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23.1.64 Limits of shrinking smooth potentialsC99F8C57CD9A

When the bounded mass property fails, corollary 13.15.18 produces smooth potentials converging to zero whose Monge–Ampère measures keep an excess mass in the limit. On the Hopf threefold we can determine exactly which excess measures occur: they are the lifts of arbitrary base measures by the fiber probability measures. The proof combines the intermediate currents of proposition 13.15.20 with averaging over the automorphism group.

On a general Hermitian manifold, the restrictions on the intermediate currents in proposition 13.15.20 are still necessary; we do not know whether they are sufficient, or what the support and pluripolar behavior of shrinking defects can be in general.

We keep the Hopf threefold $X$, its projection $\pi :X\to \mathbb P^2$, and the forms $\omega =\alpha +\beta $ of the previous sections. For $p\in \mathbb P^2$, let $h_ p$ be the Haar probability measure on the elliptic fiber $\pi ^{-1}(p)$, that is, the measure $\mathrm{d}A_ E$ transported by the action of $E$. For a finite positive measure $\mu $ on the base, define its fiberwise lift $\mathcal L\mu $ by

\[ \int _ X\varphi \, \mathrm{d}\mathcal L\mu =\int _{\mathbb P^2}\left(\int _{\pi ^{-1}(p)}\varphi \, \mathrm{d}h_ p\right)\mathrm{d}\mu (p), \qquad \varphi \in C(X). \]

Since fiber integrals of continuous functions are continuous in $p$, the map $\mathcal L$ is weakly continuous. The normalization $\int _{\pi ^{-1}(p)}\beta =2$ gives

ABD574F21AEB 23.1.65
\begin{equation} \mathcal L\mu =\tfrac 12\beta \wedge \pi ^*\mu , \qquad \mathcal L(\omega _{\mathrm{FS}}^2)=\tfrac 16\omega ^3. \end{equation}

Here $\pi ^*\mu $ denotes the pullback of a top-degree current by the submersion $\pi $; locally it is the base measure with constant coefficients in the fiber directions.

For a Hermitian form $\Omega $ on $X$, put

\[ M_\Omega (f)=\int _ X(\Omega +\mathrm{dd}^{\mathrm{c}}f)^3,\qquad V_\Omega =\int _ X\Omega ^3 \]

for smooth real $f$, and let $b$ be a real number. A positive measure $\sigma $ on $X$ is called a shrinking defect for $(\Omega ,b)$ if, for every sequence $0<\varepsilon _ j<1$ tending to zero, there are smooth $U_ j\in \mathrm{PSH}(X,\Omega )$ and numbers $0<r_ j<\varepsilon _ j$ such that

823EC0E922FD 23.1.66
\begin{equation} \sup _ XU_ j=0,\qquad w_ j=r_ jU_ j,\qquad M_\Omega (w_ j)=b,\qquad (\Omega +\mathrm{dd}^{\mathrm{c}}w_ j)^3\rightharpoonup \Omega ^3+\sigma . \end{equation}

Here weak convergence means convergence against every continuous function on $X$. We first classify the shrinking defects for the metric $\omega $, for which $V_\omega =6$; we then write $m=b-6>0$. Afterwards we pass to an arbitrary Hermitian form.

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