JongHae Keum

FUNDAMENTAL GROUPS OF OPEN K3 SURFACES ENRIQUES SURFACES AND FANO 3-FOLDS

Abstract 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 rational-connectedness conjecture in Kollar et al. (J. Algebra Geom. 1 (1992) 429) which is still open when the dimension is at least 4) would imply that every log terminal Fano variety has a finite fundamental group.

The Alternating Group of Degree 6 in the Geometry of the Leech Lattice and K3 Surfaces

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 a certain pentagon in the Leech lattice and also in the complex algebraic geometry of K3 surfaces.

Construction of singular rational surfaces of Picard number one with ample canonical divisor

Kollar gave a series of examples of rational surfaces of Picard number 1 with ample canonical divisor having cyclic singularities. In this paper, we construct several series of new examples in a geometric way, i.e., by blowing up several times inside a configuration of curves on the projective plane and then by contracting chains of rational curves. One series of our examples have the same singularities as Kollar’s examples.

Extensions of the alternating group of degree 6 in the geometry of K3 surfaces

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.

Gorenstein ℚ-homology projective planes

We present the complete list of all singularity types on Gorenstein ℚ-homology projective planes, i.e., normal projective surfaces of second Betti number one with at worst rational double points. The list consists of 58 possible singularity types, each except two types supported by an example.

ALGEBRAIC SURFACES WITH QUOTIENT SINGULARITIES - INCLUDING SOME DISCUSSION ON AUTOMORPHISMS AND FUNDAMENTAL GROUPS

We survey some recent progress in the study of algebraic varieties X with log terminal singularities, especially, the uni-ruledness of the smooth locus X^0 of X, the fundamental group of X^0 and the automorphisms group on (smooth or singular) X when dim X = 2. The full automorphism groups of a few interesting types of K3 surfaces are described, mainly by Keum-Kondo. We conjecture that when X is Q-Fano then X^0 has a finite fundamental group, which had been proved if either dim X < 3 or the Fano index is bigger than dim X - 2. We also conjecture that when X is a log Enriques (e.g. a normal K3 or a normal Enriques) surface then either pi_1(X^0) is finite or X has an abelian surface as its quasi-etale cover, which has been proved by Catanese-Keum-Oguiso under some extra conditions.