HAMILTONIAN CIRCLE ACTIONS ON EIGHT-DIMENSIONAL MANIFOLDS WITH MINIMAL FIXED SETS

Let the circle act in a Hamiltonian fashion on a closed 8-dimensional sym-plectic manifold M with exactly five fixed points, which is the smallest possible fixed set. In [GS], L. Godinho and S. Sabatini show that if M satisfies an extra “positivity condition” then the isotropy weights at the fixed points of M agree with those of some linear action on ℂℙ4. As a consequence, H *(M; ℤ) = ℤ[y]/y 5 and c(TM) = (1 + y)5. In this paper, we prove that their positivity condition holds for M. This completes the proof of the “symplectic Petrie conjecture” for Hamiltonian circle actions on 8-dimensional closed symplectic manifolds with minimal fixed sets.