Pome

Corollary 13.14.221D7A78F6BD02

Corollary 13.14.22.1D7A78F6BD02 If the bounded mass property fails, there are nonpositive potentials $w_\varepsilon \in \mathrm{PSH}(X,\omega )$, $0<\varepsilon <1$, such that

\[ \omega +\mathrm{dd}^{\mathrm{c}}w_\varepsilon \geq (1-\varepsilon )\omega , \qquad \mathcal I(pw_\varepsilon )=\mathcal O_ X\quad (p>0), \]

and such that both the integral of the pointwise determinant $\det _\gamma (\omega +\mathrm{dd}^{\mathrm{c}}w_\varepsilon )$ against $\gamma ^ n$ and the extended Monge–Ampère mass of $w_\varepsilon $ are infinite. As $\varepsilon \downarrow 0$, the potentials converge to zero in $L^1$, their currents converge to $\omega $ in the coefficient $L^1$ norm, and their full trace measures $(\omega +\mathrm{dd}^{\mathrm{c}}w_\varepsilon )\wedge \gamma ^{n-1}$ converge to $\omega \wedge \gamma ^{n-1}$ in total variation.

Proof. Take $w_\varepsilon =\varepsilon U$, with $U$ and $T=\omega +\mathrm{dd}^{\mathrm{c}}U$ as in proposition 13.14.21. Its current is

\[ T_\varepsilon =(1-\varepsilon )\omega +\varepsilon T. \]

The lower bound and the two $L^1$ convergence assertions follow immediately. The full trace measures have integrable densities, and their distance from $\omega \wedge \gamma ^{n-1}$ in total variation is at most

\[ \varepsilon \| T-\omega \| _{L^1(\gamma )}. \]

Every multiplier ideal is trivial by the exponential integrability of $U$ proved in theorem 13.14.18. Since $T_\varepsilon \geq \varepsilon T$ almost everywhere, the determinant integral of $T_\varepsilon $ is at least $\varepsilon ^ n$ times that of $T$, hence infinite. The extended mass of $w_\varepsilon $ is infinite by the proof of corollary 13.14.19. $\square$

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