• 충청수학회지
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• 제24권2호
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• pp.319-335
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• 2011

We classify equivariant topoligical complex vector bundles over real projective plane under a compact Lie group (not necessarily effective) action. It is shown that nonequivariant Chern classes and isotropy representations at (at most) three points are sufficient to classify equivariant vector bundles over real projective plane except one case. To do it, we relate the problem to classification on two-sphere through the covering map because equivariant vector bundles over two-sphere have been already classified.

• 대한수학회보
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• 제54권5호
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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.

• Kyungpook Mathematical Journal
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• 제45권2호
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• pp.167-169
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• 2005

Here we introduce and study vector bundles, E, on a smooth projective curve X having many "spread" sections and for which $E^{*}\;{\otimes}{\omega}X$ has many "spread" sections. We show that no such bundle exists on X if the gonality of X is too low.

• 대한수학회지
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• 제60권3호
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• pp.479-501
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• 2023

We study the rationality and symmetry of the Gromov-Witten invariants of the projective line twisted by certain line bundles.

• 대한수학회지
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• 제56권2호
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• pp.387-419
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• 2019

A fullness conjecture of Kuznetsov says that if a smooth projective variety X admits a full exceptional collection of line bundles of length l, then any exceptional collection of line bundles of length l is full. In this paper, we show that this conjecture holds for X as the blow-up of ${\mathbb{P}}^3$ at a point, a line, or a twisted cubic curve, i.e., any exceptional collection of line bundles of length 6 on X is full. Moreover, we obtain an explicit classification of full exceptional collections of line bundles on such X.

• 대한수학회논문집
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• 제29권4호
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• pp.569-579
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• 2014

We consider a condition under which the projectivization $P(E^k)$ of a complex k-bundle $E^k{\rightarrow}M$ over an even-dimensional manifold M can have the hard Lefschetz property, affected by [10]. It depends strongly on the rank k of the bundle $E^k$. Our approach is purely algebraic by using rational Sullivan minimal models [5]. We will give some examples.

• 대한수학회보
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• 제54권5호
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• pp.1699-1718
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• 2017

We give a construction of the second Chern number of a vector bundle over a smooth projective surface by means of adelic transition matrices for the vector bundle. The construction does not use an algebraic K-theory and depends on the canonical ${\mathbb{Z}}-torsor$ of a locally linearly compact k-vector space. Analogs of certain auxiliary results for the case of an arithmetic surface are also discussed.

• 호남수학학술지
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• 제9권1호
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• pp.135-147
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• 1987

It is well known that there are two ways to define Chern classes of complex vector bundles. One gives the definition of Chern classes by the five axioms ([2]. [3], [4]). and an other defines Chern classes with the associated projective space bundle of a given bundle ([1]. [5]). The purpose of this paper is to describe the latter way in detail and to give new proofs of that our Chern classes satisfy the five axioms with respect to Chern classes (for example Theorem 5).

• 대한수학회논문집
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• 제34권1호
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• pp.279-286
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• 2019

Complex Grassmann manifolds $G_{n,k}$ are a generalization of complex projective spaces and have many important features some of which are captured by the $Pl{\ddot{u}}cker$ embedding $f:G_{n,k}{\rightarrow}{\mathbb{C}}P^{N-1}$ where $N=\(^n_k\)$. The problem of existence of cross sections of fibrations can be studied using the Gottlieb group. In a more generalized context one can use the relative evaluation subgroup of a map to describe the cohomology of smooth fiber bundles with fiber the (complex) Grassmann manifold $G_{n,k}$. Our interest lies in making use of techniques of rational homotopy theory to address problems and questions involving applications of Gottlieb groups in general. In this paper, we construct the Sullivan minimal model of the (complex) Grassmann manifold $G_{n,k}$ for $2{\leq}k<n$, and we compute the rational evaluation subgroup of the embedding $f:G_{n,k}{\rightarrow}{\mathbb{C}}P^{N-1}$. We show that, for the Sullivan model ${\phi}:A{\rightarrow}B$, where A and B are the Sullivan minimal models of ${\mathbb{C}}P^{N-1}$ and $G_{n,k}$ respectively, the evaluation subgroup $G_n(A,B;{\phi})$ of ${\phi}$ is generated by a single element and the relative evaluation subgroup $G^{rel}_n(A,B;{\phi})$ is zero. The triviality of the relative evaluation subgroup has its application in studying fibrations with fibre the (complex) Grassmann manifold.