arXiv subjects
Publications and source records attributed to Keiji Oguiso.
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 consider a compact Kaehler manifold whose dual Kaehler cone contains a rational interior point. The general problem we have in mind is how far the manifold is from being projective; i.e. we ask for the algebraic dimension. We prove e.g. that if the interior point is represented by an effective curve in dimension 3, then the manifold has algebraic dimension at least 2 unless the variety is a so-called simple non-Kummer 3-fold. We also investigate 3-folds containing curves with ample normal bundle. It seems likely that examples with algebraic dimension 2 really exist.
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 complex algebraic geometry of $K3$ surfaces.
We shall give a Counting Formula for the number of Fourier-Mukai partners of a K3 surface and consider three applications.
We shall characterize the Fermat K3 surface, among all complex K3 surfaces, by means of its finite group symmetries.
We find a concise relation between the moduli $τ, ρ$ of a rational Narain lattice $Γ(τ,ρ)$ and the corresponding momentum lattices of left and right chiral algebras via the Gauss product. As a byproduct, we find an identity which counts the cardinality of a certain double coset space defined for isometries between the discriminant forms of rank two lattices.
Narain lattices are unimodular lattices {\it in} $\R^{r,s}$, subject to certain natural equivalence relation and rationality condition. The problem of describing and counting these rational equivalence classes of Narain lattices in $\R^{2,2}$ has led to an interesting connection to binary forms and their Gauss products, as shown in [HLOYII]. As a sequel, in this paper, we study arbitrary rational Narain lattices and generalize some of our earlier results. In particular in the case of $\R^{2,2}$, a new interpretation of the Gauss product of binary forms brings new light to a number of related objects -- rank 4 rational Narain lattices, over-lattices, rank 2 primitive sublattices of an abstract rank 4 even unimodular lattice $U^2$, and isomorphisms of discriminant groups of rank 2 lattices.
We shall give a complete geometrical description of the FM partners of a K3 surface of Picard number 1 and its applications.
We apply our earlier results on Fourier-Mukai partners to answer definitively a question about Kummer surface structures, posed by T. Shioda 25 years ago.
We consider autoequivalences of the bounded derived category of coherent sheaves on a K3 surface. We prove that the image of the autoequivalences has index at most two in the group of the Hodge isometries of the Mukai lattice. Motivated by homological mirror symmetry we also consider the mirror counterpart, i.e. symplectic version of it. In the case of $ρ(X)=1$, we find an explicit formula which reproduces the number of Fourier-Mukai (FM) partners from the monodromy problem of the mirror K3 family. We present an explicit example in which a monodromy action does not come from an autoequivalence of the mirror side.
Based on the result on derived categories on K3 surfaces due to Mukai and Orlov and the result concerning almost-prime numbers due to Iwaniec, we remark the following fact: For any given positive integer N, there are N (mutually non-isomorphic) projective complex K3 surfaces such that their Picard lattices are not isomorphic but their transcendental lattices are Hodge isometric, or equivalently, their derived categories are mutually equivalent. After reviewing some finiteness result, we also give an explicit formula for the cardinality of the isomorphism classes of projective K3 surfaces having derived categories equivalent to the one of X with Picard number 1 in terms of the degree of X.
A few facts concerning the phrase "the automorphism groups become larger at special points of the moduli of K3 surfaces" are presented. It is also shown that the automorphism groups are of infinite order over a dense subset in any one-dimensional non-trivial family of projective K3 surfaces.
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.
The aim of this note is to characterize a K3 surface of Klein-Mukai type in terms of its symmetry.
We remark the density of the jumping loci of the Picard number of a hyperkähler manifold under small one-dimensional deformation and provide some applications for the Mordell-Weil groups of Jacobian K3 surfaces.
The aim of this note is to study local and global Seshadri constants for a family of smooth surfaces with prescribed polarization. We shall first observe that given $α$ being smaller than the square root of the degree of polarization, the set of local Seshadri constants in the range $(0, α]$ is finite. This in particular implies that the square root of the degree of polarization is the only possible accumulation point of the set of local Seshadri constants. Next we shall remark the Zariski closedness of the set of points whose local Seshadri constants are in any given interval $(0, a]$. As applications, we shall also add a few remarks on the lower semi-continuity of both local and global Seshadri constants with respect to parameters involved, and on the minimality and the maximality of their infimum and supremum.
We study non-isotrivial projective families of elliptic surfaces of Kodaira dimension one, over complex projective curves. If the base is an elliptic curve, we show that the family must have a singular fibre, and that over the projective line it must have at least three singular fibres. Similar results, for families of surfaces of general type, have been obtained by Migliorini and Kovács, and they are well-known for projective families of surfaces of Kodaira dimension zero. Revised version: We corrected some minor errors and ambiguities, and we completed the list of references.
We give an elementary proof of the following projectivity criterion of Huybrechts: a compact Kähler surface is projective if and only if the dual Kähler cone contains an inner integral point.