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 ) (M,ω ) which satisfies H 2 i ( M ; R ) = H 2 i ( C P n , R ) H²ⁱ(M;R) = H²ⁱ({ C}{ P}^n,R) for all i i . Is H j ( M ; Z ) = H j ( C P n ; Z ) H^j(M;Z) = H^j({ C}{ P}^n;Z) for all j j ? Is the total Chern class of M M determined by the cohomology ring H ∗ ( M ; Z ) H^*(M;Z) ? We answer these questions in the six-dimensional case by showing that