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,ω) which satisfies H²ⁱ(M;ℝ) = H ²ⁱ(ℂℙⁿ,ℝ) for all i. Is Hj (M; ℤ) = H^j (ℂℙⁿ; ℤ) for all j? Is the total Chern class of M determined by the cohomology ring H*(M; ℤ)? We answer these questions in the sixdimensional case by showing that H^j (M; ℤ) is equal to H^j (ℂℙ³; 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; ℤ) is isomorphic to H *(ℂℙ³; ℤ) or H* (G̃₂(ℝ⁵); 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.