First, we 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 confirm the… (More)
In this paper we study automorphisms g of order p of K3-surfaces defined over an algebraically closed field of characteristic p > 0. We divide all possible actions in the following cases according to the structure of the set of fixed points X g : X g is a finite set, X g contains a one-dimensional part D which is a positive divisor of Kodaira dimension κ(X,… (More)
We shall determine the uniquely existing extension of the alternating group of degree 6 (being normal in the group) by a cyclic group of order 4, which can act on a complex K3 surface.
We find generators of the group of birational automorphisms of the Hessian surface of a general cubic surface. Its nonsingular minimal model is a K3 surface with the Picard lattice of rank 16 which embeds naturally in the even unimodular lattice II 1,25 of rank 26 and signature (1, 25). The generators are related to reflections with respect to some Leech… (More)
We investigate when the fundamental group of the smooth part of a K3 surface or Enriques surface with Du Val singularities, is finite. As a corollary we give an effective upper bound for the order of the fundamental group of the smooth part of a certain Fano 3-fold. This result supports Conjecture A below, while Conjecture A (or alternatively the… (More)
The alternating group of degree 6 is located at the junction of three series of simple non-commutative groups : simple sporadic groups, alternating groups and simple groups of Lie type. It plays a very special role in the theory of finite groups. We shall study its new roles both in a finite geometry of certain pentagon in the Leech lattice and also in a… (More)
A fake projective plane is a compact complex surface (a compact complex manifold of dimension 2) with the same Betti numbers as the complex projective plane, but not isomorphic to the complex projective plane. As was shown by Mumford, there exists at least one such surface. In this paper we prove the existence of a fake projective plane which is birational… (More)
We shall give, in an optimal form, a sufficient numerical condition for the finiteness of the fundamental group of the smooth locus of a normal K3 surface. We shall moreover prove that, if the normal K3 surface is elliptic and the above fundamental group is not finite, then there is a finite covering which is a complex torus.
A normal projective complex surface is called a rational homology projective plane if it has the same Betti numbers with the complex projective plane CP 2. It is known that a rational homology projective plane with quotient singularities has at most 5 singular points. But all known examples have at most 4 singular points. In this paper, we prove that a… (More)