Related papers: Circle actions on almost complex manifolds with 4 …

In an earlier paper, the second author resolved a question of McDuff by constructing a non-Hamiltonian symplectic circle action on a closed, connected six-dimensional symplectic manifold with exactly 32 fixed points. In this paper, we…

We study the problem of determining which diffeomorphism classes of K\"{a}hler manifolds admit a Hamiltonian circle action. Our main result is the following: Let $M$ be a closed symplectic manifold, diffeomorphic to a complete intersection…

Let the circle act in a Hamiltonian fashion on a connected compact symplectic manifold $(M, \omega)$ of dimension $2n$. Then the $S^1$-action has at least $n+1$ fixed points. In a previous paper, we study the case when the fixed point set…

For every compact almost complex manifold (M,J) equipped with a J-preserving circle action with isolated fixed points, a simple algebraic identity involving the first Chern class is derived. This enables us to construct an algorithm to…

We construct a non-Hamiltonian symplectic circle action on a closed, connected, six-dimensional symplectic manifold with exactly 32 fixed points.

Consider a Hamiltonian circle action on a closed $8$-dimensional symplectic manifold $M$ with exactly five fixed points, which is the smallest possible fixed set. In their paper, L. Godinho and S. Sabatini show that if $M$ satisfies an…

Let $(M, \omega)$ 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 $\omega$ is integral with a generalized moment map $\mu$. We first prove…

Consider a compact symplectic manifold of dimension $2n$ with a Hamiltionan circle action. Then there are at least $n+1$ fixed points. Motivated by recent works on the case that the fixed point set consists of precisely $n+1$ isolated…

Let $G$ be a compact Lie group acting effectively by isometries on a compact Riemannian manifold $M$ with nonempty fixed point set $Fix(M,G)$. We say that the action is \emph{fixed point homogeneous} if $G$ acts transitively on a normal…

We prove that if the circle group acts smooth and unitary on 2n-dimensional stably complex manifold with two isolated fixed points and it is not bound equivariantly, then n=1 or 3. Our proof relies on the rigid Hirzebruch genera.

An almost complex torus manifold is a $2n$-dimensional compact connected almost complex manifold equipped with an effective action of a real $n$-dimensional torus $T^n \simeq (S^1)^n$ that has fixed points. For an almost complex torus…

We study smooth, closed orientable $S^1$-manifolds $M$ with exactly $3$ fixed points. We show that the dimension of $M$ is of the form $4\cdot 2^a$ or $8\cdot(2^a+2^b)$ with $a,b\geq 0$ and $a\neq b$. Moreover, under the extra assumption…

Following the idea of Lusztig, Atiyah-Hirzebruch and Kosniowski, we note that the Dolbeault-type operators on compact, almost-complex manifolds are rigid. When the circle action has isolated fixed points, this rigidity result will produce…

The study of fixed points is a classical subject in geometry and dynamics. If the circle acts in a Hamiltonian fashion on a compact symplectic manifold M, then it is classically known that there are at least 1 + dim(M)/2 fixed points; this…

Let $G$ be a compact Lie group acting isometrically on a compact Riemannian manifold $M$ with nonempty fixed point set $M^G$. We say that $M$ is fixed-point homogeneous if $G$ acts transitively on a normal sphere to some component of $M^G$.…

Let a $k$-dimensional torus $T^k$ act on a $2n$-dimensional compact connected almost complex manifold $M$ with isolated fixed points. As for circle actions, we show that there exists a (directed labeled) multigraph that encodes weights at…

Let $M$ be a $2n$-dimensional closed symplectic manifold admitting a Hamiltonian circle action with isolated fixed points. We show that if $M$ contains an $S^1$-invariant symplectic hypersurface $D$ such that $M\setminus D$ is a homology…

Let $(M,J)$ be a compact, connected, almost complex manifold of dimension $2n$ endowed with a $J$-preserving circle action with isolated fixed points. In this note we analyse the `geography problem' for such manifolds, deriving equations…

We study $3$-folds with an action of a algebraic torus $T$ and finite fixed point set. In particular, assuming the torus action has (exactly) $6$ fixed points we show that aside from Mori fibre spaces, the topology of such spaces is…

We obtain a general lower bound for the number of fixed points of a circle action on a compact almost complex manifold $M$ of dimension $2n$ with nonempty fixed point set, provided the Chern number $c_1c_{n-1}[M]$ vanishes. The proof…