The Taub-NUT Metric Is Not Projectively Induced

Abstract

LeBrun's K\"ahler 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 $\alpha g_m$ admits a K\"ahler immersion into a finite- or infinite-dimensional complex projective space when $m>\alpha/2$, and conjectured that the same holds for every $m>0$. We prove the conjecture. The restriction of the K\"ahler potential to the axis $z_2=0$ is governed by the Lambert $W$ function, so $\exp(\alpha\Phi_m)$ 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 K\"ahler immersions. In statistical terms the axis restriction of $g_m$ would be a natural exponential family with mean domain $(0,\infty)$ and variance function $\mu/(1+2m\mu)$; 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 $\mu/(1+c\mu)$ for any $c>0$. The result confirms onemore case of the conjecture of Loi, Salis and Zuddas that Ricci-flat projectively induced K\"ahler metrics are flat. The analytic core of the proof has been machine-checked in Lean~4.