

A simple remark on a flat projective morphism with a Calabi-Yau fiber

If a K3 surface is a fiber of a flat projective morphisms over a connected noetherian scheme over the complex number field, then any smooth connected fiber is also a K3 surface. Observing this, Professor Nam-Hoon Lee asked if the same is true for higher dimensional Calabi-Yau fibers. We shall give an explicit negative answer to his question as well as a proof of his initial observation.

math.AG↗
arXiv2009

On volume preserving complex structures on real tori

A basic problem in the classification theory of compact complex manifolds is to give simple characterizations of complex tori. It is well known that a compact Kähler manifold $X$ homotopically equivalent to a a complex torus is biholomorphic to a complex torus. The question whether a compact complex manifold $X$ diffeomorphic to a complex torus is biholomorphic to a complex torus has a negative answer due to a construction by Blanchard and Sommese. Their examples have however negative Kodaira dimension, thus it makes sense to ask the question whether a compact complex manifold $X$ with trivial canonical bundle which is homotopically equivalent to a complex torus is biholomorphic to a complex torus. In this paper we show that the answer is positive for complex threefolds satisfying some additional condition, such as the existence of a non constant meromorphic function.

math.CV↗
arXiv2009

Multiple fibers of holomorphic Lagrangian fibrations

We determine all possible multiplicities of general singular fibers of a holomorphic Lagrangian fibration, under the assumption that all components of the fibers are of Fujiki class. The multiplicities are at most 6 and the possible values are intricately related to the Kodaira type of the characteristic cycle.

math.AG↗
arXiv2009

The third smallest Salem number in automorphisms of K3 surfaces

We realize the logarithm of the third smallest known Salem number as the topological entropy of a K3 surface automorphism with a Siegel disk and a pointwisely fixed curve at the same time. We also show the logarithm of the Lehmer number, the smallest known Salem number, is not realizable as the topological entropy of any Enriques surface automorphism. These results are entirely inspired by McMullen's works and Mathematica programs.

math.AG↗
arXiv2009

Picard number of the generic fiber of an abelian fibered hyperkaehler manifold

We shall show that the Picard number of the generic fiber, in the sense of scheme, of an abelian fibered hyperkähler manifold over the projective space is always one. We then give a few applications for the Mordell-Weil group. In particular, by deforming O'Grady's 10-dimensional manifold, we construct an abelian fibered hyperkähler manifold of Mordell-Weil rank 20, which is the maximum possible among all known ones.

math.AG↗
arXiv2008

Non-algebraic Hyperkaehler manifolds

We study the algebraic dimension a(X) of a compact hyperkaehler manfold of dimension 2n. We show that a(X) is at most n unless X is projective. If a compact Kaehler manifold with algebraic dimension 0 and Kodaira dimension 0 has a minimal model, then only the values 0,n and 2n are possible. In case of middle dimension, the algebraic reduction is holomorphic Lagrangian. If n = 2, then - without any assumptions - the algebraic dimension only takes the values 0,2 and 4. The paper gives structure results for "generalised hyperkaehler" manifolds and studies nef lines bundles.

math.DG↗
arXiv2007

Characteristic foliation on the discriminantal hypersurface of a holomorphic Lagrangian fibration

We give a Kodaira-type classification of general singular fibers of a holomorphic Lagrangian fibration in Fujiki's class $\mathcal C$. Our approach is based on the study of the characteristic vector field of the discriminantal hypersurface, which naturally arises from the defining equation of the hypersurface via the symplectic form. As an application, we show that the characteristic foliation of the discriminantal hypersurface has algebraic leaves which are either rational curves or smooth elliptic curves.

math.AG↗
arXiv2007

Shioda-Tate formula for an abelian fibered variety and applications

We give an explicit formula for the Mordell-Weil rank of an abelian fibered variety and some of its applications for an abelian fibered hyperkähler manifold. As a byproduct, we also give an explicit example of an abelian fibered variety in which the Picard number of the generic fiber in the sense of scheme is different from the Picard number of generic closed fibers.

math.AG↗
arXiv2007

Conjecture of Tits type for complex varieties and Theorem of Lie-Kolchin type for a cone

First, we shall formulate and prove Theorem of Lie-Kolchin type for a cone and derive some algebro-geometric consequences. Next, inspired by a recent result of Dinh and Sibony we pose a conjecture of Tits type for a group of automorphisms of a complex variety and verify its weaker version. Finally, applying Theorem of Lie-Kolchin type for a cone, we shall confirm the conjecture of Tits type for complex tori, hyperkähler manifolds, surfaces, and minimal threefolds.

math.AG↗
arXiv2006

Mordell-Weil groups of a hyperkähler manifold - a question of F. Campana

Mordell-Weil groups of different abelian fibrations of a hyperkähler manifold may have non-trivial relation even among elements of infinite order, but have essentially no relation, as its birational transformation. Precise definition of the terms "essentially no relation" will be given in Introduction.

math.AG↗
arXiv2005

Automorphisms of hyperkähler manifolds in the view of topological entropy

First we show that any group of automorphisms of null-entropy of a projective hyperkähler manifold $M$ is almost abelian of rank at most $ρ(M) - 2$. We then characterize automorphisms of a K3 surface with null-entropy and those with positive entropy in algebro-geometric terms. We also give an example of a group of automorphisms which is not almost abelian in each dimension.

math.AG↗
arXiv2005

Tits alternative in hypekähler manifolds

We show an analogous result of the famous Tits alternative for a group G of birational automorphisms of a projective hyperkäher manifold: Either G contains a non-commutative free group or G is an almost abelian group of finite rank. As an application, we show that the automorphism groups of the so-called singular K3 surfaces contain non-commutative free groups.

math.AG↗
arXiv2005

Bimeromorphic automorphism groups of non-projective hyperkähler manifolds - a note inspired by C. T. McMullen

Being inspired by a work of Curtis T. McMullen about a very impressive automorphism of a K3 surface of Picard number zero, we shall clarify the structure of the bimeromorphic automorphism group of a non-projective hyperkähler manifold, up to finite group factor. We also discuss relevant topics, especially, new counterexamples of Kodaira's problem about algebraic approximation of a compact Kähler manifold.

math.AG↗