Classification of six-dimensional monotone symplectic manifolds admitting semifree circle actions II

Abstract

This is the second of the series of papers on the classification of six-dimensional closed monotone symplectic manifold admitting a semifree Hamiltonian [Formula: see text]-action. In [Y. Cho, Classification of six-dimensional monotone symplectic manifolds admitting semifree circle actions I, Int. J. Math. 6 1950032], we dealt with the case where at least one of the extremal fixed point is isolated and proved that every such manifold is Kähler Fano. In this paper, we show that if the maximal and the minimal fixed components are both two-dimensional, then the manifold is [Formula: see text]-equivariantly symplectomorphic to some Kähler Fano manifold with a certain holomorphic Hamiltonian [Formula: see text]-action. We also give a complete list of such Fano manifolds together with an explicit description of the [Formula: see text]-actions.

Symplectic and Kähler structures on $$\mathbb CP^1$$-bundles over $$\mathbb CP^2$$

“…One could try to prove this claim using the fact that the Hamiltonian S 1 -action induced by the Hamiltonian φ * (y) is semi-free. This permits one to apply the classification result of Gonzales [18] in the spirit of the work [6][7][8]. We believe that this should allow one to prove that any two such manifolds are at least equivarianlty S 1 -symplectomorphic.…”

Classification of six-dimensional monotone symplectic manifolds admitting semifree circle actions I

“…Moreover, it turned out that there are 18 types and 21 types of such manifolds up to S 1 -equivariant symplectomorphism, respectively. In this paper (sequel to [Cho1,Cho2]), we complete the classification and prove the following.…”

“…In a series of papers [Cho1,Cho2], the author studied the existence of Kähler structure on a six-dimensional closed symplectic manifold (M, ω) admitting a Hamiltonian circle action. More precisely, the author proved that any six-dimensional closed monotone semifree 1 (M, ω) Hamiltonian S 1 -manifold (M, ω) is S 1 -equivariantly symplectomorphic to some Kähler Fano manifold with a certain holomorphic Hamiltonian circle action in the case where • (at least) one of extremal fixed components is an isolated point, or • any extremal fixed component is two dimensional.…”

“…Among those manifolds, it is known that CP 2 , S 2 ×S 2 , and X k for 1 ≤ k ≤ 4 are known to be symplectically rigid (see Definition 2.4). We crucially used this fact in the previous works [Cho1,Cho2] where no X k (with k > 4) appears as a reduced space.…”

“…On the other hand, unlike the case of [Cho1,Cho2], X k (k > 4) appears as a reduced space so that the Gonzalez's theorem cannot be applied directly. (See (II-1-4.k), k=2 ∼ 8 in Figure 1.)…”