发表机构
Kspectra Research Inc.（Kspectra 研究公司）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明了Taub-NUT度量的Kähler实现不存在任何倍数的Kähler浸入复射影空间，通过Lambert W函数和Vivanti-Pringsheim定理给出一般障碍，并证实了相关猜想。
AI 中文摘要
LeBrun对$\mathbb{C}^2$上Taub-NUT度量的Kähler实现$g_m$是完备的、Ricci平坦的且非平坦的。Loi、Zedda和Zuddas证明了当$m>\alpha/2$时，没有倍数$\alpha g_m$允许嵌入到有限维或无限维复射影空间中的Kähler浸入，并猜想对每个$m>0$该结论同样成立。我们证明了这个猜想。Kähler势在轴$z_2=0$上的限制由Lambert $W$函数控制，因此$\exp(\alpha\Phi_m)$作为$|z_1|^2$的幂级数具有有限收敛半径，尽管它在整个半直线上是实解析的；Vivanti-Pringsheim定理禁止非负Taylor系数，且Calabi准则失效。我们将Arezzo、Loi、Placini和Zedda最近用于径向度量的机制表述为Kähler浸入的一般障碍。在统计术语中，$g_m$的轴限制将是一个自然指数族，其均值域为$(0,\infty)$，方差函数为$\mu/(1+2m\mu)$；该论证给出了已知事实的一个初等证明，即Bar-Lev、Bshouty和Enis证明了对于任何$c>0$，不存在方差函数为$\mu/(1+c\mu)$的这样的族。该结果证实了Loi、Salis和Zuddas猜想的一个更多情形，即Ricci平坦的射影诱导Kähler度量是平坦的。证明的解析核心已在Lean 4中通过机器检验。
英文摘要
LeBrun's Kähler realization $g_m$ of the Taub--NUT metric on $\mathbb{C}^2$ is complete, Ricci-flat and not flat. Loi, Zedda and Zuddas proved that no multiple $αg_m$ admits a Kähler immersion into a finite- or infinite-dimensional complex projective space when $m>α/2$, and conjectured that the same holds for every $m>0$. We prove the conjecture. The restriction of the Kähler potential to the axis $z_2=0$ is governed by the Lambert $W$ function, so $\exp(αΦ_m)$ has a finite radius of convergence as a power series in $|z_1|^2$ although it is real analytic on the whole half-line; the Vivanti--Pringsheim theorem forbids nonnegative Taylor coefficients, and Calabi's criterion fails. We state the mechanism, which Arezzo, Loi, Placini and Zedda recently used for radial metrics, as a general obstruction to Kähler immersions. In statistical terms the axis restriction of $g_m$ would be a natural exponential family with mean domain $(0,\infty)$ and variance function $μ/(1+2mμ)$; the argument gives an elementary proof of the known fact, due to Bar-Lev, Bshouty and Enis, that no such family exists with variance function $μ/(1+cμ)$ for any $c>0$. The result confirms onemore case of the conjecture of Loi, Salis and Zuddas that Ricci-flat projectively induced Kähler metrics are flat. The analytic core of the proof has been machine-checked in Lean~4.
