Related papers: Circle actions on 8-dimensional almost complex man…

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 $(M, \omega)$ be a connected, compact 6-dimensional symplectic manifold equipped with a semi-free Hamiltonian $S^1$ action such that the fixed point set consists of isolated points or surfaces. Assume dim $H^2(M)<3$, in \cite{L}, we…

This is a slightly altered version of the authors thesis from 2014. In the first main part we show that the quotient space of a compact, simply connected and nonnegatively curved Riemannian 4-manifold by an effective, isometric…

The aim of this paper is to study compact 5--manifolds which admit fixed point free circle actions. The first result implies that the torsion in the second homology and the second Stiefel--Whitney class have to satisfy strong restrictions.…

We construct metrics of positive scalar curvature on manifolds with circle actions. One of our main results is that there exist $S^1$-invariant metrics of positive scalar curvature on every $S^1$-manifold which has a fixed point component…

Recall that an effective circle action is semifree if the stabilizer subgroup of each point is connected. We show that if $(M, \om)$ is a coadjoint orbit of a compact Lie group $G$ then every element of $\pi_1(G)$ may be represented by a…

Let $(M,\omega)$ be an eight-dimensional closed symplectic manifold equipped with a Hamiltonian circle action with only isolated fixed points. In this article, we will show that the Betti numbers of $M$ are unimodal, i.e. $b_0(M) \leq…

We construct, for $m\geq 6$ and $2n\leq m$, closed manifolds $M^{m}$ with finite nonzero $\varphi(M^{m},S^{n}$), where $\varphi(M,N)$ denotes the minimum number of critical points of a smooth map $M\to N$. We also give some explicit…

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

Let $M$ be a smooth $4k$-dimensional orientable closed manifold, and assume that $M$ has at most two non-zero Pontrjagin numbers which are associated to the top dimensional Pontrjagin class and the square of the middle dimensional…

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…

In this paper, we study $6$-dimensional GKM manifolds with $4$ fixed points. We classify all possible GKM graphs, and for each type of graph we construct a manifold, proving the existence. We show that six types occur. (P1) complex…

In this paper, we study fixed-point sets of $S^{1}$-actions and compatible complex structures on quaternionic manifolds. We obtain an equation involving the first Chern classes of the fixed-point set and of a quaternionically flat manifold…

Let $(X,\sigma,J)$ be a compact K\"{a}hler Calabi-Yau manifold equipped with a symplectic circle action. By Frankel's theorem \cite{F}, the action on $X$ is non-Hamiltonian and $X$ does not have any fixed point. In this paper, we will show…

On a smooth closed oriented $4$-manifold $M$ with a smooth action by a compact Lie group $G$, we define a $G$-monopole class as an element of $H^2(M;\Bbb Z)$ which is the first Chern class of a $G$-equivariant Spin$^c$ structure which has a…

Let the circle act effectively in a Hamiltonian fashion on a compact symplectic manifold $(M, \omega)$. Assume that the fixed point set $M^{S^1}$ has exactly two components, $X$ and $Y$, and that $\dim(X) + \dim(Y) +2 = \dim(M)$. We first…

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…

Let $M^m$ be a compact oriented smooth manifold which admits a smooth circle action with isolated fixed points which are isolated as singularities as well. Then all the Pontryagin numbers of $M^m$ are zero and its Euler number is…

Let $M^{2n}$ be a unitary torus $(2n)$-manifold, i.e., a $(2n)$-dimensional oriented stable complex connected closed $T^n$-manifold having a nonempty fixed set. In this paper we show that $M$ bounds equivariantly if and only if the…

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…