Prescribed Lelong Numbers for One-Pole Green Functions on Complex Projective Space
Abstract
Let ωFS be the normalized Fubini--Study form on Pn, with n≥2. We prove that the one-pole Lelong-number range in DMA(Pn,ωFS) 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 ℛα(x) and the Seshadri interval [0,ε(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.
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BibTeX
@misc{qin2026prescribed,
title = {{Prescribed Lelong Numbers for One-Pole Green Functions on Complex Projective Space}},
author = {Xiangsen Qin},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.02513},
eprint = {2609.02513},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.02513},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - Prescribed Lelong Numbers for One-Pole Green Functions on Complex Projective Space
AU - Xiangsen Qin
PY - 2026
DA - 2026/09/02
PB - arXiv
DO - 10.48550/arxiv.2609.02513
UR - https://arxiv.org/abs/2609.02513
AB - Let ω_FS be the normalized Fubini--Study form on P^n, with n≥2. We prove that the one-pole Lelong-number range in DMA(P^n,ω_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 ℛ_α(x) and the Seshadri interval [0,ε(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.
ER -