• Bulletin of the Korean Mathematical Society
• /
• v.36 no.1
• /
• pp.209-216
• /
• 1999

Let X be a minimal symplectic four-manifold with ${b_2}^{+}$=1 and $c_1(K)^2\;\geq\;0$. Then we show that there are no symple tic structures $\omega$ such that $$c_1(K)$\cdot\omega$ > 0, if X contains an embedded symplectic submanifold $\Sigma$ satisfying $\int_\Sigmac_1$(K)<0.

• 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.47 no.4
• /
• pp.675-689
• /
• 2010

For a closed symplectic 4-manifold X, let $Diff_0$(X) be the group of diffeomorphisms of X smoothly isotopic to the identity, and let Symp(X) be the subgroup of $Diff_0$(X) consisting of symplectic automorphisms. In this paper we show that for any finitely given collection of positive integers {$n_1$, $n_2$, $\ldots$, $n_k$} and any non-negative integer m, there exists a closed symplectic (or K$\ddot{a}$hler) 4-manifold X with $b_2^+$ (X) > m such that the homologies $H_i$ of the quotient space $Diff_0$(X)/Symp(X) over the rational coefficients are non-trivial for all odd degrees i = $2n_1$ - 1, $\ldots$, $2n_k$ - 1. The basic idea of this paper is to use the local invariants for symplectic 4-manifolds with contact boundary, which are extended from the invariants of Kronheimer for closed symplectic 4-manifolds, as well as the symplectic compactifications of Stein surfaces of Lisca and Mati$\acute{c}$.

• Communications of the Korean Mathematical Society
• /
• v.11 no.2
• /
• pp.463-469
• /
• 1996

We construct closed orientable four-manifolds which have nontrivial Seiberg-Witten invariants, but do not admit any symplectic structure.

• Bulletin of the Korean Mathematical Society
• /
• v.38 no.4
• /
• pp.741-751
• /
• 2001

In this paper when $(M, \omega)$ is a compact weakly exact symplectic manifold with nonempty boundary satisfying $c_1｜{\pi}_2(M)$ = 0, we construct an action spectrum bundle over the group of Hamil-tonian diffeomorphisms of the manifold M generated by the time-dependent Hamiltonian vector fields, whose fibre is nowhere dense and invariant under symplectic conjugation.

• Journal of the Korean Mathematical Society
• /
• v.43 no.4
• /
• pp.691-701
• /
• 2006

In this paper, we construct a new exotic smooth 4-manifold X which is homeomorphic, but not diffeomorphic, to ${\mathbb{C}}\mathbb{P}^2{\sharp}13\overline{\mathbb{C}\mathbb{P}}^2$. Moreover the manifold X has vanishing Seiberg-Witten invariants for all $Spin^c$-structures of X and has no symplectic structure.

• Journal of the Chungcheong Mathematical Society
• /
• v.25 no.4
• /
• pp.739-745
• /
• 2012

We show that if a symplectic diffeomorphism has the $C^1$-robustly orbital shadowing property, then the diffeomorphism is Anosov.

• Journal of the Korean Mathematical Society
• /
• v.41 no.5
• /
• pp.809-820
• /
• 2004

On $R^{2n}$, n$\geq$2, with the standard symplectic structure we construct compatible almost K hler metrics with negative scalar curvature on a polydisc which are Euclidean away from the polydisc.c.

• Journal of the Chungcheong Mathematical Society
• /
• v.36 no.1
• /
• pp.1-8
• /
• 2023

Let C be a smooth projective curve and W a symplectic bundle over C of degree w. Let π : 𝕃𝔾(W) → C be the relative Lagrangian Grassmannian over C and S_d(W) be the space of Lagrangian subbundles of degree w -d. Then Kontsevich's space ${\bar{\mathcal{M}}}_g$(𝕃𝔾(W), β_d) of stable maps to 𝕃𝔾(W) is a compactification of S_d(W). In this article, we give an upper bound on the dimension of ${\bar{\mathcal{M}}}_g$(𝕃𝔾(W), β_d), which is an analogue of a result in [8] for the relative Lagrangian Grassmannian.

• Communications of the Korean Mathematical Society
• /
• v.29 no.2
• /
• pp.285-293
• /
• 2014

Let M be a closed, symplectic connected Riemannian manifold and f a symplectomorphism on M. We prove that if f is $C^1$-stably weak shadowable on M, then the whole manifold M admits a partially hyperbolic splitting.