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The constant-rank hypothesis in lemma 12.25.1 can be removed. For a closed semipositive form $B$, the intersection polynomial $\int _ X(B+\epsilon \Omega )^ n$ has a leading term of order $\epsilon ^{n-R}$, where $R$ is the largest rank that $B$ attains anywhere on $X$. Combined with the comparison of quotients, this gives the order $n-R$ for every background that is comparable to $B$. The criterion applies to backgrounds that vanish on an open set, and to positive variable multiples of arbitrary closed semipositive generators.
Proposition 12.26.1.94DF4FEA70D3 Let $X$ be a compact Kähler manifold of dimension $n$, let $B$ be a smooth closed semipositive $(1,1)$-form, and put $R=\max _ X\operatorname {rank}B$. Suppose a smooth nef background $\theta $ satisfies
\[ \theta \leq CB,\qquad \theta +\mathrm{dd}^{\mathrm{c}}\varphi \geq cB \]
as currents, for constants $c,C>0$ and a real function $\varphi \in L^1(X)$. For every smooth positive probability form $\mu $ and Hermitian form $h$, there are constants $L,U>0$ such that
\[ L\epsilon ^{n-R}\leq \mathcal V_\mu (\theta +\epsilon h) \leq U\epsilon ^{n-R},\qquad 0<\epsilon \leq 1. \]
Thus $\theta $ has common order $n-R$, without any constant-rank hypothesis.
Proof.
Fix a Kähler form $\Omega $. The first part of lemma 12.25.1, which does not use the rank, shows that $\lambda _\mu (\theta +\epsilon h)$ and $\lambda _\mu (B+\epsilon \Omega )$ differ by bounded positive factors. Since $B^{R+1}=0$, the closed nef formula following theorem 12.11.1 gives
\[ \mathcal V_\mu (B+\epsilon \Omega ) =\sum _{k=0}^{R}\binom nk\epsilon ^{n-k} \int _ X B^ k\wedge \Omega ^{n-k}. \]
All coefficients are nonnegative. If $R>0$, then at a point where $B$ has rank $R$, some $R\times R$ minor of its coefficient matrix is positive and stays positive nearby, so $B^ R\wedge \Omega ^{n-R}$ has positive integral. If $R=0$, this integral is the Kähler volume. Hence the coefficient of $\epsilon ^{n-R}$ is positive. For $0<\epsilon \leq 1$, every other power of $\epsilon $ that occurs is at most $\epsilon ^{n-R}$, so the polynomial is bounded below by its leading term and above by the sum of its coefficients times $\epsilon ^{n-R}$. Raising the comparison of quotients to the $n$th power gives the asserted bounds, and taking logarithms gives the order.
$\square$
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.
$\square$
The constants depend on the parameters $t_ j$ and need not stay bounded as a positive parameter tends to zero. The rank of a sum records how the generators overlap, and it need not equal the sum of their ranks. Members of the family are not closed when $\sum _ jt_ j\, \mathrm df_ j\wedge B_ j\ne 0$.
Example 12.26.3.698A2684C823 Let $X=\mathbb P^1\times \mathbb P^ q$, $q\geq 1$, with projections $p_1,p_2$. Choose a smooth nonnegative function $b\not\equiv 0$ on $\mathbb P^1$ that vanishes on a nonempty open set, and a smooth strictly positive nonconstant function $f$ on $\mathbb P^ q$. With $\omega _1$ the normalized Fubini–Study form on $\mathbb P^1$, set
\[ B=p_1^*(b\omega _1),\qquad \theta =(f\circ p_2)B. \]
The smooth nef background $\theta $ is not closed, vanishes on a nonempty open set, and has common order $q$, with two-sided bounds by multiples of $\epsilon ^ q$.
Proof.
The form $b\omega _1$ is closed, because the first factor is a curve, and semipositive. Its pullback has rank one where $b>0$ and rank zero where $b=0$, so its maximal rank is one. Since $n=q+1$, corollary 12.26.2 gives the order $q$. A nonconstant smooth function $f$ on the connected second factor has $\mathrm df\ne 0$ somewhere. At a point of the product where $b>0$ and $\mathrm df\neq 0$, the horizontal two-form $B$ wedges nontrivially with the vertical one-form $p_2^*\mathrm df$. Thus $\mathrm d\theta =p_2^*\mathrm df\wedge B\ne 0$.
$\square$
These are geometric sufficient conditions for an integral common order, in the direction of 12.33.4. They do not cover backgrounds without a closed semipositive comparison form, coefficient functions that are allowed to vanish, or the comparison of orders under residuals or non-Kähler modifications in general.
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