Proposition 6.14.8AFD530210C3A

Proof. Multiplication by $q\in \mathbb Q_{>0}$ replaces $c$ by $qc$. Sums and finite maxima of quasi-plurisubharmonic functions are quasi-plurisubharmonic. Indeed, let $\varphi _ i=u_ i+\sigma _ i$ near a point, with $u_ i$ plurisubharmonic and $\sigma _ i$ smooth. On a small coordinate ball about the point, the smooth functions $\sigma _ i+N|z|^2$ have positive semidefinite complex Hessians for a large constant $N$, hence are plurisubharmonic by proposition 5.9.2(1). By proposition 5.9.5, $\varphi _1+\varphi _2$ is a plurisubharmonic function minus $2N|z|^2$, and $\max _ i\varphi _ i$ is a plurisubharmonic function minus $N|z|^2$. For a sum or a maximum of two functions with analytic singularities, choose presentations about a given point, with coefficients $c$ and $d$, and a connected coordinate neighborhood $V$ of the point contained in both domains; by the identity theorem, neither tuple vanishes identically on $V$. Since $c$ and $d$ are rational, we can choose $c_0\in \mathbb Q_{>0}$ such that $c/c_0$ and $d/c_0$ are positive integers. For an integer $k\geq 1$ the multinomial formula gives

again a sum of squared moduli of holomorphic functions, not all identically zero on $V$. We may therefore assume that both presentations have the coefficient $c_0$, say $c_0\log A+b$ and $c_0\log B+b'$ with $A=\sum _ i|g_ i|^2$ and $B=\sum _ l|g'_ l|^2$. The sum is $c_0\log (AB)+b+b'$, and $AB=\sum _{i,l}|g_ ig'_ l|^2$, where some product $g_ ig'_ l$ is not identically zero on the connected set $V$. For the maximum we use

which shows that $\max (c_0\log A+b,c_0\log B+b')$ differs from $c_0\log (A+B)$ by a bounded function.

For the pullback, let $f\colon Y\to X$ be holomorphic, and suppose that $\varphi \circ f$ is not identically $-\infty $ on any connected component of $Y$. Near each point of $Y$, write $\varphi \circ f=u\circ f+\sigma \circ f$ on a connected coordinate neighborhood $W$, where $\varphi =u+\sigma $ near the image point with $u$ plurisubharmonic and $\sigma $ smooth; by proposition 5.9.3, $u\circ f$ is plurisubharmonic or identically $-\infty $ on $W$. Let $S$ be the open set of points of $Y$ near which $\varphi \circ f$ is identically $-\infty $. If $y$ lies in the closure of $S$ and $W$ is as above for $y$, then $u\circ f=-\infty $ on the nonempty open set $W\cap S$; since plurisubharmonic functions are locally integrable, $u\circ f$ is identically $-\infty $ on $W$, so $y\in S$. Thus $S$ is open and closed, and by hypothesis it contains no component of $Y$; hence $S=\varnothing $, and $\varphi \circ f$ is quasi-plurisubharmonic. If $\varphi =c\log \sum _ i|g_ i|^2+b$ on $V$, then on every connected coordinate neighborhood $W\subset f^{-1}(V)$ we have $\varphi \circ f=c\log \sum _ i|g_ i\circ f|^2+b\circ f$, with $b\circ f$ bounded, and the functions $g_ i\circ f$ do not all vanish identically on $W$, because $S=\varnothing $.

Now let $\pi $ be a log resolution of $\mathfrak a$ as in the statement. If $g_1,\ldots ,g_ r$ generate $\mathfrak a$, their pullbacks generate $\mathcal O_ Y(-D)$, so locally $g_ i\circ \pi =s_ Dh_ i$ with holomorphic $h_ i$, and $s_ D=\sum _ ik_ i(g_ i\circ \pi )$ for holomorphic $k_ i$. Since $s_ D$ is not a zero divisor, $\sum _ ik_ ih_ i=1$, so the $h_ i$ have no common zero and $\log \sum _ i|h_ i|^2$ is smooth. Hence

which proves both assertions about the remainder. $\square$