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23.1.32 Higher-dimensional Hopf manifolds28A754700E01

The threefold mass formula locates the source of unbounded mass, but it does not by itself give a construction in higher dimension. The same heat potentials do work there: the radial–vertical block is unchanged, and each extra angular direction contributes a factor whose lower bound is uniform where the threefold mass diverges.

For $n\geq 3$, let

\[ H_ n=(\mathbb C^ n\setminus \{ 0\} )/ \langle z\mapsto \mathrm e^{-1}z\rangle , \qquad p_ n:H_ n\longrightarrow \mathbb P^{n-1}. \]

Put $s=\log |z|^2$, $\alpha _ n=\mathrm{dd}^{\mathrm{c}}s=p_ n^*\omega _{\mathrm{FS}}$, $\beta _ n=\mathrm ds\wedge \mathrm d^{\mathrm c}s$ and $\omega _ n=\alpha _ n+\beta _ n$, where $\int _{\mathbb P^{n-1}}\omega _{\mathrm{FS}}^{n-1}=1$. The forms descend and $\omega _ n$ is positive for the same reasons as in dimension three. The fiber is the same elliptic curve $E$, with $\int _ E\beta _ n=2$. Since $\alpha _ n$ has rank $n-1$ and $\beta _ n^2=0$,

\[ \int _{H_ n}\omega _ n^ n=2n,\qquad \mathrm{dd}^{\mathrm{c}}(\omega _ n^{n-1})=-(n-1)\alpha _ n^ n=0. \]

Thus $\omega _ n$ is Gauduchon.

On the affine chart write $\xi \in \mathbb C^{n-1}$, $x=\log |\xi |^2$, $w=\log z_0\pmod\Lambda $ and $\mu =\mathrm e^ x/(1+\mathrm e^ x)$ as before. For a smooth real function $V(x,w)$ with the endpoint properties of lemma 23.1.7, let $J_ n(V;x,w)$ be the density obtained by integrating $(\omega _ n+\mathrm{dd}^{\mathrm{c}}V)^ n$ over the angular variables, with angular probability measure. Thus

\[ \int _{H_ n}(\omega _ n+\mathrm{dd}^{\mathrm{c}}V)^ n =\int _{\mathbb R\times E}J_ n(V;x,w)\, \mathrm dx\, \mathrm dA_ E. \]

The endpoints and the complement of the affine chart have zero smooth volume.

Lemma 23.1.33.264893FAE204 The positivity criterion (23.1.8) holds in every dimension $n\geq 3$, and

D216F8E7C5AA 23.1.34
\begin{equation} J_ n(V;x,w)=\frac{n(n-1)}6 (\mu +V_ x)^{n-3}J_3(V;x,w). \end{equation}

Here $J_3$ is evaluated for the same function $V$ on the threefold.

Proof. Choose $n-2$ horizontal angular vectors, one horizontal radial vector and the vertical vector $\partial _ w$. In the normalization of the proof of lemma 23.1.7, each angular entry is $A=(\mu +V_ x)/\mu $, and the radial–vertical block is

\[ K_ V=\begin{pmatrix} (\mu +V_{xx})/[\mu (1-\mu )] & (\mu +V_{x\bar w})/\sqrt{\mu (1-\mu )} \\ (\mu +V_{wx})/\sqrt{\mu (1-\mu )} & 1+V_{w\bar w} \end{pmatrix}. \]

This follows by differentiating $s=w+\bar w+\log (1+\mathrm e^ x)$: the radial part of $\partial \bar\partial x$ is zero, whereas its angular part contributes $V_ x/\mu $. The Schur complement of the last entry gives the stated positivity criterion. At $\xi =0$ the same endpoint argument as in dimension three applies.

Set $d=n-1$. The radial Fubini–Study probability density is $d\mu ^{d-1}\mathrm d\mu =d\mu ^ d(1-\mu )\mathrm dx$; it follows from the polar-coordinate determinant of $\mathrm{dd}^{\mathrm{c}}\log (1+|\xi |^2)$ and its integral is one. The background block $K_0$ has determinant one. Using the fiber mass two, therefore,

\[ J_ n(V)=2nd\mu ^ d(1-\mu )A^{n-2}\det K_ V. \]

For $n=3$ this is $12\mu ^2(1-\mu )A\det K_ V$. Comparing the two expressions proves (23.1.34). $\square$

Theorem 23.1.35.D3E13E0C86B3 For every $n\geq 3$, the Gauduchon metric $\omega _ n$ admits smooth potentials $V_\eta $ with $\omega _ n+\mathrm{dd}^{\mathrm{c}}V_\eta >0$ whose total Monge–Ampère masses tend to infinity as $\eta \downarrow 0$.

Proof. Use exactly the functions $V_\eta =W+g_\eta $ constructed in the proof of theorem 23.1.1, including the heat scale $\varepsilon =\eta ^{1/\eta ^2}$. Their endpoint properties give smooth potentials on every $H_ n$, and lemma 23.1.33 gives strict positivity. The slope bounds in lemma 23.1.25 and $-m_\eta \leq g_\eta '\leq 0$ imply

\[ \mu +(V_\eta )_ x\geq \tfrac 14 \qquad (x\geq 0) \]

for all sufficiently small $\eta $.

We must first localize the divergent threefold mass to this region. On $x<0$, the heat potential is $W=F_\eta (\varepsilon ,w)$ and has no $x$ derivatives. The determinant formula in the preceding proof and $\int _ E F_{w\bar w}\, \mathrm dA_ E=0$ give

\[ \int _ E J_3(V_\eta ;x,w)\, \mathrm dA_ E =12(\mu +g_\eta ') \bigl (g_\eta ''+\mu (1-\mu )\bigr ). \]

Here $0<\mu +g_\eta '\leq \mu $, and $g_\eta ''=r_\eta -m_\eta \psi $ with $\int r_\eta =m_\eta $ and $\int \psi =1$. Hence

\[ \int _{x<0}\! J_3(V_\eta )\, \mathrm dx\, \mathrm dA_ E \leq 12\int _{-\infty }^0\mu ^2(1-\mu )\, \mathrm dx +12\int _{\mathbb R}|g_\eta ''|\, \mathrm dx \leq \tfrac 32+24m_\eta . \]

No bound on the amplitude of the fiber Hessian is required; its mean has been integrated exactly. Since the total threefold mass tends to infinity, its mass on $x\geq 0$ tends to infinity. The dimensional comparison now gives

\[ \int _{H_ n}(\omega _ n+\mathrm{dd}^{\mathrm{c}}V_\eta )^ n \geq \frac{n(n-1)}6\, 4^{-(n-3)} \int _{x\geq 0}J_3(V_\eta )\, \mathrm dx\, \mathrm dA_ E \longrightarrow +\infty . \]
$\square$

Corollary 23.1.36.77FC30E16968 Every Hermitian metric on $H_ n$, $n\geq 3$, fails the bounded mass property.

Proof. For a Hermitian form $h$, compactness gives $h\geq c\omega _ n$ for some $c>0$. The potentials $cV_\eta $ satisfy $h+\mathrm{dd}^{\mathrm{c}}(cV_\eta )\geq c(\omega _ n+\mathrm{dd}^{\mathrm{c}}V_\eta )>0$, and positivity of determinants bounds their masses below by $c^ n$ times the divergent masses in theorem 23.1.35. $\square$

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