Hamiltonian circle actions with dimension constraints (Q2875837)

In this survey article, the author discusses some recent progress (mainly since 2010) on the classification of Hamiltonian and symplectic circle actions on compact, connected symplectic manifolds. This was solved in dimension 4, for Hamiltonian actions, by Audin, Ahara-Hattori and Karshon, and this paper is focused on the higher-dimensional case.NEWLINENEWLINEFor Hamiltonian actions, various constraints on the cohomology of such manifolds have been proved, under various assumptions on NEWLINE{\parindent=6mmNEWLINE\begin{itemize}\item[{\(\bullet\)}] the fixed point set of the action (usually assumed to have 2 connected components); NEWLINE\item[{\(\bullet\)}] integrality of the cohomology class of the symplectic form; NEWLINE\item[{\(\bullet\)}] the equivariant Euler class of the normal bundle to the minimal fixed point component (the moment map being of pure type).NEWLINENEWLINE\end{itemize}} NEWLINEIn addition, restrictions on the stabilisers of such an action have been proved.NEWLINENEWLINEFor symplectic actions, the author proved a uniqueness result for actions with three fixed points. Finally, the author discusses progress on the question if a symplectic circle action with only isolated (and at least one) fixed points is always Hamiltonian.NEWLINENEWLINEThis paper is a concise and interesting overview of the state of the art in this area of symplectic geometry.NEWLINENEWLINEFor the entire collection see [Zbl 1284.00072].