Prescribed Lelong Numbers for One-Pole Green Functions on Complex Projective Space

Abstract

Let $ω_{\mathrm{FS}}$ be the normalized Fubini--Study form on $\mathbf P^n$, with $n\geq2$. We prove that the one-pole Lelong-number range in $DMA(\mathbf P^n,ω_{\mathrm{FS}})$ is exactly $[0,1]$: for every $λ$ in this interval there is a Green function with a single pole, Monge--Ampère measure equal to the Dirac mass at that pole, and Lelong number $λ$. This answers Question~9 in the survey of Dinew--Guedj--Zeriahi, where the problem is attributed to Coman and Guedj. The construction adapts Li and Xia's local zero-Lelong-number scheme through variable degrees and homogeneous finite stages, while also establishing the exact Lelong number and membership in the global $DMA$ class. We then study the relation between the one-pole range $\mathcal R_α(x)$ and the Seshadri interval $[0,\varepsilon(α,x)]$. An application of Koike's equivalence theorem gives a point on a degree-one del Pezzo surface where the Seshadri endpoint is not attained. Conversely, a finite-pullback criterion and an explicit finite morphism show that every ample rational class on a product of projective spaces realizes its full Seshadri interval at every point.