Theorem 17.2.53E698B84D63D

Proof. Step 1: a repeated factor and a common background. Let $(\varphi _1,\varphi _2)$ and $(\psi _1,\psi _2)$ be presentations of $a$ and $b$, and suppose that these four functions and a further potential $\eta $ all belong to $\mathrm{PSH}(X,\theta )$ for one smooth closed real $(1,1)$-form $\theta $. For $0\leq t\leq 1$ the functions $\psi _ j^ t=(1-t)\eta +t\psi _ j$ are $\theta $-psh, and $\psi _1^ t-\psi _2^ t=tb$ is bounded. Lemma 17.2.3, applied to $\varphi _1,\varphi _2,\psi _1^ t,\psi _2^ t$, gives

Since $\theta _{\psi _ j^ t}=(1-t)\theta _\eta +t\theta _{\psi _ j}$, multilinearity expands every measure here as a polynomial in $t$ with finite measures as coefficients, so both sides are polynomials in $t$. We compare their coefficients of $t$. On the left this coefficient is

where we used that (17.2.1) may be computed in any smooth background with respect to which the presentation is psh. On the right it is $\int _ Xb(S_1^0-S_2^0)$, and $S_ j^0=n\langle \theta _{\varphi _ j}\wedge \theta _\eta ^{n-1}\rangle $, so it equals $n\int _ Xb\, \mathrm{dd}^{\mathrm{c}}a\wedge \langle \theta _\eta ^{n-1}\rangle $. This proves (17.2.6) when $T_1=\cdots =T_{n-1}=\theta _\eta $ and all potentials share the background $\theta $. For $n=1$ there are no currents $T_ j$, and (17.2.6) is (17.2.4) for a Kähler background with respect to which the four functions are psh.

Step 2: arbitrary currents. Let $n>1$. By theorem 10.2.15, write $T_ j=\theta _ j+\mathrm{dd}^{\mathrm{c}}\eta _ j$ with $\theta _ j$ smooth closed and $\eta _ j\in \mathrm{PSH}(X,\theta _ j)$. For positive numbers $r_1,\ldots ,r_{n-1}$ with $\sum _ jr_ j=1$ put $\theta _ r=\sum _ jr_ j\theta _ j$ and $\eta _ r=\sum _ jr_ j\eta _ j$, so that $\sum _ jr_ jT_ j=\theta _ r+\mathrm{dd}^{\mathrm{c}}\eta _ r$. The potential $\eta _ r$ is $(\theta _ r+s\omega )$-psh for every $s\geq 0$, since $\theta _ r+s\omega +\mathrm{dd}^{\mathrm{c}}\eta _ r=\sum _ jr_ jT_ j+s\omega \geq 0$. Let $\kappa $ be a multiple of $\omega $ such that the presentations of $a$ and $b$ are $\kappa $-psh. Since the forms $\theta _ r$ are convex combinations of the finitely many smooth forms $\theta _ j$, there is $s_0$ with $\theta _ r+s\omega \geq \kappa $ for all $r$ and all $s\geq s_0$, and then the presentations are $(\theta _ r+s\omega )$-psh as well. Step 1 applied to the current $\sum _ jr_ jT_ j+s\omega =(\theta _ r+s\omega )+\mathrm{dd}^{\mathrm{c}}\eta _ r$ gives

where, by the independence of (17.2.1) from the background, the Hessian factors may be computed in one fixed Kähler background. Expanding the repeated current by multilinearity, both sides become polynomials in $s$, and comparing constant coefficients gives

Both sides are homogeneous polynomials of degree $n-1$ in $(r_1,\ldots ,r_{n-1})$. Since they agree on the open simplex, homogeneity makes them agree on the open positive cone of $\mathbb R^{n-1}$. Hence their coefficients agree: a polynomial that vanishes on a nonempty open set is zero, because all its partial derivatives vanish at a point $r_0$ of that set and it equals its finite Taylor expansion at $r_0$. The coefficient of $r_1\cdots r_{n-1}$ is $(n-1)!$ times the corresponding side of (17.2.6). $\square$