PomeProposition 12.25.11.57BC641DA62E Replacing some plain factors by linear blowups as above preserves the order of every background in the fixed cone. More generally, for every modification $\pi :Y\to X$ with $Y$ a compact Kähler manifold, the pullback $\pi ^*\theta $ has common order $n-R(\theta )$, and satisfies two-sided power bounds with this exponent for every smooth positive probability form and every Hermitian perturbation on $Y$.
Proposition 12.25.1157BC641DA62E
Proof. For a product of linear blowups, pull back the coefficient functions and take the new coefficients of the forms $Q_ i$ to be zero. A form $H_ i$ that occurs on a plain factor becomes a form $H_ i$ that occurs on a blown-up factor, with the same rank $m_ i$, and factors in which nothing occurs remain so. Hence theorem 12.25.8, applied on both products, gives the same order.
For a general compact Kähler manifold $Y$, pull back the semipositive smooth remainders and the divisor identities used in the proof above. Each nonzero holomorphic section remains nonzero under the surjective map $\pi $, and the logarithm of the squared norm of its pullback is locally integrable, with $\mathrm{dd}^{\mathrm{c}}$ equal to the current of its effective zero divisor minus the pulled-back curvature. Hence
\[ \pi ^*\theta \leq C\pi ^*B_\theta ,\qquad \pi ^*\theta +\mathrm{dd}^{\mathrm{c}}(\varphi \circ \pi )\geq c\pi ^*B_\theta . \]
Fix a Kähler form $\Omega _ Y$ on $Y$, and apply the first part of lemma 12.25.1, which needs no rank hypothesis; the pulled-back background is semipositive and hence nef. Put $R=R(\theta )$. Pullback does not increase rank, so $(\pi ^*B_\theta )^{R+1}=0$, while
\[ \int _ Y(\pi ^*B_\theta )^ R\wedge \Omega _ Y^{n-R}>0: \]
for $R>0$ the integrand is positive on the nonempty set where $\pi $ is biholomorphic, and for $R=0$ the integral is the Kähler volume. For any smooth positive probability form $\mu _ Y$, the closed nef formula following theorem 12.11.1 gives
\[ \mathcal V_{\mu _ Y}(\pi ^*B_\theta +\epsilon \Omega _ Y) =\sum _{k=0}^{R}\binom nk\epsilon ^{n-k} \int _ Y(\pi ^*B_\theta )^ k\wedge \Omega _ Y^{n-k}. \]
All coefficients are nonnegative and the coefficient for $k=R$ is positive, so this polynomial is bounded above and below by positive multiples of $\epsilon ^{n-R}$ for $0<\epsilon \leq 1$. The comparison of quotients gives the same bounds for $\pi ^*\theta +\epsilon h_ Y$. In particular, the order upstairs equals the order downstairs. $\square$
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