复射影空间上单极点格林函数的指定Lelong数

中文摘要 AI 辅助

设$\omega_{\mathrm{FS}}$为$\mathbf P^n$上的正规化Fubini--Study形式，其中$n\geq2$。我们证明$DMA(\mathbf P^n,\omega_{\mathrm{FS}})$中单极点的Lelong数取值范围恰好是$[0,1]$：对该区间内的任意$\lambda$，都存在一个单极点格林函数，其Monge--Ampère测度等于该极点处的Dirac质量，且Lelong数为$\lambda$。这解答了Dinew--Guedj--Zeriahi综述中的问题9，该问题由Coman和Guedj提出。我们的构造通过可变次数和齐次有限阶段调整了Li和Xia的局部零Lelong数方案，同时确立了精确的Lelong数以及全局$DMA$类的隶属关系。随后我们研究了单极点取值范围$\mathcal R_\alpha(x)$与Seshadri区间$[0,\varepsilon(\alpha,x)]$之间的关系。应用Koike等价定理，我们在一次del Pezzo曲面上得到了一个Seshadri区间端点不可达的点。反过来，有限拉回准则和一个显式有限态射表明，射影空间乘积上的每个丰沛有理类在每一点都能实现其完整的Seshadri区间。

英文摘要

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.

发表机构

• Chern Institute of Mathematics and LPMC, Nankai University（南开大学陈省身数学研究所和LPMC）

机构由 AI 辅助整理，请以论文原文为准。