In the paper "S^1-actions on homotopy complex projective spaces" by Petrie (Bulletin of the AMS, 1972), Petrie constructs a smooth circle action on $\mathbb{CP}^{3}$ (page 148). The fixed point set consists of $4$ points and the weights at these fixed points cannot be equal to any linear circle action (A linear circle action corresponds to a subgroup $S^{1} \hookrightarrow PGL_{4}(\mathbb{C})$ acting on projective space in the usual way).

**Question**: Does this circle action preserve an almost complex structure $J$ on $\mathbb{CP}^{3}$?

**Remarks.** 1. It is a classical result that holomorphic $S^{1}$-actions on complex projective spaces are linear (with the standard complex structure), so $J$ cannot be integrable and have a compatible Kähler metric (by Hirzebruch-Kodaira theorem).

- It follows from Tolmans result ("A symplectic generalisation of Petries Conjecture", Transactions of the AMS, 2010), that if the circle action preserves a symplectic form then it is linear, so there is no $S^{1}$-invariant symplectic form.