A few explicit examples of complex dynamics of inertia groups on surfaces - a question of Professor Igor Dolgachev

We give a few explicit examples which answer an open minded question of Professor Igor Dolgachev on complex dynamics of the inertia group of a smooth rational curve on a projective K3 surface and its variants for a rational surface and a non-projective K3 surface.Comment: final form, to appear in Transformation Grou

Tits alternative in hypek\"ahler manifolds

We show an analogous result of the famous Tits alternative for a group G of birational automorphisms of a projective hyperk\"aher 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.Comment: 9pages; all the previous results have been generalized and accordingto this the title has been change

Free automorphisms of positive entropy on smooth K\"ahler surfaces

We prove that there is a projective K3 surface admitting a (fixed point) free automorphism of positive entropy and that no smooth compact K\"ahler surface other than projective K3 surfaces and their blow up admits such an automorphism.Comment: 10 pages, some typos are corrected, one reference together with a very important comment from Professor Bert van Geemen is adde

Picard numbers in a family of hyperk\"ahler manifolds - A supplement to the article of R. Borcherds, L. Katzarkov, T. Pantev, N. I. Shepherd-Barron

We remark the density of the jumping loci of the Picard number of a hyperk\"ahler manifold under small one-dimensional deformation and provide some applications for the Mordell-Weil groups of Jacobian K3 surfaces.Comment: 12 pages, AMS Te

A characterization of the Fermat quartic K3 surface by means of finite symmetries

We shall characterize the Fermat K3 surface, among all complex K3 surfaces, by means of its finite group symmetries.Comment: 22 page

A question of Doctor Malte Wandel on automorphisms of the punctural Hilbert schemes of K3 surfaces

We present a sufficient condition for the punctural Hilbert scheme of length two of a K3 surface with finite automorphism group to have automorphism group of infinite order in geometric terms (Theorem 2.1). We then give concrete examples (Theorem 1.2). We also discuss about Mori dream space (MDS) structures under an extermal crepant resolution (Theorems 1.2, 5.2, 4.1) from the viewpoint of automorphisms. These affirmatively answer a question of Doctor Malte Wandel.Comment: 14pages, expanded semi-final version to be submitte

Isomorphic quartic K3 surfaces in the view of Cremona and projective transformations

We show that there is a pair of smooth complex quartic K3 surfaces $S_1$ and $S_2$ in ${\mathbf P}^3$ such that $S_1$ and $S_2$ are isomorphic as abstract varieties but not Cremona isomorphic. We also show, in a geometrically explicit way, that there is a pair of smooth complex quartic K3 surfaces $S_1$ and $S_2$ in ${\mathbf P}^3$ such that $S_1$ and $S_2$ are Cremona isomorphic, but not projectively isomorphic. This work is much motivated by several e-mails from Professors Tuyen Truong and J\'anos Koll\'ar.Comment: 14pages, final version, to appear in TJM special issu

Some aspects of explicit birational geometry inspired by complex dynamics

Our aim is to illustrate how one can effectively apply the basic ideas and notions of topological entropy and dynamical degrees, together with recent progress of minimal model theory in higher dimension, for an explicit study of birational or biregular selfmaps of projective or compact K\"ahler manifolds, through concrete examples.Comment: A survey article for ICM2014 proceedings (Invited talk at Section 4). Format, typos, references, ambiguities are correcte

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 hyperk\"ahler 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.Comment: 10 page

A surface in odd characteristic with discrete and non-finitely generated automorphism group

It was proved by Tien-Cuong Dinh and me that there is a smooth complex projective surface whose automorphism group is discrete and not finitely generated. In this paper, we will show that there is a smooth projective surface, birational to some K3 surface, such that the automorphism group is discrete and not finitely generated, over any algebraically closed field of odd characteristic except precisely an algebraic closure of the prime field.Comment: 16 pages, Theorem 2.4 is new and expanded after a suggestion by the referee, the final form accepted by Advances in Mat