Fine–Panov conjecture on six-dimensional monotone Hamiltonian circle manifolds

Let $(M,\omega)$ be a six dimensional closed monotone symplectic manifold, meaning that $\langle c_1(TM),[\Sigma]\rangle>0$ for every symplectic surface $\Sigma$, equipped with an effective Hamiltonian circle action. Fine–Panov conjecture. $(M,\omega)$ is $S^1$-equivariantly symplectomorphic to some Kähler manifold $(X,\omega_X,J)$ with a certain holomorphic Hamiltonian $S^1$-action. This conjecture concerns whether six-dimensional monotone symplectic manifolds with effective Hamiltonian circle actions must arise from Kähler geometry; the source presents it as an ongoing conjecture in the context of the still-unknown existence of closed monotone non-Kähler manifolds in dimension six.