New Tools for Classifying Hamiltonian Circle Actions with Isolated Fixed Points

For every compact almost complex manifold \((\mathsf {M},\mathsf {J})\) equipped with a \(\mathsf {J}\)-preserving circle action with isolated fixed points, a simple algebraic identity involving the first Chern class is derived. This enables us to construct an algorithm to obtain linear relations among the isotropy weights at the fixed points. Suppose that \(\mathsf {M}\) is symplectic and the action is Hamiltonian. If the manifold satisfies an extra so-called positivity condition, then this algorithm determines a family of vector spaces that contain the admissible lattices of weights. When the number of fixed points is minimal, this positivity condition is necessarily satisfied whenever \(\dim (\mathsf {M})\le 6\) and, when \(\dim (\mathsf {M})=8\), whenever the \(S^1\)-action extends to an effective Hamiltonian \(T^2\)-action, or none of the isotropy weights is \(1\). Moreover, there are no known examples with a minimal number of fixed points contradicting this condition, and their existence is related to interesting questions regarding fake projective spaces. We run the algorithm for \(\dim (\mathsf {M})\le 8\), quickly obtaining all the possible families of isotropy weights. In particular, we simplify the proofs of Ahara and Tolman for \(\dim (\mathsf {M})=6\) and, when \(\dim (\mathsf {M})=8\), we prove that the equivariant cohomology ring, Chern classes, and isotropy weights agree with those of \({\mathbb {C}}P^4\) with the standard \(S^1\)-action (thereby proving the symplectic Petrie conjecture in this setting).