Proposition 12.14.13F8EA88FD522E

Proof. Choose a nonzero smooth function $f_\rho \geq 0$ on $B$, supported in $\{ x<\rho \} $ and normalized by $\int _ Bf_\rho \omega ^ m=1$. The form

is smooth, strictly positive and closed: the first summand has top degree on the base, and $h$ is Kähler. The positive-root construction in the proof of theorem 11.6.1 gives a unique smooth metric $b_\rho $ with $b_\rho ^ m=\Gamma _\rho $, and $b_\rho $ is balanced because $\Gamma _\rho $ is closed. Since $\int _ Xh^{m+1}=n$, its $h$-trace is one.

Put $C_\theta =\int _ X\theta \wedge h^ m$. Since $a<\rho /t$ on the support of $f_\rho $,

The lower bound in the proof of theorem 12.14.6 gives $D_\mu (b_\rho )\leq n^{-1/n}\int _ X\theta \wedge \Gamma _\rho $, so the determinant integrals also tend to zero. The probability measures $f_\rho \omega ^ m$ converge to the Dirac mass at $p$ because their supports shrink to that point. Fiber integration, as in the preceding proof, gives the current limit $[Z]$, while the term $\rho h^ m$ tends to zero. $\square$