• Journal of the Korean Mathematical Society
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• v.60 no.6
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• pp.1137-1169
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• 2023

Let C be a curve and V → C an orthogonal vector bundle of rank r. For r ≤ 6, the structure of V can be described using tensor, symmetric and exterior products of bundles of lower rank, essentially due to the existence of exceptional isomorphisms between Spin(r, ℂ) and other groups for these r. We analyze these structures in detail, and in particular use them to describe moduli spaces of orthogonal bundles. Furthermore, the locus of isotropic vectors in V defines a quadric subfibration Q_V ⊂ ℙV . Using familiar results on quadrics of low dimension, we exhibit isomorphisms between isotropic Quot schemes of V and certain ordinary Quot schemes of line subbundles. In particular, for r ≤ 6 this gives a method for enumerating the isotropic subbundles of maximal degree of a general V , when there are finitely many.

• Journal of the Korean Mathematical Society
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• v.40 no.3
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• pp.341-371
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• 2003

This article discusses in detail how the study of proper holomorphic rational mappings between balls in different dimensions relates to positivity conditions and to isometric imbedding of holomorphic bundles. The first chapter discusses rational proper mappings between balls; the second chapter discusses seven distinct positivity conditions for real-valued polynomials in several complex variables; the third chapter reveals how these issues relate to an isometric imbedding theorem for holomorphic vector bundles proved by the author and Catlin.

• Korean Journal of Mathematics
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• v.6 no.2
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• pp.303-311
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• 1998

We study the conformally flat unit vector bundle $E_1$ of constant scalar curvature for the bundle ${\pi}:E^{n+2}{\rightarrow}M^n$ over an Einstein manifold M.

• Journal of the Korean Mathematical Society
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• v.39 no.3
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• pp.331-349
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• 2002

Let G be a reductive algebraic group and let B, F be G-modules. We denote by $VEC_{G}$ (B, F) the set of isomorphism classes in algebraic G-vector bundles over B with F as the fiber over the origin of B. Schwarz (or Karft-Schwarz) shows that $VEC_{G}$ (B, F) admits an abelian group structure when dim B∥G = 1. In this paper, we introduce a stable functor $VEC_{G}$ (B, $F^{\chi}$) and prove that it is an abelian group for any G-module B. We also show that this stable functor will have nice properties.

• Communications of the Korean Mathematical Society
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• v.5 no.2
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• pp.113-124
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• 1990

• Communications of the Korean Mathematical Society
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• v.5 no.1
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• pp.29-35
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• 1990

• Journal of the Korean Mathematical Society
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• v.24 no.2
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• pp.267-285
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• 1987

• Honam Mathematical Journal
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• v.11 no.1
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• pp.7-13
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• 1989

• Bulletin of the Korean Mathematical Society
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• v.54 no.5
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• pp.1743-1755
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• 2017

Let B be a simply-connected projective variety such that the first cohomology groups of all line bundles on B are zero. Let E be a vector bundle over B and $X={\mathbb{P}}(E)$. It is easily seen that a power of any endomorphism of X takes fibers to fibers. We prove that if X admits an endomorphism which is of degree greater than one on the fibers, then E splits into a direct sum of line bundles.

• Communications of the Korean Mathematical Society
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• v.9 no.4
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• pp.927-931
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• 1994

Let G be a cyclic group of order 2 and let $S^1$ denote the unit circle in $R^2$ with the standard metric. We consider smooth G-vector bundles over $S^1$ when G acts on $S^1$ by reflection. Then the fixed point set of G on $S^1$ is two points ${z_0, z_1}$. Let $E$\mid$_{z_0} and E$\mid$_{z_1}$ be the fiber G-representation spaces at $z_0$ and $z_1$ respectively. We associate an orthogonal G-representation $\rho_i : G \to O(n)$ to $E$\mid$_{z_i}, i = 0, 1$. Let det $p\rho_i(g), g \neq 1$, be denoted by det $E$\mid$_{z_i}$ since det $\rho_i(g)$ is independent of choice of $\rho_i$.