• Journal of the Korean Mathematical Society
• /
• v.43 no.5
• /
• pp.1065-1080
• /
• 2006

For 3-dimensional Bieberbach groups, we study the de-formation spaces in the group of isometries of $R^3$. First we calculate the discrete representation spaces and the automorphism groups. Then for each of these Bieberbach groups, we give complete descriptions of $Teichm\ddot{u}ller$ spaces, Chabauty spaces, and moduli spaces.

• Kyungpook Mathematical Journal
• /
• v.46 no.4
• /
• pp.591-595
• /
• 2006

In this paper, we define ${\Delta}^m$-Lacunary strongly convergent sequences defined by sequence of moduli and give various properties and inclusion relations on these sequence spaces.

• Journal of the Korean Mathematical Society
• /
• v.43 no.4
• /
• pp.765-782
• /
• 2006

We describe the moduli spaces of stable vector bundles of rank 2 on Enriques surfaces. They all have the structure of the fibrations reflecting those of Enriques surfaces.

• Journal of the Korean Mathematical Society
• /
• v.58 no.2
• /
• pp.501-523
• /
• 2021

We study the geometry of the moduli space of elliptic stable maps to projective space. The main component of the moduli space of elliptic stable maps is singular. There are two different ways to desingularize this space. One is Vakil-Zinger's desingularization and the other is via the moduli space of logarithmic stable maps. Our main result is a proof of the direct geometric relationship between these two spaces. For degree less than or equal to 3, we prove that the moduli space of logarithmic stable maps can be obtained by blowing up Vakil-Zinger's desingularization.

• Communications of the Korean Mathematical Society
• /
• v.27 no.3
• /
• pp.571-581
• /
• 2012

The moduli spaces of stable quasimaps unify various moduli appearing in the study of Gromov-Witten theory. This note is a survey article on the moduli of stable quasimaps, based on papers [9, 11, 18] as well as the author's talk at Kinosaki Algebraic Geometry Symposium 2010.

• Journal of the Korean Mathematical Society
• /
• v.54 no.6
• /
• pp.1759-1786
• /
• 2017

We examine the moduli space of oriented locally homogeneous manifolds of Type ${\mathcal{A}}$ which have non-degenerate symmetric Ricci tensor both in the setting of manifolds with torsion and also in the torsion free setting where the dimension is at least 3. These exhibit phenomena that is very different than in the case of surfaces. In dimension 3, we determine all the possible symmetry groups in the torsion free setting.

• Journal of the Korean Mathematical Society
• /
• v.37 no.3
• /
• pp.455-461
• /
• 2000

The quantum hyperplane section theorem is explained for nonnegative decomposable concavex bundle spaces over generalized flag manifolds.

• Journal of the Korean Mathematical Society
• /
• v.44 no.1
• /
• pp.35-54
• /
• 2007

Let $M_c$ = M(2, 0, c) be the moduli space of O(l)-semistable rank 2 torsion-free sheaves with Chern classes $c_1=0\;and\;c_2=c$ on a K3 surface X, where O(1) is a generic ample line bundle on X. When $c=2n\geq4$ is even, $M_c$ is a singular projective variety equipped with a holomorphic symplectic structure on the smooth locus. In particular, $M_c$ has trivial canonical divisor. In [22], O'Grady asks if there is any symplectic desingularization of $M_{2n}$ for $n\geq3$. In this paper, we show that there is no crepant resolution of $M_{2n}$ for $n\geq3$. This obviously implies that there is no symplectic desingularization.

• Journal of the Korean Mathematical Society
• /
• v.59 no.4
• /
• pp.651-684
• /
• 2022

Let G be an arbitrary, connected, simply connected and unimodular Lie group of dimension 3. On the space 𝔐(G) of left-invariant Lorentzian metrics on G, there exists a natural action of the group Aut(G) of automorphisms of G, so it yields an equivalence relation ≃ on 𝔐(G), in the following way: h₁ ≃ h₂ ⇔ h₂ = 𝜙^*(h₁) for some 𝜙 ∈ Aut(G). In this paper a procedure to compute the orbit space Aut(G)/𝔐(G) (so called moduli space of 𝔐(G)) is given.

• Kyungpook Mathematical Journal
• /
• v.63 no.2
• /
• pp.199-223
• /
• 2023

We introduce the constants E[t, X], C_NJ[X] and J[t, X] to describe the asymmetry of the norm. They can be seen as the skew version of the Gao's parameter, von Neumann-Jordan constant and Milman's moduli, respectively. We establish basic properties of these constants, relating them other well known constants, and use these properties to calculate the constants for specific spaces. We then use these constants to study Hilbert spaces, uniformly non-square spaces and their normal structures. With the Banach-Mazur distance, we use them to study isomorphic Banach spaces.