The Taub–NUT Metric Is Not Projectively Induced

Shaosai Huang

doi:10.2139/ssrn.7484238

Working paper · September 2026

The Taub–NUT Metric Is Not Projectively Induced

Shaosai Huang

∗ Working paper. Comments welcome.

Kspectra Research Inc., Toronto, Canada · [email protected]

Abstract

LeBrun’s Kähler realization $g_m$ of the Taub–NUT metric on $\C ^2$ is complete, Ricci-flat and not flat. Loi, Zedda and Zuddas proved that no multiple $\alpha g_m$ admits a Kähler 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 Kähler 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 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 $\mu /(1+2m\mu )$ ; 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 one more 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.

Contents

1Introduction
2Preliminaries
3Proof of Theorem A
4The Pringsheim obstruction
5Exponential families
6Interpretation and scope
References
How to cite

2020 Mathematics Subject Classification. Primary 53C55; Secondary 32Q15, 53C25, 30B10, 62E10.

Keywords. Taub–NUT metric; projectively induced Kähler metrics; Calabi’s diastasis; Ricci-flat Kähler metrics; Vivanti–Pringsheim theorem; Lambert $W$ function; natural exponential families; variance functions.

1Introduction

This note settles a conjecture of Loi, Zedda and Zuddas about the Taub–NUT metric. The question is whether LeBrun’s Kähler model of this metric on $\C ^2$ , or any constant multiple of it, can be induced holomorphically and isometrically from a complex projective space. We show that it cannot. The obstruction is a single branch point of the Lambert $W$ function, and it has a natural counterpart for exponential families in statistics.

A Kähler metric $g$ on a complex manifold $M$ is projectively induced if $(M,g)$ admits a Kähler immersion, that is a holomorphic isometric immersion, into a complex projective space $(\CP ^N,g_{\FS })$ with $N\le \infty$ . Calabi’s diastasis [Cal53] reduces the existence of such an immersion near a point to the positive semidefiniteness of an explicit Hermitian matrix, and much of the subsequent theory asks which canonical metrics pass this test; see the monograph [LZ18]. For Einstein metrics the answer is expected to be very rigid. Umehara [Ume87] showed that Einstein Kähler submanifolds of $\C ^N$ and of complex hyperbolic space are totally geodesic, Hulin [Hul00] showed that a compact Kähler–Einstein submanifold of a finite-dimensional projective space has positive scalar curvature, and Loi, Salis and Zuddas formulated the following conjecture [LSZ18, Conjecture 1].

A Ricci-flat projectively induced Kähler metric is flat.

The conjecture is known for immersions into finite-dimensional projective spaces [ALL25]; for metrics with a Kähler potential depending only on $|z_1|^2,\dots ,|z_n|^2$ in coordinates centred at a point, this case already follows from Salis’s theorem that such a Kähler–Einstein metric, if finitely projectively induced, has positive Einstein constant [Sal17, Theorem 1.1]. In infinite dimension it is known for radial metrics, first under a stability assumption [LSZ18] and then in general [ALPZ26], for Calabi’s Ricci-flat metrics on line bundles over compact Kähler–Einstein manifolds, in particular for the Eguchi–Hanson metric [LZZ21], and for Kähler cones over regular complete Sasakian manifolds [MTZ24]. For Stenzel’s metrics on the complexifications of $\CP ^n$ and $\mathbb {HP}^n$ , $n\ge 2$ , it is known for the multiples $cg$ with $0<c\le 1$ [Zed21, Theorem 1].

LeBrun [LeB91] realized the Taub–NUT metric as a complete Ricci-flat Kähler metric on $\C ^2$ , showing that such metrics need not be flat. For $m\ge 0$ let $u,v\ge 0$ be defined implicitly by

\begin{equation}\label {eq:lebrun-coordinates} |z_1|=\ee ^{m(u^2-v^2)}u, \qquad |z_2|=\ee ^{m(v^2-u^2)}v, \end{equation}

and put

\begin{equation}\label {eq:lebrun-potential} \Phi _m(u,v)=u^2+v^2+m\,(u^4+v^4), \qquad \omega _m=\tfrac {i}{2}\partial \bar \partial \Phi _m . \end{equation}

For $m=0$ this is the flat metric $g_0$ . For $m>0$ the metric $g_m$ is complete, Ricci-flat, not flat, has the same volume form as $g_0$ , and is isometric up to scale to the Taub–NUT gravitational instanton [Haw77, LeB91, LZZ12].

The question whether the multiples $\alpha g_m$ are projectively induced has a history of partial answers. Loi, Zedda and Zuddas [LZZ12] proved that $\alpha g_m$ is not projectively induced when $m>\alpha /2$ , by restricting to the axis $z_2=0$ and reading off the coefficient of $|z_1|^4$ in Calabi’s expansion. The axis computation goes back to Zedda’s thesis [Zed09, Remark 4.4.5], which treats $m>1/2$ for $\alpha =1$ , gives the Taylor coefficients of $u^2$ as a function of $|z_1|^2$ and the first few coefficients of the exponentiated potential, and records the expectation that a negative coefficient occurs for every $m>0$ . Loi, Zedda and Zuddas also conjectured in [LZZ12] that $\alpha g_m$ is not projectively induced for any $m>0$ , adding that they had computer evidence but no proof; the statement is recorded as [LZ18, Conjecture 7.3.2]. Zedda [Zed21] notes that the coefficient method reaches smaller values of $m$ but that “it is hard to prove it for values of $m$ approaching to $0$ ”. For finite-dimensional targets the conjecture follows from [Sal17], since $g_m$ is Ricci-flat and its potential depends only on $|z_1|^2$ and $|z_2|^2$ ; the open case is $N=\infty$ . We settle it.

Theorem A. For every $m>0$ and every $\alpha >0$ , the Kähler manifold $(\C ^2,\alpha g_m)$ admits no Kähler immersion into $(\CP ^N,g_{\FS })$ for any $N\le \infty$ . Already the restriction of $\alpha g_m$ to the axis $\{z_2=0\}$ fails Calabi’s criterion at the origin.

The proof is short and uses nothing beyond Calabi’s criterion, the Lambert $W$ function and a classical fact about power series. On the axis $z_2=0$ the relation (1) reads $|z_1|^2=U\ee ^{2mU}$ with $U=u^2$ , so $U$ is the Lambert $W$ function of $2m|z_1|^2$ up to scale. On the closed disc of radius $1/(2m\ee )$ its only singularity is a branch point at $|z_1|^2=-1/(2m\ee )$ , and we show that this singularity survives in $\exp (\alpha \Phi _m)$ . Hence the Taylor series of $\exp (\alpha \Phi _m)$ in $|z_1|^2$ has finite radius of convergence $1/(2m\ee )$ , while the function itself is real analytic on the whole half-line $[0,\infty )$ . By the theorem of Vivanti and Pringsheim, a power series with nonnegative coefficients is singular at the positive point of its circle of convergence. So the coefficients are not all nonnegative, and Calabi’s criterion fails. The coefficient of $|z_1|^4$ is $\tfrac {\alpha }{2}(\alpha -2m)$ , which gives the case $m>\alpha /2$ of [LZZ12]. For smaller $m$ no fixed finite set of coefficients can decide the question, since each coefficient tends to $\alpha ^k/k!>0$ as $m\to 0$ ; in the exact computations of Remark 3.3 the first negative coefficient appears late and is extremely small.

The same argument applies whenever a Kähler manifold contains a complex curve with a rotation-invariant induced metric whose exponentiated potential has a finite radius of convergence but continues analytically along the positive axis. The mechanism is not new. Arezzo, Loi, Placini and Zedda [ALPZ26, Lemma 4.1 and the proof of Theorem 1.1(3)] use it to show that a radial projectively induced Kähler–Einstein metric with negative Einstein constant is a multiple of the complex hyperbolic metric: in the remaining case of their proof, a germ with a branch singularity on the negative real axis continues analytically along the positive axis beyond the modulus of that singularity, so it cannot have nonnegative Taylor coefficients. Theorem B isolates the mechanism for a rotation-invariant complex curve inside an arbitrary Kähler manifold, which is what the non-radial Taub–NUT metric requires; there the curve is a coordinate axis and the branch point is that of the Lambert $W$ function.

Theorem B (Pringsheim obstruction). Let $(M,g)$ be a Kähler manifold, $p\in M$ , and let $\iota \colon \{|z|^2<\rho \}\to M$ , $0<\rho \le \infty$ , be a holomorphic embedding of a disc with $\iota (0)=p$ such that $\iota ^*g$ has a Kähler potential $\varphi (|z|^2)$ , where $\varphi (0)=0$ and $\varphi$ is real analytic on $[0,\rho )$ , that is, $\varphi$ extends holomorphically to a complex neighbourhood of $[0,\rho )$ . If the Taylor series of $\ee ^{\varphi (X)}$ at $X=0$ has radius of convergence $r<\rho$ , then no neighbourhood of $p$ admits a Kähler immersion into $(\CP ^N,g_{\FS })$ for any $N\le \infty$ . In particular $g$ is not projectively induced.

Finally, the axis restriction has an exact meaning in statistics. Write the axis potential $\Phi _m|_{z_2=0}$ as a function $K(\theta )$ of $\theta =\log |z_1|^2$ . Then $K$ is formally the cumulant function of a natural exponential family with mean $\mu =u^2$ and variance function $V(\mu )=\mu /(1+2m\mu )$ , which behaves like the Poisson variance function $\mu$ near $\mu =0$ and tends to the constant $1/(2m)$ as $\mu \to \infty$ . For a torus-invariant metric, Calabi’s criterion asks precisely that the exponentiated potential, as a function of $\theta _j=\log |z_j|^2$ , be the Laplace transform of a positive measure on the lattice $\N _0^n$ . The argument of Theorem A therefore gives an elementary proof of the following known fact, which excludes positive measures of any kind.

Corollary C (Bar-Lev, Bshouty and Enis; Letac and Mora). Let $c>0$ . There is no natural exponential family on $\R$ whose mean domain is $(0,\infty )$ and whose variance function is $V(\mu )=\mu /(1+c\mu )$ ; more generally, no natural exponential family has this variance function on an open interval of positive means. Equivalently, there is no positive measure on $\R$ whose Laplace transform equals $\exp K_c$ on a nonempty open interval, where $K_c(\theta )=p+\tfrac c2p^2$ with $p=c^{-1}\Wo (c\ee ^{\theta })$ .

Corollary C itself is not new. In the full-mean-domain case it is the instance $P(\mu )=\mu$ , $Q(\mu )=1+c\mu$ of an example of Bar-Lev, Bshouty and Enis [BLBE91, Remark 4.1 and Example (f)(i)]. They show that a rational function $V=P/Q$ that is positive on $(0,\infty )$ , with $P$ and $Q$ coprime, $P(0)=0$ , $\deg P\le \deg Q+1$ and $Q$ without zeros on $(0,\infty )$ , is the variance function of a family with mean domain $(0,\infty )$ only if it is quadratic, which under these hypotheses means $V(\mu )=a\mu$ , the Poisson case; this extends the earlier treatment by Bar-Lev and Bshouty [BLB89] of rational variance functions on bounded mean domains that vanish at the endpoints. Their proof starts, as ours does, from the divergence of $\int \dd \mu /V$ at both ends of the mean domain, which forces the natural domain to be $\R$ ; the mean function is then meromorphic in $\C$ , and a Nevanlinna-theoretic argument [BLBE91, Theorem 4.1] leaves only quadratic variance functions. The extension to an open interval of positive means follows from the maximality of the mean domain proved by Letac and Mora [LM90, Theorem 3.1]. The full-mean-domain case $c=1$ was also found by Bryc and Ismail [BI05, Remark 2.6] through an explicit negative coefficient.

We nevertheless include a self-contained proof, because it is elementary and makes the link with Theorem A exact: the mean function is $c^{-1}\Wo (c\ee ^{\theta })$ , and the branch point of the Lambert $W$ function at $\ee ^{\theta }=-1/(c\ee )$ is incompatible with a positive representing measure on a full natural domain, which is precisely the failure of Calabi’s criterion in Theorem A. For $c=2m$ the function $1/V(\mu )=1/\mu +c$ is exactly the Gibbons–Hawking harmonic function of Taub–NUT along the axis, and its constant term is the value at infinity that makes Taub–NUT asymptotically locally flat; Section 6 explains this identity and the scope of the method.

Section 2 recalls Calabi’s criterion, the Vivanti–Pringsheim theorem and the facts about the Lambert $W$ function that we use. Section 3 proves Theorem A, Section 4 proves Theorem B, Section 5 proves Corollary C, and Section 6 discusses the Gibbons–Hawking picture and the scope of the method. Section 5 uses nothing from the geometric part except the facts about $\Wo$ in Section 2.3, the definitions (6) and Lemmas 3.1 and 3.2, so readers mainly interested in exponential families can read Section 2.3 and Section 3 up to Lemma 3.2 and then go straight to Section 5.

2Preliminaries

The proofs combine three classical ingredients. Calabi’s criterion turns the existence of a Kähler immersion into a positivity condition on Taylor coefficients. The Vivanti–Pringsheim theorem forces a power series with nonnegative coefficients and finite radius of convergence to be singular on the positive axis. The Lambert $W$ function describes the Taub–NUT potential on a coordinate axis. We recall them in turn.

2.1Calabi’s criterion

The criterion is phrased through Calabi’s diastasis, a Kähler potential that is canonically normalized at a given point. Let $(M,g)$ be a Kähler manifold with real analytic metric, $p\in M$ , and let $\Phi$ be a real analytic Kähler potential near $p$ , so that the Kähler form is $\frac i2\partial \bar \partial \Phi$ . Polarize $\Phi$ to a holomorphic function $\hat \Phi (z,\bar w)$ of the coordinates of $z$ and the conjugate coordinates of $w$ . Calabi’s diastasis is

\begin{equation}\label {eq:diastasis} D_p(z)=\hat \Phi (z,\bar z)+\hat \Phi (p,\bar p)-\hat \Phi (z,\bar p)-\hat \Phi (p,\bar z); \end{equation}

it depends on the metric alone. In local coordinates centred at $p$ , expand

\begin{equation}\label {eq:calabi-matrix} \ee ^{D_p(z)}-1=\sum _{j,k}B_{jk}\,z^{j}\bar z^{k} \end{equation}

over multi-indices. The metric is $1$ -resolvable of rank at most $N$ at $p$ if the Hermitian matrix $(B_{jk})$ is positive semidefinite of rank at most $N$ .

Theorem 2.1 (Calabi [Cal53]; see [LSZ18, Lemma 2.1], [LZ18, Chapter 2]). A neighbourhood of $p$ admits a Kähler immersion into $(\CP ^N,g_{\FS })$ , $N\le \infty$ , if and only if $g$ is $1$ -resolvable of rank at most $N$ at $p$ .

Two consequences are used below. First, if $F\colon (M,g)\to \CP ^N$ is a Kähler immersion and $S\subset M$ is a complex submanifold, then $F|_S$ is a Kähler immersion of $(S,g|_S)$ , so $g|_S$ is $1$ -resolvable at every point of $S$ . Second, in one complex variable, if the potential has the form $\varphi (|z|^2)$ with $\varphi$ real analytic near $0$ and $\varphi (0)=0$ , then $\hat \Phi (z,\bar w)=\varphi (z\bar w)$ , so (3) at $p=0$ gives $D_0(z)=\varphi (|z|^2)$ , and (4) reads

\begin{equation}\label {eq:radial-matrix} \ee ^{\varphi (|z|^2)}-1=\sum _{k\ge 1}c_k\,|z|^{2k}, \qquad B_{jk}=c_k\,\delta _{jk}. \end{equation}

Thus $1$ -resolvability at the origin is equivalent to $c_k\ge 0$ for all $k\ge 1$ , where $c_k$ are the Taylor coefficients of $X\mapsto \ee ^{\varphi (X)}$ at $X=0$ . This is the form of the criterion used in [LZZ12].

2.2The Vivanti–Pringsheim theorem

For a rotation-invariant potential in one variable, Calabi’s criterion asks that a power series have nonnegative coefficients, as in (5). The following classical theorem is the only fact about such series that we need.

Theorem 2.2 (Vivanti [Viv93], Pringsheim [Pri94]; see [Tit39, §7.21], [FS09, Theorem IV.6]). Let $\sum _{k\ge 0}c_kX^k$ be a power series with real coefficients $c_k\ge 0$ and radius of convergence $R\in (0,\infty )$ . Then $X=R$ is a singular point of its sum: the sum has no analytic continuation to any neighbourhood of $R$ .

2.3The Lambert $W$ function

On the axis $z_2=0$ of Taub–NUT, recovering $u^2$ from $|z_1|^2$ amounts to inverting $w\mapsto w\ee ^{w}$ after a rescaling; see (8) below. Let $\Wo$ denote the principal branch of the Lambert $W$ function, the inverse of $w\mapsto w\ee ^{w}$ that is real and increasing on $[-1/\ee ,\infty )$ with $\Wo (0)=0$ . We use the following facts from [CGH $^{+}$ 96, Sections 3–4].

$\Wo$ is holomorphic on $\C \setminus (-\infty ,-1/\ee ]$ and continuous on $\C \setminus (-\infty ,-1/\ee )$ , with $\Wo (-1/\ee )=-1$ . In particular $\Wo$ is holomorphic on the open disc $|y|<1/\ee$ , continuous on the closed disc, and real analytic on $(-1/\ee ,\infty )$ .
The Taylor series at the origin is $\Wo (y)=\sum _{n\ge 1}(-n)^{n-1}y^n/n!$ , with radius of convergence $1/\ee$ .
$y=-1/\ee$ is a branch point: the derivative of $w\mapsto w\ee ^w$ vanishes at $w=-1$ , and $\Wo$ has no analytic continuation to any neighbourhood of $-1/\ee$ .

We only need the third item in the following elementary form. If $h(w)=w\ee ^{w}$ and $W$ is holomorphic on an open set $\Omega \ni y_0$ with $h(W(y))=y$ on $\Omega$ , then differentiating gives $h'(W(y))W'(y)=1$ , so $h'(W(y))\ne 0$ on $\Omega$ ; hence $W(y_0)\ne -1$ .

3Proof of Theorem A

The proof has two steps. We first express the axis potential through the Lambert $W$ function and show that the branch point of $\Wo$ survives exponentiation (Lemmas 3.1 and 3.2). Calabi’s criterion and the Vivanti–Pringsheim theorem then turn this singularity into a negative Taylor coefficient. We work with a general parameter $c>0$ , because Section 5 uses the same functions; the axis of Taub–NUT is the case $c=2m$ .

Fix $c>0$ . Set

\begin{equation}\label {eq:Uc} R_c=\frac 1{c\ee }, \qquad U_c(X)=\frac {\Wo (cX)}{c}, \qquad \varphi _c(X)=U_c(X)+\frac c2\,U_c(X)^2 . \end{equation}

By the properties of $\Wo$ , the function $U_c$ is holomorphic on $\C \setminus (-\infty ,-R_c]$ , in particular on the disc $D_c=\{|X|<R_c\}$ and near every point of $(-R_c,\infty )$ , and it is continuous on $\overline {D_c}$ with $U_c(-R_c)=-1/c$ . It satisfies

\begin{equation}\label {eq:inverse} U_c(X)\,\ee ^{cU_c(X)}=X , \end{equation}

and on $[0,\infty )$ it is the inverse of the increasing bijection $h_c(U)=U\ee ^{cU}$ of $[0,\infty )$ onto itself. The function $\varphi _c$ has the same regularity.

For $m>0$ the restriction of LeBrun’s potential to the axis $z_2=0$ is, by (1)–(2) with $v=0$ and $U=u^2$ ,

\begin{equation}\label {eq:axis} |z_1|^2=U\ee ^{2mU}, \qquad \Phi _m|_{z_2=0}=U+mU^2=\varphi _{2m}(|z_1|^2), \end{equation}

which is the setting of the axis computation in [LZZ12]. Thus the axis is the case $c=2m$ of (6). The whole argument rests on the following formula for the derivative of $\varphi _c$ .

Lemma 3.1 (mean identity). On $\C \setminus (-\infty ,-R_c]$ ,

\begin{equation}\label {eq:mean} \varphi _c'(X)=\ee ^{-cU_c(X)}=\frac {U_c(X)}{X}\qquad (X\ne 0). \end{equation}

Consequently

\varphi _c'

is holomorphic on

D_c

, bounded and nowhere zero on

D_c

, and real analytic on

(-R_c,\infty )

Proof.Differentiating (7) gives $U_c'(X)\,(1+cU_c)\,\ee ^{cU_c}=1$ , so $U_c'=\ee ^{-cU_c}/(1+cU_c)$ wherever $1+cU_c\ne 0$ , which holds on $\C \setminus (-\infty ,-R_c]$ since $\Wo \ne -1$ there. Hence $\varphi _c'=(1+cU_c)U_c'=\ee ^{-cU_c}$ , and $\ee ^{-cU_c}=U_c/X$ by (7). Boundedness follows from the boundedness of $U_c$ on $\overline {D_c}$ .∎

In statistical language, (9) says that with $X=\ee ^{\theta }$ the derivative of $\theta \mapsto \varphi _c(\ee ^\theta )$ is $U_c$ : the mean parameter of the axis family is LeBrun’s $u^2$ .

The next lemma is the heart of the proof: the branch point of $U_c$ at $-R_c$ survives in $\exp (\alpha \varphi _c)$ . The idea is that a continuation of $\exp (\alpha \varphi _c)$ to a neighbourhood of $-R_c$ would, through (9), give one of $U_c$ , which is impossible because $U_c$ inverts a map with a critical point at $-1/c$ .

Lemma 3.2 (radius of convergence). Let $\alpha >0$ and $G_c=\ee ^{\alpha \varphi _c}$ . Then $G_c$ is holomorphic on $\C \setminus (-\infty ,-R_c]$ and has no analytic continuation to any neighbourhood of $-R_c$ . Consequently the Taylor series of $G_c$ at $X=0$ has radius of convergence exactly $R_c$ .

Proof.Holomorphy on $\C \setminus (-\infty ,-R_c]$ is inherited from $U_c$ . Suppose $\tilde G$ is holomorphic on a disc $\Delta$ centred at $-R_c$ and agrees with $G_c$ on $\Delta \cap D_c$ . Since $\varphi _c$ is bounded on $D_c$ , $|G_c|$ is bounded below by a positive constant on $D_c$ , so $\tilde G(-R_c)\ne 0$ . Shrinking $\Delta$ , the function $\tilde G$ has no zeros on $\Delta$ , so it has a holomorphic logarithm on $\Delta$ . On the connected set $\Delta \cap D_c$ the difference $\alpha ^{-1}\log \tilde G-\varphi _c$ is continuous with values in $(2\pi i/\alpha )\mathbb Z$ , hence constant, and we choose the logarithm so that it vanishes. Thus $\varphi _c$ extends holomorphically to $\Delta$ , hence so does $\varphi _c'$ , and by Lemma 3.1 the extension of $\varphi _c'$ is nowhere zero on $\Delta$ after shrinking $\Delta$ once more, since it is continuous and equals $\ee ^{-cU_c(-R_c)}=\ee$ at the centre.

The same argument applied to $\varphi _c'=\ee ^{-cU_c}$ shows that $U_c$ extends to a holomorphic function $\tilde U$ on $\Delta$ , and $\tilde U(-R_c)=U_c(-R_c)=-1/c$ by continuity of $U_c$ on $\overline {D_c}$ . Applying $h_c$ , we get $h_c(\tilde U(X))=X$ on $\Delta \cap D_c$ , hence on $\Delta$ . Differentiating at $X=-R_c$ gives $h_c'(-1/c)\,\tilde U'(-R_c)=1$ , which is impossible because $h_c'(U)=(1+cU)\ee ^{cU}$ vanishes at $U=-1/c$ . This proves that $G_c$ has no analytic continuation to a neighbourhood of $-R_c$ .

The Taylor series of $G_c$ at $0$ converges on $D_c$ , so its radius is at least $R_c$ . If the radius were larger, the sum of the series would be an analytic continuation of $G_c$ to a neighbourhood of $-R_c$ . Hence the radius equals $R_c$ .∎

Proof of Theorem A. Let $m>0$ , $\alpha >0$ , and suppose that $F\colon (\C ^2,\alpha g_m)\to \CP ^N$ is a Kähler immersion for some $N\le \infty$ . The axis $S=\{z_2=0\}$ is a complex submanifold, and by (8) the induced metric $\alpha g_m|_S$ has the Kähler potential $\alpha \varphi _{2m}(|z_1|^2)$ , which is real analytic on $S$ and vanishes at the origin. By Theorem 2.1 applied to $F|_S$ and by (5), the Taylor coefficients $c_k$ of $G_{2m}(X)=\ee ^{\alpha \varphi _{2m}(X)}$ at $X=0$ satisfy $c_k\ge 0$ for all $k\ge 1$ ; also $c_0=1$ . By Lemma 3.2 the series $\sum c_kX^k$ has radius of convergence $R_{2m}=1/(2m\ee )\in (0,\infty )$ . Theorem 2.2 then says that its sum has no analytic continuation to a neighbourhood of $X=R_{2m}$ . But the sum equals $G_{2m}$ on $D_{2m}$ , and by Lemma 3.2 $G_{2m}$ is holomorphic on $\C \setminus (-\infty ,-R_{2m}]$ , which contains a disc centred at $R_{2m}$ ; so $G_{2m}$ continues the sum analytically across $R_{2m}$ . This contradiction shows that no such $F$ exists, and that already $\alpha g_m|_S$ is not $1$ -resolvable at the origin.∎

Remark 3.3. The coefficient of $|z_1|^4$ in $\ee ^{\alpha \varphi _{2m}}$ is $\frac \alpha 2(\alpha -2m)$ , as in [LZZ12], so for $m\le \alpha /2$ the obstruction sits at a higher index. Exact computation of the coefficients, using the Lagrange series $U_{2m}(X)=\sum _{n\ge 1}n^{n-1}(-2m)^{n-1}X^n/n!$ , gives the following values for $\alpha =1$ ; the script is provided as an ancillary file.

$m$	$1/(2m)$	first $k$ with $c_k<0$	$c_k$	$R_{2m}$
$1/2$	$1$	$4$	$-2.5\times 10^{-1}$	$0.37$
$1/4$	$2$	$4$	$-1.0\times 10^{-2}$	$0.74$
$1/10$	$5$	$8$	$-2.0\times 10^{-6}$	$1.84$
$1/20$	$10$	$14$	$-1.1\times 10^{-13}$	$3.68$
$1/50$	$25$	$28$	$-1.2\times 10^{-36}$	$9.20$
$1/100$	$50$	$54$	$-1.1\times 10^{-83}$	$18.39$

In every case, in the range computed, all coefficients of even index from the first negative one onwards are negative and all coefficients of odd index are positive, the alternation expected from a singularity on the negative axis. In these six computations with $\alpha =1$ the first negative index roughly tracks $1/(2m)$ ; we do not claim a proof of such a law. The exact scaling $\alpha \varphi _{2m}(X)=\varphi _{2m/\alpha }(\alpha X)$ , which follows from $\Wo$ being the inverse of $w\mapsto w\ee ^{w}$ , gives $c_k(m,\alpha )=\alpha ^kc_k(m/\alpha ,1)$ , so the first negative index depends only on $m/\alpha$ .

For fixed parameters the singularity at $-R_{2m}$ does determine the asymptotics. The expansion of $\Wo$ at its branch point [CGH $^{+}$ 96, Section 4] gives, with $c=2m$ and $t=1+X/R_c$ ,

\varphi _c(X)=A(t)-\frac {2\sqrt 2}{3c}\,t^{3/2}+O(t^{5/2}), \qquad A\text { holomorphic near }0,\quad A(0)=-\frac 1{2c},

and the transfer theorem [FS09, Chapter VI] yields

c_k\sim (-1)^{k+1}\,\frac {\alpha }{c\sqrt {2\pi }}\,\ee ^{-\alpha /(2c)}\, R_c^{-k}\,k^{-5/2}\qquad (k\to \infty ),

in agreement with the observed alternation. The amplitude

\ee ^{-\alpha /(2c)}=\ee ^{-\alpha /(4m)}

is exponentially small in

\alpha /m

. This is consistent with the late and tiny negative coefficients observed for small

m

, and it helps explain why the computer evidence mentioned in [LZZ12] could not be turned into a proof by inspection; the fixed-parameter asymptotics do not, however, locate the first sign change uniformly as

m\to 0

Remark 3.4. The proof does not depend on the size of $\alpha$ , which only rescales $\varphi _{2m}$ . Balanced metrics in the sense of Donaldson are projectively induced (see [LZZ12]), so Theorem A also recovers the statement of [LZZ12] that $\alpha g_m$ is never balanced for $m>0$ . Immersions of $(\C ^2,\alpha g_m)$ into complex hyperbolic space, and of $(\C ^2,g_m)$ into $\C ^N$ , were already excluded in [LZZ12]; since $\alpha \Phi _m(z)=\Phi _{m/\alpha }(\sqrt \alpha \,z)$ by (1)–(2), the metric $\alpha g_m$ is the pull-back of $g_{m/\alpha }$ under a dilation, so the Euclidean statement covers every multiple as well.

4The Pringsheim obstruction

The proof of Theorem A used only two features of the exponentiated axis potential: its Taylor series at the origin has a finite radius of convergence, and the function itself continues analytically along the whole positive axis. Theorem B isolates these two features for a rotation-invariant complex curve in an arbitrary Kähler manifold; for radial metrics the same mechanism appears in [ALPZ26, Lemma 4.1].

Proof of Theorem B. Suppose an open neighbourhood $\mathcal U$ of $p$ admits a Kähler immersion into $(\CP ^N,g_{\FS })$ . Since $\iota (0)=p$ , there is $\varepsilon \in (0,\rho )$ with $\iota (\{|z|^2<\varepsilon \})\subset \mathcal U$ . Composing the immersion with $\iota$ on this subdisc gives a Kähler immersion of $(\{|z|^2<\varepsilon \},\iota ^*g)$ , so $\iota ^*g$ is $1$ -resolvable at $z=0$ by Theorem 2.1. Its potential is $\varphi (|z|^2)$ with $\varphi (0)=0$ , so by (5) the Taylor coefficients $c_k$ of $\ee ^{\varphi (X)}$ satisfy $c_k\ge 0$ for $k\ge 1$ , and $c_0=1$ . Since $\varphi$ is analytic at $0$ , the radius of convergence $r$ is positive, and by hypothesis $r<\rho$ . By Theorem 2.2, the sum of $\sum c_kX^k$ is singular at $X=r$ .

On the other hand $\varphi$ is real analytic on $[0,\rho )\ni r$ , so $\ee ^{\varphi }$ is holomorphic on a connected complex neighbourhood $\Omega$ of $[0,r]$ . The sum and $\ee ^{\varphi }$ agree near $0$ , hence, by the identity theorem, on the connected component of $\Omega \cap \{|X|<r\}$ that contains $[0,r)$ ; this component contains $\Delta \cap \{|X|<r\}$ for every disc $\Delta \subset \Omega$ centred at $r$ , because that intersection is convex and meets $[0,r)$ . So $\ee ^{\varphi }$ is an analytic continuation of the sum to a neighbourhood of $r$ , a contradiction. Hence no neighbourhood of $p$ admits a Kähler immersion into $(\CP ^N,g_{\FS })$ ; in particular there is no global one.∎

For torus-invariant metrics on $\C ^n$ each coordinate axis is a rotation-invariant complex curve, so Theorem B gives a test that looks at one variable at a time.

Corollary 4.1. Let $g$ be a Kähler metric on $\C ^n$ invariant under the standard action of the torus $(S^1)^n$ , with a Kähler potential $\Phi (|z_1|^2,\dots ,|z_n|^2)$ that is real analytic on $[0,\infty )^n$ and vanishes at the origin. If for some $j$ the power series of $X\mapsto \exp \Phi (0,\dots ,0,X,0,\dots ,0)$ at $X=0$ has finite radius of convergence, then $g$ is not projectively induced.

Proof.Apply Theorem B to the $j$ -th coordinate axis, with $\rho =\infty$ .∎

Theorem A is the case $n=2$ , $\Phi =\alpha \Phi _m$ of Corollary 4.1. The hypothesis that $\Phi$ be real analytic on the closed quadrant holds for LeBrun’s potential because the map $(U,W)\mapsto (U\ee ^{2m(U-W)},W\ee ^{2m(W-U)})$ , with $U=u^2$ and $W=v^2$ , is a bijection of $[0,\infty )^2$ onto itself with Jacobian determinant $1+2m(U+W)>0$ ; only the axis is needed in the proof.

5Exponential families

We now turn to statistics and prove Corollary C. The proof parallels that of Theorem A. The variance function determines the cumulant function, which turns out to be $\varphi _c(\ee ^\theta )$ up to normalization. The generating measure must then sit on the lattice $\N _0$ with nonnegative weights, which plays the role of Calabi’s criterion, and Lemma 3.2 gives the contradiction. We first recall the definitions.

Let $\nu$ be a positive Borel measure on $\R$ , not a point mass, whose Laplace transform $L(\theta )=\int \ee ^{\theta x}\,\nu (\dd x)$ is finite on a nonempty open interval. Let $\Theta$ be the interior of $\{L<\infty \}$ , an open interval, and $K=\log L$ on $\Theta$ . The natural exponential family generated by $\nu$ is $\{\ee ^{\theta x-K(\theta )}\nu :\theta \in \Theta \}$ ; its mean map $\mu =K'$ is an increasing real analytic bijection of $\Theta$ onto the mean domain $M=K'(\Theta )$ , and its variance function is $V(\mu )=K''(\theta (\mu ))$ . Together with the mean domain, the variance function determines the family [Mor82, LM90]. We use one standard fact [BN78, Chapters 8–9]: if $\Theta =\R$ the family is regular, hence steep, and then $M$ is the interior of the convex hull of $\supp \nu$ .

Proof of Corollary C. Suppose $\nu$ generates a natural exponential family with mean domain $M=(0,\infty )$ and variance function $V(\mu )=\mu /(1+c\mu )$ . Put $p=K'$ on $\Theta$ . Then $p'=K''=V(p)=p/(1+cp)$ , so $(1/p+c)\,p'=1$ and

\begin{equation}\label {eq:theta-of-p} \theta =\log p+cp-\theta _0 \end{equation}

for a constant

\theta _0

. The right side of (10) is an increasing bijection of

p\in (0,\infty )

onto

\R

, and

p

ranges over

M=(0,\infty )

, so

\Theta =\R

. Replacing

\nu

by an exponential tilt we may take

\theta _0=0

. Then

p\ee ^{cp}=\ee ^{\theta }

, that is

p=U_c(\ee ^\theta )

in the notation (6), and integrating

K'=p

with the help of Lemma 3.1 gives

\begin{equation}\label {eq:K-lambert} K(\theta )=\varphi _c(\ee ^{\theta })+a \qquad (\theta \in \R ) \end{equation}

for a constant

a

; indeed

\frac {\dd }{\dd \theta }\varphi _c(\ee ^\theta )=\ee ^\theta \varphi _c'(\ee ^\theta )=U_c(\ee ^\theta )=p

Since $\Theta =\R$ , the family is steep and $M=(0,\infty )$ is the interior of the convex hull of $\supp \nu$ , so $\supp \nu \subset [0,\infty )$ . Put $u=\ee ^{\theta }>0$ and $\Lambda (u)=L(\theta )=\int _{[0,\infty )}u^x\,\nu (\dd x)$ . By (11), $\Lambda (u)=\ee ^{a}G(u)$ with $G=\ee ^{\varphi _c}$ , which is holomorphic on $D_c$ with $G(0)=1$ ; write $G(u)=\sum _{n\ge 0}f_nu^n$ for $|u|<R_c$ . For $t>-\log R_c$ we therefore have

\int _{[0,\infty )}\ee ^{-tx}\,\nu (\dd x)=\ee ^{a}\sum _{n\ge 0}f_n\ee ^{-nt},

and the right side is the Laplace transform of the signed measure

\eta =\ee ^{a}\sum _nf_n\delta _n

, whose total variation transform

\sum |f_n|\ee ^{-nt}

converges for the same

t

. Fix

t_0>-\log R_c

; the finite signed measure

\ee ^{-t_0x}(\nu -\eta )

[0,\infty )

has vanishing Laplace transform at every positive argument, so it is zero by the uniqueness theorem for Laplace–Stieltjes transforms [Wid41, Chapter II, §6], and

\nu =\eta

. Hence

f_n\ge 0

for all

n

and

\nu

is supported on the nonnegative integers.

Now $L(\theta )=\ee ^{a}\sum _nf_n\ee ^{n\theta }$ is finite for every $\theta \in \R$ , so the series $\sum f_nu^n$ converges for every $u>0$ and its radius of convergence is infinite. Its sum is an entire function that agrees with $G$ on $D_c$ ; in particular $G$ extends analytically to a neighbourhood of $-R_c$ . This contradicts Lemma 3.2 (with $\alpha =1$ ). Hence no such $\nu$ exists. The remaining assertions follow from Remark 5.1: a family with this variance function on an open interval of positive means, and likewise a positive measure whose Laplace transform equals $\exp K_c$ on an open interval (its family has variance function $\mu /(1+c\mu )$ on the corresponding interval of means), has natural domain $\R$ and mean domain $(0,\infty )$ , so the argument above applies.∎

Remark 5.1. Suppose the variance function $\mu /(1+c\mu )$ is only assumed on some open interval $J\subset (0,\infty )$ of means, and let $I=(K')^{-1}(J)\subset \Theta$ . The differential equation holds on $I$ , so on $I$ we obtain (10) and

K(\theta )=\varphi _c(\ee ^{\theta +\theta _0})+a .

Both sides of this identity are real analytic on the interval

\Theta

, the right side being real analytic on all of

\R

, so by the identity theorem the identity holds on all of

\Theta

. Hence

L=\ee ^{K}

continues analytically across every finite endpoint of

\Theta

. By Landau’s theorem, the Laplace–Stieltjes transform of a positive measure is singular at each finite endpoint of its interval of convergence [Wid41, Chapter II, §5]; therefore

\Theta =\R

. Then (10), now valid on all of

\R

, shows that the mean domain

K'(\R )

is all of

(0,\infty )

and that the variance function is

\mu /(1+c\mu )

on all of it, so the proof of Corollary C applies. This is a special case of the maximality of the mean domain proved by Letac and Mora [LM90, Theorem 3.1]; we include the short argument to keep the proof self-contained.

Remark 5.2.The lattice step in the proof is the familiar fact that a family whose inverse mean map has the form $\theta =q\log (\mu -\mu _0)+(\text {analytic})$ at an endpoint $\mu _0$ of its mean domain is concentrated on $\mu _0+q^{-1}\N _0$ ; see [LM90, Proposition 4.4] for a precise criterion, stated there for families on $\N _0$ . The Poisson family is the case $V(\mu )=\mu$ . What is specific here is the second step: the constant $c$ in $1/V(\mu )=1/\mu +c$ produces the critical point $p=-1/c$ of $p\mapsto p\ee ^{cp}$ , hence a branch point at $u=-R_c$ of the mean $p=U_c(u)$ as a function of $u=\ee ^\theta$ , and a lattice family with a full natural domain cannot accommodate a singularity of its generating function at a negative real point. For $c<0$ the critical point moves to the positive axis and the argument says nothing. Indeed $\mu /(1+c\mu )=\sum _{n\ge 1}|c|^{n-1}\mu ^n$ is then the variance function of an infinitely divisible family with mean domain $(0,1/|c|)$ [LM90, Corollary 3.3]; Bryc and Ismail give the generating law explicitly, up to a dilation [BI05, Theorem 2.5].

6Interpretation and scope

We close with the Gibbons–Hawking description of Taub–NUT, which makes the statistical picture exact on the axis, and with remarks on how far the method reaches.

Taub–NUT is the Gibbons–Hawking metric [Haw77, GH78] over $\R ^3$ with harmonic function $1+a/|x|$ , where $a>0$ is proportional to the mass; it is asymptotically locally flat because of the constant term. In LeBrun’s Kähler picture the circle $(z_1,z_2)\mapsto (\ee ^{i\phi }z_1,\ee ^{-i\phi }z_2)$ is the Gibbons–Hawking circle, the origin is the nut, and the quotient map $\C ^2\to \R ^3$ is its hyperkähler moment map. In the normalization (2) this map is

(z_1,z_2)\longmapsto x=\bigl (\tfrac 12(u^2-v^2),\,z_1z_2\bigr )\in \R \times \C , \qquad |x|=\tfrac 12(u^2+v^2).

A direct computation, checked in Lean and by an ancillary script, puts $g_m$ in the Weyl–Papapetrou form used by Li and Sun [LS25, §7.1.1]: $g_m$ is the Gibbons–Hawking metric of the axisymmetric harmonic function

H(x)=2m+\frac 1{2|x|}.

This is the one-centre case, with constant term $2m$ , of the multi-Taub–NUT metrics [Haw77, GH78], in the normalization of Li and Sun. In their language the two half-lines into which the nut divides the axis $\R \times \{0\}$ are the two rods of Taub–NUT [LS25, Example 3.16], and the coordinate axes $z_j=0$ are their preimages. On $z_2=0$ , the preimage of the positive half-line, we have $v=0$ and $|x|=\tfrac 12u^2=\tfrac 12\mu$ , where $\mu =u^2$ is the mean of the axis family of Section 5, so

H=2m+\frac 1\mu =\frac 1{V(\mu )} .

The inverse variance function of the axis family is therefore exactly the Gibbons–Hawking function along that half-line. Its pole at the nut alone would give the Poisson family, and its constant term $2m$ is the value at infinity. For the flat metric $m=0$ the constant term vanishes, the axis family is Poisson, its generating function $\exp (\ee ^{\theta })$ is an entire function of $\ee ^{\theta }$ , and indeed $\C ^2$ is projectively induced. Theorem A thus says that the constant term alone destroys projective inducibility, however small it is relative to the pole. The proofs do not use this section.

The same computation applies verbatim to any torus-invariant Kähler metric on $\C ^2$ one of whose axis families has inverse variance $1/\mu$ plus a positive constant, and Theorem B applies to any Kähler manifold containing a rotation-invariant complex curve whose exponentiated potential has a finite radius of convergence but is analytic along the positive axis. For Calabi’s metrics on line bundles, which include Eguchi–Hanson, non-inducibility was proved in [LZZ21]; for Stenzel’s metrics on the complexifications of $\CP ^n$ and $\mathbb {HP}^n$ it was proved for the multiples $cg$ with $0<c\le 1$ [Zed21, Theorem 1]. We hope to return elsewhere to Ricci-flat torus-invariant metrics studied through their exponential families, including multi-centre Gibbons–Hawking metrics with collinear centres. In the harmonic-map framework of Li and Sun [LS25], which describes torus-invariant Ricci-flat four-manifolds whose torus action has a fixed point by axisymmetric harmonic maps into the hyperbolic plane, these multi-centre metrics correspond to rod structures of degree zero [LS25, Theorem 1.2].

Formal verification. The analytic core of the paper has been checked in Lean 4 with Mathlib. This covers Lemma 3.1, and Lemma 3.2 in a real-axis form that suffices for the formal proofs: no function analytic at $-R_c$ agrees with $G_c$ on an interval $(-R_c,-R_c+\varepsilon )$ , and the Taylor radius of $G_c$ at $0$ is at most $R_c$ . It also covers the Vivanti–Pringsheim theorem in the real-analytic form needed and the conclusions of Theorems A and B that some Taylor coefficient is negative. Among the explicit computations it checks the identities (8), the coefficient of $|z_1|^4$ , the scaling identity and the asymptotic constants of Remark 3.3, the dilation identity of Remark 3.4, the Jacobian of Section 4, the variance function of Section 5, and the Gibbons–Hawking form of Section 6, including the harmonicity of $H$ and the identity $H=1/V(\mu )$ on the axis $z_2=0$ . Calabi’s criterion, the complex-analytic facts about $\Wo$ , the transfer theorem, the measure-theoretic steps of Section 5, the standard expressions of a torus-invariant Kähler metric and of its moment maps in logarithmic coordinates, and the chain rule relating derivatives in logarithmic coordinates to derivatives in $(u^2,v^2)$ are taken as inputs. The source files are provided as ancillary files.

AI-use disclosure. The author used Anthropic Claude Code and OpenAI Codex as interactive research and writing tools. They assisted with exploratory discussion, testing and refinement of ideas, literature and source organization, code development and verification including the Lean formalization, mathematical error checking, and editorial revision.

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How to cite

Shaosai Huang (2026). The Taub–NUT Metric Is Not Projectively Induced. Working paper, version of September 2026. Kspectra Research. SSRN 7484238 (doi:10.2139/ssrn.7484238). https://kspectra.ai/papers/taub-nut-not-projectively-induced/

@misc{huang2026taub,
  author = {Huang, Shaosai},
  title  = {{The Taub--NUT Metric Is Not Projectively Induced}},
  year   = {2026},
  month  = sep,
  note   = {Working paper, version of September 2026},
  doi    = {10.2139/ssrn.7484238},
  url    = {https://kspectra.ai/papers/taub-nut-not-projectively-induced/}
}

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