On a symplectic generalization of Petrie's conjecture

Motivated by the Petrie conjecture, we consider the following questions: Let a circle act in a Hamiltonian fashion on a compact symplectic manifold $(M,\omega)$ which satisfies $H^{2i}(M;\mathbb{R}) = H^{2i}({\mathbb{C}}{\mathbb{P}}^n,\mathbb{R})$ for all $i$. Is $H^j(M;\mathbb{Z}) = H^j({\mathbb{C}}{\mathbb{P}}^n;\mathbb{Z})$ for all $j$? Is the total Chern class of $M$ determined by the cohomology ring $H^*(M;\mathbb{Z})$? We answer these questions in the six-dimensional case by showing that $H^j(M;\mathbb{Z})$ is equal to $H^j({\mathbb{C}}{\mathbb{P}}^3;\mathbb{Z})$ for all $j$, by proving that only four cohomology rings can arise, and by computing the total Chern class in each case. We also prove that there are no exotic actions. More precisely, if $H^*(M;\mathbb{Z})$ is isomorphic to $H^*({\mathbb{C}}{\mathbb{P}}^3;\mathbb{Z})$ or $H^*(\widetilde{G}_2(\mathbb{R}^5);\mathbb{Z})$, then the representations at the fixed components are compatible with one of the standard actions; in the remaining two cases, the representation is strictly determined by the cohomology ring. Finally, our results suggest a natural question: Do the remaining two cohomology rings actually arise? This question is closely related to some interesting problems in symplectic topology, such as embeddings of ellipsoids.