Proposition 5.34.1AAD8908A0CF9

Proof. For multiplication by a positive rational number, multiply the coefficient. Consider two presentations with coefficients $c_1,c_2$ and ideals $\mathfrak a_1,\mathfrak a_2$, and choose a positive rational number $d$ with $c_1=k d$ and $c_2=\ell d$, where $k,\ell $ are positive integers. Raising a sum of squared moduli of generators of $\mathfrak a_1$ to the power $k$ gives, up to bounded factors, the corresponding sum for $\mathfrak a_1^ k$; similarly for $\mathfrak a_2^\ell $. Products of generators of $\mathfrak a_1^ k$ and $\mathfrak a_2^\ell $ therefore present the sum with coefficient $d$, since a product of sums of squares is a sum of squares of pairwise products.

For the maximum, apply the elementary inequalities

to the two sums of squared moduli of generators of $\mathfrak a_1^ k$ and $\mathfrak a_2^\ell $. Taking logarithms shows that the maximum differs from $d$ times the logarithm of their total by a bounded amount. These are local assertions; on the compact manifold, finitely many charts give the corresponding comparisons of ordinary singularity types.

For the restriction, restrict to $Y$ the two-sided bounds that define the remainder. Since $Y$ is connected and is not contained in the zero set of $\mathfrak a$, the common zero set of the restricted generators is a proper analytic subset of $Y$, so $u|_ Y$ is finite on a dense open subset of $Y$. After a smooth local correction, the restriction theorem for plurisubharmonic functions shows that $u|_ Y$ is quasi-plurisubharmonic, and the restricted bounds give the asserted presentation. $\square$