Strong Cohomological rigidity of Bott manifolds
We show that the strong cohomological rigidity conjecture for Bott manifolds is true. Namely, any graded cohomology ring isomorphism between two Bott manifolds is induced by a diffeomorphism.
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arXiv subjects
Taekgyu Hwang
Publications and source records attributed to Taekgyu Hwang.
Gromov width of symplectic toric manifolds associated with graphs
We give an explicit formula for the Gromov width for a class of symplectic toric manifolds constructed from simple graphs. As a corollary, we show a version of non-squeezing theorem with respect to the inclusion of connected graphs.
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The Gromov width of generalized Bott manifolds
By Delzant's theorem, closed symplectic toric manifolds are classified by the images of moment maps. In the case of a generalized Bott manifold, this image is a polytope $P$ combinatorially equivalent to the product of simplices. We compute the Gromov width of generalized Bott manifolds in terms of the defining inequalities of $P$.
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Symplectic capacities from Hamiltonian circle actions
Let $M$ be a closed Fano symplectic manifold with a semifree Hamiltonian circle action with isolated maximum. We compute the Gromov width and the Hofer-Zehnder capacity of $M$ using a moment map.
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Semifree Hamiltonian circle actions on 6-dimensional symplectic manifolds with non-isolated fixed point set
Let $(M, ω)$ be a 6-dimensional closed symplectic manifold with a symplectic $S^1$-action with $M^{S^1} \neq \emptyset$ and $\dim M^{S^1} \leq 2$. Assume that $ω$ is integral with a generalized moment map $μ$. We first prove that the action is Hamiltonian if and only if $b_2^+(M_{\red})=1$, where $M_{\red}$ is any reduced space with respect to $μ$. It means that if the action is non-Hamiltonian, then $b_2^+(M_{\red}) \geq 2$. Secondly, we focus on the case when the action is semifree and Hamiltonian. We prove that if $M^{S^1}$ consists of surfaces, then the number $k$ of fixed surfaces with positive genera is at most four. In particular, if the extremal fixed surfaces are spheres, then $k$ is at most one. Finally, we prove that $k \neq 2$ and we construct some examples of 6-dimensional semifree Hamiltonian $S^1$-manifolds such that $M^{S^1}$ contains $k$ surfaces of positive genera for $k = 0$ and 4. Examples with $k=1$ and 3 were given in \cite{L2}.
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