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14.9 Normalization on the Hopf threefold44E03525BBAF

44E03525BBAF

On the Hopf threefold of section 23.1, the deviation of $b$ from the closed-background formula (14.2.3) can be computed: it is the energy of the solution along the elliptic fibers. In this section, $X$ is that threefold and $\omega $ its Gauduchon metric; we use the projection $\pi :X\to \mathbb P^2$, the probability measure $\nu =\omega ^3/6$ and the averaging operator $A$ of proposition 23.1.38, and we write $\beta =\omega -\pi ^*\omega _{\mathrm{FS}}$ for the rank-one form of that construction. As in section 23.1, the fiber derivative $u_ w$ does not depend on the local trivialization of the elliptic bundle.

For $F\in C^\infty (X,\mathbb R)$, let $(u_ F,b(F))$ be the solution of

\[ (\omega +\mathrm{dd}^{\mathrm{c}}u_ F)^3=\mathrm{e}^{F+b(F)}\omega ^3, \qquad \omega +\mathrm{dd}^{\mathrm{c}}u_ F>0,\qquad \int _ Xu_ F\, \mathrm{d}\nu =0. \]

Define

\[ D(F)=b(F)+\log \int _ X\mathrm{e}^ F\, \mathrm{d}\nu . \]

For a closed background, the corresponding quantity vanishes by (14.2.3).

Proposition 14.9.1.CA9F2EF1D149 For every smooth real $F$ on this threefold,

028038652BCC 14.9.2
\begin{align} \mathrm{e}^{D(F)}& =1+\int _ X|(u_ F)_ w|^2\, \mathrm{d}\nu , \\ \| u_ F-Au_ F\| _{L^2(X,\nu )}^2 & \leq 4\bigl (\mathrm{e}^{D(F)}-1\bigr ). \tag{14.9.3} \end{align}

Thus $D(F)\geq 0$, with equality if and only if $F=\pi ^*H$ for some $H\in C^\infty (\mathbb P^2,\mathbb R)$.

Proof. Integration of the equation and (23.1.5) give

\[ \mathrm{e}^{D(F)}=\frac16\int _ X(\omega +\mathrm{dd}^{\mathrm{c}}u_ F)^3 =1+\int _ X|(u_ F)_ w|^2\, \mathrm{d}\nu . \]

Substituting the first equality into (23.1.39) proves (14.9.3).

If $D(F)=0$, then $(u_ F)_ w=0$ everywhere, because $u_ F$ is smooth. Since $u_ F$ is real, it is constant on each connected fiber, so it descends to a smooth real function $q$ on $\mathbb P^2$, and $\omega _{\mathrm{FS}}+\mathrm{dd}^{\mathrm{c}}q>0$ as in the proof of proposition 23.1.38. Since $\beta ^2=0$ and $\omega =\pi ^*\omega _{\mathrm{FS}}+\beta $, we have

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

Dividing by $\omega ^3=3\pi ^*\omega _{\mathrm{FS}}^2\wedge \beta $ and taking logarithms proves that $F$ is a smooth base pullback.

Conversely, suppose $F=\pi ^*H$. Apply theorem 14.2.1 on $(\mathbb P^2,\omega _{\mathrm{FS}})$ to obtain a smooth real $q$ and a scalar $c$ with

\[ \omega _{\mathrm{FS}}+\mathrm{dd}^{\mathrm{c}}q>0,\qquad (\omega _{\mathrm{FS}}+\mathrm{dd}^{\mathrm{c}}q)^2=\mathrm{e}^{H+c}\omega _{\mathrm{FS}}^2. \]

Since the base form is closed, Stokes' theorem gives $\mathrm{e}^ c\int _{\mathbb P^2}\mathrm{e}^ H\omega _{\mathrm{FS}}^2=1$. The preceding cube identity shows that $\pi ^*q$, after subtraction of its $\nu $-mean, solves the equation on $X$ with scalar $c$, and uniqueness gives $b(F)=c$. Since the base measure of $\nu $ is $\omega _{\mathrm{FS}}^2$, it follows that $D(F)=0$. $\square$

In particular, if a smooth positive density $f$ satisfies $\int _ Xf\, \mathrm{d}\nu =1$, the equation $(\omega +\mathrm{dd}^{\mathrm{c}}u)^3=\mathrm{e}^ b f\omega ^3$ has $b\geq 0$, with equality if and only if $f$ is the pullback of a function on the base. Fixing the total integral of the density therefore does not determine the scalar on this nonclosed background.

Corollary 14.9.4.BFCF1480CC8B Let $G\in C^\infty (X,\mathbb R)$, let $L_0v=3\, \mathrm{dd}^{\mathrm{c}}v\wedge \omega ^2/\omega ^3$ be the linearized operator at $u=0$, and let $v$ be the unique smooth real function satisfying

\[ L_0v=G-\int _ XG\, \mathrm{d}\nu , \qquad \int _ Xv\, \mathrm{d}\nu =0. \]

Then

\[ \left.\frac{\mathrm{d}}{\mathrm{d}t}D(tG)\right|_{t=0}=0, \qquad \left.\frac{\mathrm{d}^2}{\mathrm{d}t^2}D(tG)\right|_{t=0} =2\int _ X|v_ w|^2\, \mathrm{d}\nu . \]

The second derivative vanishes if and only if $G$ is a smooth base pullback.

Proof. For real $t$ near zero put $u_ t=u_{tG}$ and $b_ t=b(tG)$. By proposition 14.4.1, this is a smooth family, with $u_0=b_0=0$. Differentiating the logarithmic equation at $t=0$ gives $L_0\dot u_0=G+\dot b_0$. Since $\omega $ is Gauduchon, $\nu $ is the adjoint probability measure of $L_0$, and integration gives $\dot b_0=-\int _ XG\, \mathrm{d}\nu $. Hence $\dot u_0=v$; this also proves the existence and uniqueness of $v$.

Smooth dependence gives $(u_ t)_ w=t v_ w+O(t^2)$ uniformly on $X$, so (14.9.2) gives

\[ \mathrm{e}^{D(tG)}=1+t^2\int _ X|v_ w|^2\, \mathrm{d}\nu +O(t^3). \]

Taking logarithms proves the derivative formulas.

If the second derivative vanishes, then $v_ w=0$, so $v=\pi ^*q$ for a smooth real function $q$ on the base. Expanding $\omega ^2=\pi ^*\omega _{\mathrm{FS}}^2+2\pi ^*\omega _{\mathrm{FS}}\wedge \beta $ gives

\[ L_0(\pi ^*q)=\pi ^*\! \left( \frac{2\mathrm{dd}^{\mathrm{c}}q\wedge \omega _{\mathrm{FS}}}{\omega _{\mathrm{FS}}^2}\right). \]

Hence $G=L_0v+\int _ XG\, \mathrm{d}\nu $ is a base pullback. Conversely, when $G$ is a base pullback, proposition 14.9.1 gives $D(tG)=0$ for every real $t$. $\square$

This is a second-variation formula at the background density only. It does not determine whether $D$ is convex globally, because the second derivatives of $b$ and of the logarithmic integral have opposite signs.

9CEA8A3A68F014.9.3
\[\begin{aligned}\begin{align} \| u_ F-Au_ F\| _{L^2(X,\nu )}^2 & \leq 4\bigl (\mathrm{e}^{D(F)}-1\bigr ). \end{align}\end{aligned}\]
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