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14.19.37 Normalization on the Hopf threefold4D6684B2CEDB

In this section, $X$ is the Hopf threefold of section 23.1, with its Hermitian form $\omega $ and projection $\pi :X\to \mathbb P^2$. Recall that $\alpha =\pi ^*\omega _{\mathrm{FS}}$ and $\beta =\omega -\alpha $, and that $\nu =\omega ^3/6$ is a probability measure. We write $E$ for the elliptic fiber, $\tau _ a$ for the translation by $a\in E$, $\mathrm{d}A_ E$ for the probability Haar measure of $E$, and $A$ for the fiber-averaging operator of proposition 23.1.38.

The minimum-mass characterization in that proposition gives a lower bound for the normalization constant, and equality forces every solution to descend to the projective base. The uniform $L^ p$ control of this chapter makes the equality case stable under limits of the data, although the densities themselves need only converge weakly.

Here we take $\mathrm{d}V=\omega ^3$; the density $f$ is still a nonnegative nonzero function in $L^ p$. By theorem 14.19.2, proposition 14.19.34, there is a unique positive constant $c_ f$ for which the equation

\[ (\omega +\mathrm{dd}^{\mathrm{c}}u)^3=c_ f f\, \omega ^3 \]

has a continuous $\omega $-plurisubharmonic solution. Put $I_ f=\int _ Xf\, \mathrm{d}\nu $.

Proposition 14.19.38.56B19584AE9E One has $c_ f I_ f\geq 1$, and every continuous solution $u$ satisfies

5789C77F73D2 14.19.39
\begin{equation} \| u-Au\| _{L^2(X,\nu )}^2\leq 4(c_ f I_ f-1). \end{equation}

Moreover, $c_ f I_ f=1$ if and only if $f=\pi ^*g$ almost everywhere with respect to $\nu $, for some nonnegative $g\in L^ p(\mathbb P^2,\omega _{\mathrm{FS}}^2)$. In this case every continuous solution is the pullback of a continuous $\omega _{\mathrm{FS}}$-plurisubharmonic function.

Proof. Integrating the equation gives $\int _ X\omega _ u^3=6c_ f I_ f$. Both the inequality $c_ fI_ f\geq 1$ and (14.19.39) therefore follow from (23.1.39), whose left-hand side is nonnegative.

Suppose $c_ f I_ f=1$. By proposition 23.1.38, $u=\pi ^*q$ for a bounded $\omega _{\mathrm{FS}}$-plurisubharmonic function $q$; local holomorphic sections of $\pi $ show that $q$ is continuous. Thus $u\circ \tau _ a=u$ for every $a\in E$. Since bounded products are natural under biholomorphisms and $\omega $ is invariant under the translations, it follows that

\[ \tau _ a^*(f\nu )=f\nu \qquad (a\in E). \]

As $f$ is defined only almost everywhere, we use this invariance as an identity of measures.

By Fubini's theorem, the fiber average $Af$ is defined almost everywhere. It equals $\pi ^*g$ for a measurable function $g$ on the base, because local trivializations differ by translations of the Haar variable. Jensen's inequality gives

\[ \int _{\mathbb P^2}|g|^ p\omega _{\mathrm{FS}}^2 \leq \int _ X|f|^ p\, \mathrm{d}\nu . \]

For every continuous real test function $\chi $ on $X$, the invariance of the measure gives $\int _ X\chi f\, \mathrm{d}\nu =\int _ X(\chi \circ \tau _ a)f\, \mathrm{d}\nu $. Integrating in $a$, and using Fubini's theorem together with the invariance of Haar measure under $a\mapsto -a$, we obtain $\int _ X\chi f\, \mathrm{d}\nu =\int _ X\chi Af\, \mathrm{d}\nu $. Hence $f=Af=\pi ^*g$ almost everywhere, with $g\geq 0$.

Conversely, suppose $f=\pi ^*g$ almost everywhere. Disintegration gives $I_ f=\int _{\mathbb P^2}g\omega _{\mathrm{FS}}^2$. Applying theorem 14.19.2 on the base, we find a continuous $\omega _{\mathrm{FS}}$-plurisubharmonic function $q$ and a constant $k>0$ with $(\omega _{\mathrm{FS}}+\mathrm{dd}^{\mathrm{c}}q)^2=kg\, \omega _{\mathrm{FS}}^2$. Bounded decreasing regularization and Stokes' theorem give $\int _{\mathbb P^2}(\omega _{\mathrm{FS}}+\mathrm{dd}^{\mathrm{c}}q)^2=1$, so $k=I_ f^{-1}$.

Put $u=\pi ^*q$. To compute its bounded product, take smooth strictly $\omega _{\mathrm{FS}}$-plurisubharmonic functions $q_ j\downarrow q$ by theorem 5.32.1. Their pullbacks decrease to $u$ and are strictly $\omega $-plurisubharmonic. Since $\alpha ^3=\beta ^2=0$, the smooth cube identity gives, for each $j$,

\[ (\omega +\mathrm{dd}^{\mathrm{c}}\pi ^*q_ j)^3 =3\pi ^*(\omega _{\mathrm{FS}}+\mathrm{dd}^{\mathrm{c}}q_ j)^2\wedge \beta . \]

In a local trivialization, the right-hand side is $6\mathrm{d}A_ E$ times the base measure $(\omega _{\mathrm{FS}}+\mathrm{dd}^{\mathrm{c}}q_ j)^2$: the horizontal components of $\beta $ vanish against the base top-degree form, and its vertical restriction is $2\mathrm{d}A_ E$. Continuity of bounded products on the base and on $X$, tested in these trivializations and patched by a partition of unity, therefore gives

\[ (\omega +\mathrm{dd}^{\mathrm{c}}u)^3=\frac{f}{I_ f}\, \omega ^3. \]

Uniqueness of the constant implies $c_ f=I_ f^{-1}$. Every continuous solution then has mass $6$, so the mass characterization in proposition 23.1.38 makes it a pullback from the base, and the descended potential is continuous by local sections, as above. $\square$

When $c_ fI_ f$ is close to one, compactness gives uniform convergence of subsequences of potentials and weak convergence of their densities, and the equality case identifies the limits.

Proposition 14.19.40.21F8F4F18D0A Fix $1<p<\infty $ and $M\geq 1$. Suppose $f_ j\geq 0$ satisfy

\[ \int _ X f_ j\, \mathrm{d}\nu =1,\qquad \| f_ j\| _{L^ p(X,\nu )}\leq M,\qquad c_{f_ j}\longrightarrow 1. \]

For each $j$, let $u_ j$ be any continuous solution of $\omega _{u_ j}^3=c_{f_ j}f_ j\omega ^3$ normalized by $\sup _ Xu_ j=0$. Then

\[ \| u_ j-Au_ j\| _{C^0(X)}\longrightarrow 0,\qquad f_ j-Af_ j\rightharpoonup 0\quad \hbox{weakly in }L^ p(X,\nu ). \]

Every subsequence has a further subsequence with $u_ j\to \pi ^*q$ uniformly and $f_ j\rightharpoonup \pi ^*g$ weakly in $L^ p$, where $q$ is a continuous $\omega _{\mathrm{FS}}$-plurisubharmonic function, $g\geq 0$, $\int _{\mathbb P^2}g\omega _{\mathrm{FS}}^2=1$, and

\[ (\omega _{\mathrm{FS}}+\mathrm{dd}^{\mathrm{c}}q)^2=g\omega _{\mathrm{FS}}^2. \]

Proof. Since the constants $c_{f_ j}$ converge, the densities $c_{f_ j}f_ j$ have a common $L^ p$ bound. By theorem 14.19.21, every subsequence has a further subsequence on which $u_ j\to u$ uniformly. Since $L^ p$ is reflexive, a further extraction gives $f_ j\rightharpoonup f$ weakly in $L^ p$. Positivity, the unit integral and the bound $\| f\| _ p\leq M$ pass to the limit, and continuity of bounded products under uniform convergence gives

\[ (\omega +\mathrm{dd}^{\mathrm{c}}u)^3=f\omega ^3. \]

Thus $c_ f=1$. Since also $I_ f=1$, proposition 14.19.38 gives $u=\pi ^*q$ and $f=\pi ^*g$. The bounded cube identity proved there, tested against pullbacks of continuous functions on the base, gives the asserted base equation. In particular, both limits are fixed by $A$.

Haar averaging is a contraction on $C^0(X)$ and on $L^ p(X,\nu )$. It is also self-adjoint with respect to $\nu $: Fubini's theorem and the invariance of $\nu $ under the fiber translations give $\int \chi Af\, \mathrm{d}\nu =\int A\chi f\, \mathrm{d}\nu $ for $\chi \in L^{p'}$. Along the extracted subsequence, therefore, $u_ j-Au_ j\to 0$ uniformly and $f_ j-Af_ j\rightharpoonup 0$ in $L^ p$. If either convergence failed for the original sequence, some subsequence would stay away from the limit (for the weak convergence, when tested against one fixed function in $L^{p'}$), and a further subsequence of the kind just constructed would give a contradiction. This proves both conclusions for the whole sequence. $\square$

Example 14.19.41.F8EB82A5AD2B Choose $0<a<1/2$, and use the function $s=\log |z|^2$ on the covering space $\mathbb C^3\setminus \{ 0\} $ from section 23.1. Since $s$ changes by $-2$ under the defining contraction, every $2$-periodic function of $s$ descends to $X$. For an integer $m\geq 1$, define the smooth normalized potential

\[ u_ m=\frac{a}{\pi ^2m^2}\bigl (\cos (\pi m s)-1\bigr ), \qquad \| u_ m-Au_ m\| _{C^0}=\frac{a}{\pi ^2m^2}\longrightarrow 0. \]

Primes below denote derivatives with respect to $s$. Since $\mathrm{dd}^{\mathrm{c}}s=\alpha $ and $\mathrm{d}s\wedge \mathrm d^{\mathrm c}s=\beta $, the chain rule gives

\[ \omega +\mathrm{dd}^{\mathrm{c}}u_ m=(1+u_ m')\alpha +(1+u_ m'')\beta , \quad u_ m'=-\frac{a}{\pi m}\sin (\pi m s),\quad u_ m''=-a\cos (\pi m s). \]

Both coefficients are bounded below by positive constants independent of $m$. The rank identities $\alpha ^3=\beta ^2=0$ imply

\[ (\omega +\mathrm{dd}^{\mathrm{c}}u_ m)^3=H_ m\omega ^3,\qquad H_ m=\left(1-\frac{a}{\pi m}\sin (\pi m s)\right)^2 \bigl (1-a\cos (\pi m s)\bigr ). \]

On each fiber, $s$ is twice the real Haar coordinate plus a constant depending on the base point, so its distribution modulo $2$ is uniform. When the periodic expression $H_ m=(1+u_ m')^2(1+u_ m'')$ is averaged, the terms $u_ m'$, $u_ m''$, $u_ m'u_ m''$ and $(u_ m')^2u_ m''$ contribute zero, since they are derivatives of periodic functions. Hence

\[ I_ m:=\int _ XH_ m\, \mathrm{d}\nu =AH_ m =1+\frac{a^2}{2\pi ^2m^2}. \]

Set $f_ m=H_ m/I_ m$. These smooth densities have unit integral and common strictly positive lower and finite upper bounds. Uniqueness of the constant gives $c_{f_ m}=I_ m\to 1$, and $Af_ m=1$. Nevertheless,

\[ \| f_ m-1+a\cos (\pi m s)\| _{C^0(X)}=O(m^{-1}), \qquad \lim _{m\to \infty }\| f_ m-Af_ m\| _{L^1(X,\nu )}=\frac{2a}{\pi }>0. \]

Here the constant in $O(m^{-1})$ depends only on $a$, and the last identity uses the fact that the average of $|\cos |$ over a period is $2/\pi $. Thus near equality does not imply strong $L^1$ convergence to the fiber average, even for uniformly positive bounded smooth densities. For any pullback $g$ of a function on the base, the contraction property gives $\| f_ m-Af_ m\| _1\leq 2\| f_ m-g\| _1$, so approximation by such pullbacks also fails in $L^1$.

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