On the Chern numbers and the Hilbert polynomial of an almost complex manifold with a circle action

Let (M,J) be a compact, connected, almost complex manifold of dimension 2n endowed with a J-preserving circle action with isolated fixed points. In this paper, we analyze the “geography problem” for such manifolds, deriving equations relating the Chern numbers to the index k0 of (M,J). We study the symmetries and zeros of the Hilbert polynomial, which imply many rigidity results for the Chern numbers when k0≠1. We apply these results to the category of compact, connected symplectic manifolds. A long-standing question posed by McDuff and Salamon asked about the existence of non-Hamiltonian actions with isolated fixed points. This question was answered recently by Tolman, with an explicit construction of a 6-dimensional manifold with such an action. One issue that this raises is whether one can find topological criteria that ensure the manifold can only support a Hamiltonian or only a non-Hamiltonian action. In this vein, we are able to deduce such criteria from our rigidity theorems in terms of relatively few Chern numbers, depending on the index. Another consequence is that, if the action is Hamiltonian, the minimal Chern number coincides with the index and is at most n + 1.