Theorem 13.8.760DE4F681F75

Proof. Suppose first that $\{ \theta \} $ is big. Choose a reference potential $\varphi _0$ as in proposition 13.8.5, which exists because the class is big, and construct the corresponding potentials $\Phi _ N$ from $\varphi $ and $\Psi _ N$ from $\psi $. If $\varphi \geq \psi -C$, then for every $x\in \Sigma _ N$ the branch defining $\Phi _ N$ is at least the branch defining $\Psi _ N$ minus $t(x)C\leq C$; hence the suprema and their regularizations satisfy $\Phi _ N\geq \Psi _ N-C$, with a constant independent of $N$. By proposition 13.8.5, both are locally bounded outside the same closed set $A$, which is the pole set of the quasi-plurisubharmonic function $\varphi _0+s_0$, and this function is locally bounded outside $A$. The product $X\times \mathbb P^ N$ is a connected compact Kähler manifold of dimension $n+N$, with the Kähler form $p_ X^*\omega +p_{\mathbb P^ N}^*\omega _{\mathrm{FS}}$, so the results of this section apply to it.

For $0\leq t\leq 1$ put

and define $f_\psi $ in the same way. Applying lemma 13.8.1 on $X\times \mathbb P^ N$, with all $n+N$ factors equal to $\widetilde\theta +\mathrm{dd}^{\mathrm{c}}\Phi _ N$, respectively $\widetilde\theta +\mathrm{dd}^{\mathrm{c}}\Psi _ N$, and cancelling the common binomial coefficient in (13.8.6), we obtain

Writing $\theta +\mathrm{dd}^{\mathrm{c}}((1-t)\varphi _0+t\varphi )=(1-t)T_0+tT_\varphi $ with $T_0=\theta +\mathrm{dd}^{\mathrm{c}}\varphi _0$ and $T_\varphi =\theta +\mathrm{dd}^{\mathrm{c}}\varphi $, multilinearity shows that $f_\varphi $ is a polynomial in $t$, in particular continuous on $[0,1]$, and $f_\varphi (1)$ is the mass of $\varphi $. The probability measures $Nt^{N-1}\, \mathrm dt$ concentrate at $1$ as $N\to \infty $, since their mass on $[0,1-\delta ]$ is $(1-\delta )^ N$. By uniform continuity the two integrals tend to $f_\varphi (1)$ and $f_\psi (1)$, which proves the theorem for big classes.

In general, $\{ \theta \} $ is pseudo-effective, because it contains $\theta +\mathrm{dd}^{\mathrm{c}}\varphi $. For $\varepsilon >0$ the class of $\theta +\varepsilon \omega $ is big, since it contains the Kähler current $\theta +\varepsilon \omega +\mathrm{dd}^{\mathrm{c}}\varphi \geq \varepsilon \omega $, and $\varphi ,\psi $ are $(\theta +\varepsilon \omega )$-plurisubharmonic. By the big case, the mass of $\varphi $ with respect to $\theta +\varepsilon \omega $ is at least that of $\psi $. By multilinearity and theorem 13.5.4, both masses are polynomials in $\varepsilon $ with finite coefficients, and letting $\varepsilon \downarrow 0$ proves the theorem. $\square$