Corollary 12.26.2.87EBA7661A8A On a compact Kähler manifold $X$ of dimension $n$, fix finitely many smooth closed semipositive $(1,1)$-forms $B_ j$ and smooth strictly positive functions $f_ j$. For $t_ j\geq 0$, put
\[ \theta =\sum _ jt_ jf_ jB_ j,\qquad B=\sum _{j:t_ j>0}B_ j,\qquad R=\max _ X\operatorname {rank}B. \]
The background $\theta $ is nef, has common order $n-R$, and satisfies the two-sided power bounds of proposition 12.26.1. For every modification $\pi :Y\to X$ with $Y$ a compact Kähler manifold, the pullback $\pi ^*\theta $ has the same order and power bounds with the same exponent, for every smooth positive probability form and Hermitian perturbation on $Y$.
Proof.
Semipositivity gives nefness. If some $t_ j$ is positive, compactness and strict positivity of the functions $f_ j$ give
\[ \begin{gathered} cB\leq \theta \leq CB,\qquad c=\min _{j:t_ j>0}\bigl (t_ j\min _ X f_ j\bigr )>0,\\ C=\max _{j:t_ j>0}\bigl (t_ j\max _ X f_ j\bigr ), \end{gathered} \]
and proposition 12.26.1 applies with $\varphi =0$. If all parameters vanish, then $B=\theta =0$, $R=0$, any $c,C>0$ may be used, and the order is $n$.
These inequalities between smooth forms are preserved by pullback, and the rank of $\pi ^*B$ is at most $R$. If $R>0$, the set where $B$ has rank $R$ is nonempty and open, so it meets the dense open set where $\pi $ is biholomorphic; hence the maximal rank of $\pi ^*B$ is $R$. This is immediate if $R=0$. Since $\dim Y=n$, the same proposition applies on $Y$, with any choice of data there. No singular potential needs to be pulled back.
$\square$
Pome
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