Point Vortex Dynamics on Kähler Twistor Spaces

The paper proves an explicit Green’s function on CP^n with the Fubini–Study metric (Theorem 5.5) by solving a 1D ODE for ϕ(r) on CROSS/locally harmonic Blaschke manifolds using the volume density V_CP^n and the integral identity in Lemma 5.4; this yields G(ξ,η) = −(2n·vol(CP^n))^{-1}[log(sin r) − Σ_{j=1}^{n−1}(2j)^{-1} sin^{-2j} r] up to an additive constant, which is acceptable since Green’s functions on compact manifolds are defined modulo constants (their property (4) only enforces a constant mean in the second argument) . The model independently derives the same core expression via the radial Laplacian and a telescoping identity and correctly obtains ΔG = δ_η − 1/vol in distribution. However, it contains two sign mistakes: (i) a sign slip in the flux step (they equate ∫_{Ω_ε}ΔG with + rather than − the boundary integral at the end), and (ii) an incorrect sign for the mean-zero normalization constant—it subtracts a positive constant inside the brackets, which increases the mean instead of canceling it. The paper’s result stands; the model’s normalization claim is wrong. The paper also clearly states the definition of Green’s function used (property (1)) and that the integral over the second variable is a constant (property (4)), so not fixing the additive constant is consistent with their framework .