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

For every integer n≥2n\geq 2, every point x∈Pnx\in\mathbf P^n, and every λ∈[0,1]\lambda\in[0,1], does there exist a function u∈DMA(Pn,ωFS)u\in DMA(\mathbf P^n,\omega_{\mathrm{FS}}) such that u−1(−∞)={x}u^{-1}(-\infty)=\{x\}, (ωFS+ddcu)n=δx(\omega_{\mathrm{FS}}+dd^c u)^n=\delta_x, and ν(u,x)=λ\nu(u,x)=\lambda? Equivalently, is the one-pole Lelong-number range on (Pn,ωFS)(\mathbf P^n,\omega_{\mathrm{FS}}) exactly [0,1][0,1]?

References

Primary source

arXiv

Additional references

algebraic-geometrycomplex-analysispluripotential-theory
Posted

Progress summary

Refreshed
Claimed solved

A September 2026 preprint claims to settle which strengths of a single prescribed singularity are possible on projective space, but the result has not been independently verified.

The problem asks whether every value in [0,1][0,1] can occur as the Lelong number of a one-pole Green function on complex projective space. The reported preprint claims an affirmative answer and connects it with Seshadri intervals on selected varieties.

Known results

  • A 2009 study treated one-pole Green functions on compact Kähler manifolds, including Pn\mathbb{P}^{n}, and characterized the maximal value 11.
  • A 2005 note characterized Seshadri constants through positively curved singular metrics with an isolated pole and prescribed Lelong number.

September 2, 2026 claimed resolution

The linked preprint claims the exact one-pole range is [0,1][0,1] and derives new Seshadri-interval results for selected varieties. This is an unrefereed claim; the scan found no independent verification, objection, or named author attribution.

Current status (as of September 2026): The exact range [0,1][0,1] is claimed by an unrefereed preprint but remains unverified; the associated Seshadri results are likewise not independently confirmed.

Sources

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