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18.36 Kernels on a fibration8EDC1869ED38

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On a fibration with compact connected fibers, every section of a pulled-back line bundle is constant along the fibers and descends to the base. If probability measures are chosen on the fibers, this descent becomes an isometry, and the kernel measures on the total space are obtained from those on the base by integration along the fibers.

Let $\pi :X\to B$ be a surjective holomorphic submersion of connected compact complex manifolds, with connected fibers. Write $n=\dim X$ and $q=\dim B\geq 1$. Let $A$ be a holomorphic line bundle on $B$ with a smooth Hermitian metric $h_ A$, choose a quasi-plurisubharmonic function $\psi $ on $B$, and set

\[ (L,h)=(\pi ^*A,\pi ^*h_ A),\qquad \varphi =\pi ^*\psi . \]

For $y\in B$ write $X_ y=\pi ^{-1}(y)$. Choose a family of smooth positive probability measures $\nu _ y$ on the fibers $X_ y$ that varies smoothly with $y$; for instance, restrict a smooth positive vertical volume density to the fibers and divide by its fiber integrals. For $F\in C(X)$ set

\[ \mathcal A F(y)=\int _{X_ y}F\, \mathrm{d}\nu _ y. \]

The function $\mathcal A F$ is continuous: a proper submersion is locally a smooth fiber bundle, the densities vary smoothly in a local trivialization, and $F$ is uniformly continuous on compact sets. For a finite positive Borel measure $\mu $ on $B$, define its lift $\mathcal L\mu $ by

\[ \int _ X F\, \mathrm{d}\mathcal L\mu =\int _ B\mathcal A F\, \mathrm{d}\mu \qquad (F\in C(X)). \]

Since $\mathcal A$ is positive and $\mathcal A1=1$, the Riesz representation theorem makes $\mathcal L\mu $ a finite positive measure with the same total mass as $\mu $. Since $\mathcal AF$ is continuous, weak convergence of finite measures on $B$ implies weak convergence of their lifts.

Proposition 18.36.1.7E88736EF73B Pullback of sections induces an isomorphism

\[ H^0\bigl (B,A^ m\otimes \mathcal I(m\psi )\bigr ) \simeq H^0\bigl (X,L^ m\otimes \mathcal I(m\varphi )\bigr ). \]

Let $K\subseteq B$ be closed and non-pluripolar, let $w\in C(K)$, and let $\mu $ be a Bernstein–Markov probability measure for $(K,w)$ on $B$. Set $K_ X=\pi ^{-1}(K)$, $w_ X=w\circ \pi $ and $\mu _ X=\mathcal L\mu $. Then pullback is an isometry from the norm $\| \cdot \| _{m,w,\mu }$ on $B$ to the norm $\| \cdot \| _{m,w_ X,\mu _ X}$ on $X$, the measure $\mu _ X$ is Bernstein–Markov for $(K_ X,w_ X)$, and the partial kernel measures on $B$ and $X$, normalized with the same integer $r$, satisfy

\[ \beta _{X,m}^{(r)}=\mathcal L\bigl (\beta _{B,m}^{(r)}\bigr ). \]

Proof. Let $s$ be a holomorphic section of $\pi ^*A^ m$. Over an open set of $B$ on which $A^ m$ is trivial, the coefficient of $s$ is a holomorphic function on the inverse image, and it is constant on each fiber, since the fibers are compact and connected. It is therefore the pullback of a function on the base, which is holomorphic because it is the composition of the coefficient with a local holomorphic section of the submersion. These local functions transform by the transition functions of $A^ m$, so they define a section of $A^ m$ whose pullback is $s$. This proves the identification without multiplier ideals.

By lemma 5.37.7, the multiplier ideal $\mathcal I(m\varphi )$ on $X$ is generated by the pullbacks of the germs in $\mathcal I(m\psi )$. A pulled-back germ belongs to it exactly when the base germ belongs to $\mathcal I(m\psi )$: pullback gives one implication, and restriction to a local holomorphic section of $\pi $ gives the other. This identifies the selected spaces.

Pointwise norms of corresponding sections agree and are constant along the fibers. Since each $\nu _ y$ has mass one,

\[ \int _{K_ X}|\pi ^*s|_{h^ m}^2\mathrm{e}^{-mw_ X}\, \mathrm{d}\mu _ X =\int _ K|s|_{h_ A^ m}^2\mathrm{e}^{-mw}\, \mathrm{d}\mu . \]

The supremum norms on $K_ X$ and $K$ also agree, because $\pi $ is surjective. Thus the Bernstein–Markov inequality transfers from $(K,w,\mu )$ to $(K_ X,w_ X,\mu _ X)$, and the integral norms on $X$ are nondegenerate. Pulling back an orthonormal basis gives $B_{X,m}=B_{B,m}\circ \pi $ on $K_ X$, where each kernel is set to zero outside its weighted set. Integrating a continuous test function first along the fibers proves the identity of measures. $\square$

18.36.2 The quotient equilibrium measure29138044FA48

On a Kähler base, the partial kernel measures converge to the Monge–Ampère measure of the partial envelope. Since lifting is continuous, the kernel measures on the total space converge to the lift of this measure, with the normalization given by the dimension of the base.

Keep the submersion and the weighted compact set $(K,w,\mu )$ above. Assume that $B$ is Kähler, put $\theta =c_1(A,h_ A)$, and take $\psi \in \mathrm{PSH}(B,\theta )$ with positive non-pluripolar Monge–Ampère mass. Write $B_{B,m}$ and $B_{X,m}$ for the partial Bergman kernels on $K$ and $K_ X$, put $\mu _ X=\mathcal L\mu $, and let

\[ E=E_{\theta ,K}[\psi ]_{\mathcal I}(w) \]

be the partial envelope of section 18.32. We call $\mathcal L(\mathrm{MA}_\theta (E))$ the quotient equilibrium measure of these data on $X$.

Theorem 18.36.3.87C196C13DDE Under these hypotheses,

\[ \frac{q!}{m^ q}B_{X,m}\, \mu _ X \rightharpoonup \mathcal L\bigl (\mathrm{MA}_\theta (E)\bigr ). \]

If $n>q$, then $(n!/m^ n)B_{X,m}\, \mu _ X\to 0$ in total variation.

Proof. Apply [Theorem 14.3.1, SGPT] on the compact Kähler manifold $B$, with trivial auxiliary twisting line bundle. Its hypotheses are exactly our assumptions: $\psi $ has positive mass, $K$ is closed and non-pluripolar, $w$ is continuous, and $\mu $ is a Bernstein–Markov probability measure. It gives

\[ \frac{q!}{m^ q}B_{B,m}\, \mu \rightharpoonup \mathrm{MA}_\theta (E). \]

By proposition 18.36.1, the corresponding measures on $X$ are the lifts of these measures, and lifting preserves weak convergence; this proves the first assertion. In particular the $q$-normalized total masses are bounded, and multiplying them by $(n!/q!)m^{q-n}\to 0$ proves the last assertion when $n>q$. $\square$

18.36.4 The Hopf equilibrium measure16C99F55F2A7

On the Hopf manifold, the quotient equilibrium measure is a Hermitian Monge–Ampère measure: the vertical part of the Hopf metric supplies the missing fiber direction. The constant relating the two measures combines the fiber probability with a coefficient of the top power.

Use the Hopf manifold $X_ n$, $n\geq 2$, its projection $\pi :X_ n\to \mathbb P^{n-1}$, the Fubini–Study form $\sigma $ on $\mathbb P^{n-1}$ with $\int \sigma ^{n-1}=1$, and the forms $\alpha =\pi ^*\sigma $, $\zeta $ and $\omega =\alpha +\zeta $ of section 21.14; recall that $\zeta ^2=0$ and that $\zeta $ has integral two on each fiber. Take $A=\mathcal O(1)$ with its Fubini–Study metric, so that $\theta =\sigma $, and keep the selecting potential $\psi $ and the weighted data $(K,w,\mu )$ above. On each elliptic fiber put $\nu _ y=\zeta |_{X_ y}/2$. The lift of a base measure $\tau $ is then

\[ \mathcal L\tau =\tfrac 12\pi ^*\tau \wedge \zeta , \]

in the fiber-integration notation of section 21.14. Since $E\in \mathrm{PSH}(\mathbb P^{n-1},\sigma )$, theorem 21.14.4 gives $\mathcal L(\mathrm{MA}_\sigma (E))=(2n)^{-1}\mathrm{MA}^{\mathrm{ext}}_\omega (\pi ^*E)$, and theorem 18.36.3 becomes

C843534C9B58 18.36.5
\begin{equation} \frac{(n-1)!}{m^{n-1}}B_{X_ n,m}\, \mu _ X \rightharpoonup \frac{1}{2n}\mathrm{MA}^{\mathrm{ext}}_\omega (\pi ^*E). \end{equation}

The factor $1/(2n)$ combines the fiber probability $1/2$ with the coefficient $n$ of $\alpha ^{n-1}\wedge \zeta $ in $\omega ^ n=(\alpha +\zeta )^ n$. The closed curvature form $\alpha $ alone gives nothing: by theorem 21.14.4, descended potentials have zero Monge–Ampère measure with respect to $\alpha $, and the nonzero limit comes from the vertical form $\zeta $. For $n=3$ this gives a nonzero limit, with the normalization $2/m^2$, on a manifold without the bounded mass property (theorem 23.1.1).

For a positive integer $k$, every $k\alpha $-plurisubharmonic function on $X_ n$ is pulled back from $\mathbb P^{n-1}$, by theorem 21.14.2. An $\omega $-plurisubharmonic function, by contrast, can have an isolated pole, and then it does not descend. Its selected sections still descend, so the pole is recorded by a vanishing condition on the base. The next two sections determine this condition and its effect on the kernels.

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