Lemma 6.14.4066780FF9DDD

Proof. At points where $X$ has dimension zero, which are isolated, all functions are smooth and $\mathrm{dd}^{\mathrm{c}}$ vanishes, so every assertion is trivial there; we may therefore assume that $X$ has positive dimension.

(1) Near each point, $\varphi =u+f$ with $u$ plurisubharmonic and $f$ smooth, and $u$ is locally integrable by theorem 5.9.1. Moreover $u$ is subharmonic, by the same theorem, so at every point it is the limit of its averages over small balls, by (3.4.13); the continuous function $f$ has the same property, and hence so has $\varphi $. Now let $\varphi $ and $\psi $ be quasi-plurisubharmonic on $V$ with $\varphi \leq \psi $ almost everywhere, and let $x\in V$. Since both functions are locally integrable, their averages over small coordinate balls centered at $x$, in a chart at $x$, are finite, and those of $\varphi $ are at most those of $\psi $; letting the radius tend to zero gives $\varphi (x)\leq \psi (x)$. Applying this in both directions, we see that two quasi-plurisubharmonic functions that agree almost everywhere agree everywhere.

(2) Let $\varphi $ be quasi-plurisubharmonic on $V$. If $\varphi +h$ is plurisubharmonic, then $\theta +\mathrm{dd}^{\mathrm{c}}\varphi =\mathrm{dd}^{\mathrm{c}}(\varphi +h)\geq 0$ by theorem 5.9.1. Conversely, suppose that $\mathrm{dd}^{\mathrm{c}}(\varphi +h)\geq 0$ on $V$. On a coordinate ball $B\subset V$, theorem 5.9.1 gives a plurisubharmonic function on $B$ that agrees almost everywhere with $\varphi +h$; both are quasi-plurisubharmonic on $B$, so they agree everywhere on $B$ by (1). Hence $\varphi +h$ is plurisubharmonic on $V$.

(3) Let $B$ be a coordinate ball and $h$ a smooth real function on $B$ with $\mathrm{dd}^{\mathrm{c}}h=\theta $ (lemma 6.14.2). By (2) the functions $\varphi _ i+h$ are plurisubharmonic on $B$, hence so is $\max _ i\varphi _ i+h=\max _ i(\varphi _ i+h)$, by proposition 5.9.5. Thus $\max _ i\varphi _ i$ is quasi-plurisubharmonic on $B$, and $\theta +\mathrm{dd}^{\mathrm{c}}\max _ i\varphi _ i\geq 0$ there by (2). Such balls cover $X$.

(4) By compactness there are finitely many open sets $V_\kappa \Subset V_\kappa ^+$ such that the $V_\kappa $ cover $X$ and $\varphi =u_\kappa +f_\kappa $ on $V_\kappa ^+$, with $u_\kappa $ plurisubharmonic and $f_\kappa $ smooth. By lemma 6.14.1, applied to the compact set $\overline{V_\kappa }$ and the form $\mathrm{dd}^{\mathrm{c}}f_\kappa $, there are constants $C_\kappa \geq 0$ with $\mathrm{dd}^{\mathrm{c}}f_\kappa \geq -C_\kappa \omega $ on $V_\kappa $, and $\mathrm{dd}^{\mathrm{c}}u_\kappa \geq 0$ by theorem 5.9.1. Hence $\mathrm{dd}^{\mathrm{c}}\varphi \geq -C\omega $ with $C=\max _\kappa C_\kappa $.

(5) Replacing $C$ by $\max \{ C,0\} $, we may assume that $C\geq 0$. Let $B$ be a coordinate ball whose closure lies in a chart. By lemma 6.14.1, applied in the chart to the compact set $\overline B$, the Hermitian form $\mathrm{dd}^{\mathrm{c}}|z|^2$ and the form $-\omega $, there is $C'$ with $\omega \leq C'\mathrm{dd}^{\mathrm{c}}|z|^2$ on $B$, so $\mathrm{dd}^{\mathrm{c}}(\varphi +A|z|^2)\geq 0$ on $B$ for $A=CC'$. By theorem 5.9.1, $\varphi +A|z|^2$ agrees almost everywhere on $B$ with a plurisubharmonic function, and subtracting $A|z|^2$ gives a quasi-plurisubharmonic function $\varphi _ B$ on $B$ that agrees almost everywhere with $\varphi $. If $B'$ is a second such ball, then $\varphi _ B$ and $\varphi _{B'}$ agree almost everywhere on $B\cap B'$, and both are quasi-plurisubharmonic, so they agree everywhere on $B\cap B'$ by (1). Since such balls cover $X$, the functions $\varphi _ B$ define a quasi-plurisubharmonic function on $X$ that agrees almost everywhere with $\varphi $. $\square$