In this paper, we provide tools to study the dynamics of point vortex dynamics on $\mathbb{ CP}^n$ and the flag manifold $\mathbb{ F}_{ 1,2}(\mathbb{ C}^3)$. These are the only K\"ahler twistor spaces arising from 4-manifolds. We give an explicit expression for Green's function on $\mathbb{ CP}^n$ which enables us to determine the Hamiltonian $H$ and the equations of motions for the point vortex problem on $\mathbb{ CP}^n$. Moreover, we determine the momentum map $\mu:\mathbb{ F}_{ 1,2}(\mathbb{ C}^3)\to \mathfrak{ su}^*(3)$ on the flag manifold.
Categories: math.SG math-ph math.DG math.DS math.MP