Lemma 10.8.259C08A2D9AA88

Proof. We begin with the inequalities, which are pointwise. Fix a point and suppose that $a\geq b$ there; the other case is symmetric. If $a=-\infty $, then $b=-\infty $ as well, and all three expressions equal $-\infty $. Otherwise $\log (\mathrm e^ a+\mathrm e^ b)=a+\log (1+\mathrm e^{b-a})$, and $0\leq \log (1+\mathrm e^{b-a})\leq \log 2$.

Put $H(x,y)=\log (\mathrm e^ x+\mathrm e^ y)$ for $x,y\in [-\infty ,\infty )$, with the same conventions. On $\mathbb R^2$ the function $H$ is smooth and increasing in each variable: its partial derivatives are $p$ and $1-p$, where $p=\mathrm e^ x/(\mathrm e^ x+\mathrm e^ y)$, and its Hessian matrix is $p(1-p)\left(\begin{smallmatrix}1& -1\\ -1& 1\end{smallmatrix}\right)$. On $[-\infty ,\infty )^2$ it is still increasing in each variable, and it is continuous, with values in $[-\infty ,\infty )$. Indeed, by the inequalities just proved, $\max \{ x,y\} \leq H(x,y)\leq \max \{ x,y\} +\log 2$, so $H(x,y)\to -\infty $ when $\max \{ x,y\} \to -\infty $; and $H(x,y)=y+\log (1+\mathrm e^{x-y})\to y_0$ when $x\to -\infty $ and $y\to y_0\in \mathbb R$.

The first assertion is local. Let $B\Subset U$ be a coordinate ball, let $s$ be a smooth real function on $B$ with $\mathrm{dd}^{\mathrm{c}}s=\theta $ (lemma 6.14.2), and let $B'\Subset B$ be a smaller concentric ball. Then $a+s$ and $b+s$ are plurisubharmonic on $B$, by lemma 6.14.4(2), and $\log (\mathrm e^ a+\mathrm e^ b)+s=H(a+s,b+s)$. If $f,g$ are smooth plurisubharmonic functions, the chain rule gives

with $p$ evaluated at $(f,g)$; hence $H(f,g)$ is plurisubharmonic, by proposition 5.9.2(1). In general, let $f_ k$ and $g_ k$ be the regularizations of $a+s$ and $b+s$ by radial mollifiers whose radii decrease to zero and are smaller than the distance from $B'$ to $\partial B$. By proposition 5.9.2(2), they are smooth plurisubharmonic functions on $B'$, and they decrease to $a+s$ and $b+s$. Since $H$ is continuous and increasing in each variable on $[-\infty ,\infty )^2$, the functions $H(f_ k,g_ k)$ decrease to $H(a+s,b+s)$. This limit is at least $a+s$, so it is not identically $-\infty $, and it is plurisubharmonic on $B'$ by theorem 5.10.1. Consequently $\log (\mathrm e^ a+\mathrm e^ b)=H(a+s,b+s)-s$ is quasi-plurisubharmonic on $B'$ and $\theta +\mathrm{dd}^{\mathrm{c}}\log (\mathrm e^ a+\mathrm e^ b)=\mathrm{dd}^{\mathrm{c}}H(a+s,b+s)\geq 0$ there. Every point of $U$ is the center of such a ball $B'$, so the first assertion follows. $\square$