Compact symplectic manifolds with free circle actions, and Massey products (Q803958)

The main result of this paper is the following theorem: Let \((M^{2n},F_0)\) be a compact symplectic manifold with a free symplectic \(S^1\)-action that induces the fibration \(\pi: M^{2n}\to B^{2n-1}\). Then, there exist \(r>0\), a compact manifold \(U\), and a diffeomorphism \(\phi: U\to U\) such that: (i) \(B\cong U\times\mathbb{R}|_{\{(x,t)\sim (\phi (x),t-r)\}}\) (the mapping torus); (ii) \(U\) has a family \(\Omega_t\) of symplectic forms, where \(\phi^*\Omega_{t-r}=\Omega_t\) and \([\Omega_t] = [\Omega_0]+t\xi\) for a \(\phi\)-invariant class \(\xi \in H^2(U,\mathbb{Z}),\) which is the restriction to \(U\times \{0\}\) of the Euler class of \(\pi\); (iii) \(M\) has an \(S^1\)-invariant symplectic form \(F=\pi^*(\Omega)+\pi^*(\alpha)\wedge \eta\), where \(\eta\) is a connection form on \(M\), \(\alpha\) is the pullback of the form \(d\theta/2\pi\) of \(S^1\), and \(\Omega =\{\Omega_t\}\) seen as a form on \(B\). Furthermore, \(U\), \(\phi\), \(\eta\) can be chosen such that \(F\) is arbitrarily close to \(F''_0\). The other topic of the paper is the study of non-existence of positive definite Kähler metrics on the symplectic manifolds that are like in the previous theorem.