Point vortex dynamics on Kähler twistor spaces

In this paper, we provide tools to study the dynamics of point vortex dynamics on ${\mathbb{C}\mathbb{P}}^n$ and the flag manifold ${\mathbb{F}}_{1,2}({\mathbb{C}}^3)$. These are the only Kähler twistor spaces arising from 4-manifolds. We give an explicit expression for Green’s function on ${\mathbb{C}\mathbb{P}}^n$ which enables us to determine the Hamiltonian $H$ and the equations of motions for the point vortex problem on ${\mathbb{C}\mathbb{P}}^n$. Moreover, we determine the momentum map $\mu :{\mathbb{F}}_{1,2}({\mathbb{C}}^3)\to {\mathfrak{s}\mathfrak{u}}^{\ast }(3)$ on the flag manifold.