On a symplectic generalization of Petrie's conjecture

Abstract

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 $H^{2i}(M;\R) = H^{2i}(\CP^n,\R)$ for all $i$. Is $H^j(M;\Z) = H^j(\CP^n;\Z)$ for all $j$? Is the total Chern class of $M$ determined by the cohomology ring $H^*(M;\Z)$? We answer these questions in the six dimensional case by showing that $H^j(M;\Z)$ is equal to $H^j(\CP^3;\Z)$ 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 $H^*(\CP^3;\Z)$ or $H^*(\Tilde{G}_2(\R^5);\Z)$, then the representations at the fixed components are compatible with one of the standard actions; in the remaining two case, 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.