• Journal of the Korean Mathematical Society
• /
• v.40 no.4
• /
• pp.747-756
• /
• 2003

We establish a vanishing theorem on the square-integrable cohomology associated to the Cauchy-Riemann complex on some complete Kaehler manifolds. The hypothesis needed for this result is a growth condition on a primitive of the Kaehler form.

• Communications of the Korean Mathematical Society
• /
• v.11 no.2
• /
• pp.391-405
• /
• 1996

In this article, we calculate the cohomology groups of flat vector bundles on some manifolds.

• Journal of the Korean Mathematical Society
• /
• v.34 no.1
• /
• pp.167-179
• /
• 1997

In the study of a manifold M, the exterior algebra $\Lambda^* M$ plays an important role. In fact, the de Rham cohomology theory gives many informations of a manifold. Another important object in the study of a manifold is its Clifford algebra (Cl(M), generated by the tangent space.

• Bulletin of the Korean Mathematical Society
• /
• v.52 no.6
• /
• pp.1855-1899
• /
• 2015

We discuss some additivity properties of the simplicial volume for manifolds with boundary: we give proofs of additivity for glueing amenable boundary components and of superadditivity for glueing amenable submanifolds of the boundary, and we discuss doubling of 3-manifolds.

• Bulletin of the Korean Mathematical Society
• /
• v.57 no.2
• /
• pp.449-457
• /
• 2020

The aim of this short paper is to show some rigidity results for the actions of certain finitely presented groups by homeomorphisms. As an interesting and special case, we show that the actions of Higman-Thompson groups by homeomorphisms on a cohomology manifold with a non-zero Euler characteristic should be trivial. This is related to the wellknown Zimmer program and shows that the actions by homeomorphism could be very much different from those by diffeomorphisms.

• Bulletin of the Korean Mathematical Society
• /
• v.35 no.4
• /
• pp.669-681
• /
• 1998

In this paper, we prove that on the complete Kahler manifold, if ${\rho}(x){\geq}-\frac{1}{2}{\lambda}_0$ and either ${\rho}(x_0)>-\frac{1}{2}{lambda}_0$ at some point $x_0$ or Vol(M)=${\infty}$, then the Clifford $L^2$ cohomology group $L^2{\mathcal H}^{\ast}(M,S)$ is trivial, where $\rho(x)$ is the least eigenvalue of ${\mathcal R}_x + \bar{{\mathcal R}}(x)\;and\;{\lambda}_0$ is the infimum of the spectrum of the Laplacian acting on $L^2$-functions on M.

• Journal of the Korean Mathematical Society
• /
• v.42 no.4
• /
• pp.621-634
• /
• 2005

Let$P{\rightarrow}M$ be an oriented $S^2-fiber$ bundle over a closed manifold M and let Q be its associated SO(3)-bundle, then we investigate the ring structure of the cohomology of the total space P by constructing the coupling form TA induced from an SO(3) connection A. We show that the cohomology ring of total space splits into those of the base space and the fiber space if and only if the Pontrjangin class $p_1(Q)\;{\in}\;H^4(M;\mathbb{Z})$ vanishes. We apply this result to the twistor spaces of 4-manifolds.

• Journal of the Korean Mathematical Society
• /
• v.53 no.2
• /
• pp.331-346
• /
• 2016

The primary aim of this paper is to generalize a theorem of Hirzebruch for the complex 2-dimensional Bott manifolds, usually called Hirzebruch surfaces, to more general Bott towers of height n. To do so, we first show that all complex vector bundles of rank 2 over a Bott manifold are classified by their total Chern classes. As a consequence, in this paper we show that two Bott manifolds $B_n({\alpha}_1,{\ldots},{\alpha}_{n-1},{\alpha}_n)$ and $B_n({\alpha}_1,{\ldots},{\alpha}_{n-1},{\alpha}_n^{\prime})$ are isomorphic to each other, as Bott towers if and only if both ${\alpha}_n{\equiv}{\alpha}_n^{\prime}$ mod 2 and ${\alpha}_n^2=({\alpha}_n^{\prime})^2$ hold in the cohomology ring of $B_{n-1}({\alpha}_1,{\ldots},{\alpha}_{n-1})$ over integer coefficients. This result will complete a circle of ideas initiated in [11] by Ishida. We also give some partial affirmative remarks toward the assertion that under certain condition our main result still holds to be true for two Bott manifolds just diffeomorphic, but not necessarily isomorphic, to each other.

• Journal of the Korean Mathematical Society
• /
• v.60 no.5
• /
• pp.1109-1133
• /
• 2023

We prove that the two-step flag variety 𝓕ℓ(1, n; n+1) carries a non-displaceable and non-monotone Lagrangian Gelfand-Zeitlin fiber diffeomorphic to S³ × T^2n-4 and a continuum family of non-displaceable Lagrangian Gelfand-Zeitlin torus fibers when n > 2.

• Journal of the Korean Mathematical Society
• /
• v.46 no.2
• /
• pp.363-447
• /
• 2009

The author previously defined the spectral invariants, denoted by $\rho(H;\;a)$, of a Hamiltonian function H as the mini-max value of the action functional ${\cal{A}}_H$ over the Novikov Floer cycles in the Floer homology class dual to the quantum cohomology class a. The spectrality axiom of the invariant $\rho(H;\;a)$ states that the mini-max value is a critical value of the action functional ${\cal{A}}_H$. The main purpose of the present paper is to prove this axiom for nondegenerate Hamiltonian functions in irrational symplectic manifolds (M, $\omega$). We also prove that the spectral invariant function ${\rho}_a$ : $H\;{\mapsto}\;\rho(H;\;a)$ can be pushed down to a continuous function defined on the universal (${\acute{e}}tale$) covering space $\widetilde{HAM}$(M, $\omega$) of the group Ham((M, $\omega$) of Hamiltonian diffeomorphisms on general (M, $\omega$). For a certain generic homotopy, which we call a Cerf homotopy ${\cal{H}}\;=\;\{H^s\}_{0{\leq}s{\leq}1}$ of Hamiltonians, the function ${\rho}_a\;{\circ}\;{\cal{H}}$ : $s\;{\mapsto}\;{\rho}(H^s;\;a)$ is piecewise smooth away from a countable subset of [0, 1] for each non-zero quantum cohomology class a. The proof of this nondegenerate spectrality relies on several new ingredients in the chain level Floer theory, which have their own independent interest: a structure theorem on the Cerf bifurcation diagram of the critical values of the action functionals associated to a generic one-parameter family of Hamiltonian functions, a general structure theorem and the handle sliding lemma of Novikov Floer cycles over such a family and a family version of new transversality statements involving the Floer chain map, and many others. We call this chain level Floer theory as a whole the Floer mini-max theory.