PomeA528EDFC5F96We now subtract effective divisors from members of the fixed cone. Let $\mathcal O(H_ i)$ and $\mathcal O(Q_ i)$ be the pullbacks of $\mathcal O(1)$ under the projections defining $H_ i$ and $Q_ i$, with their Fubini–Study metrics, whose curvature forms are $H_ i$ and $Q_ i$. Consider line bundles that are tensor products of integer powers of these bundles, negative powers included, and finite nonnegative real combinations of zero divisors of their nonzero holomorphic sections $s_\nu $:
Here $\eta _ D$ is the corresponding combination of the curvature forms of the fixed metrics, so that $[D]=\eta _ D+\mathrm{dd}^{\mathrm{c}}\Phi _ D$. The terms involving $Q_ i$ are omitted for plain factors.
Lemma 12.25.13.5DAA9B186CEE The coefficients $(\delta _ i,\rho _ i)$ that occur for divisors $D$ of this form are exactly those satisfying
Proof. It suffices to treat one nonzero holomorphic section $s$, since nonnegative real combinations preserve the inequalities. Choose a point at which $s$ is nonzero and which lies outside all exceptional divisors, and fix the coordinates of the other factors. A projective line through that point inside a fiber of $q_ i$ has $H_ i$-degree one and $Q_ i$-degree zero. The restriction of $s$ to it is a nonzero section, so the exponent of $\mathcal O(H_ i)$ is nonnegative. On a plain factor, a line through the point gives the same inequality.
On a blown-up factor, write the chosen point as $([z:w],[w])$ with $w\ne 0$, and choose a linear map $A$ with $Aw=z$. The section of $q_ i$ given by $[t]\mapsto ([At:t],[t])$ passes through the point, and both $H_ i$ and $Q_ i$ have degree one on each projective line in its image. Such a line through the point exists because $r_ i\geq 1$, and $s$ restricts nontrivially to it. Its degree shows that the sum of the two exponents is nonnegative.
Conversely, let $F_ i$ be the pullback under $q_ i$ of a hyperplane. An admissible pair $(\delta _ i,\rho _ i)$ is realized by $\delta _ iE_ i+(\delta _ i+\rho _ i)F_ i$, since $E_ i$ has curvature $H_ i-Q_ i$, and coefficients on plain factors are realized by hyperplanes. These are allowed nonnegative combinations and give every coefficient tuple in the stated range. $\square$
Theorem 12.25.14.F79036F002E1 Let $\theta $ be in the fixed cone, with parameters $a_ i,b_ i,c_ i,d_ i$, and let $D$ be a divisor of the above form. Assume
Then the residual $\beta =\theta -\eta _ D$ lies in the fixed cone, and the common orders $a(\beta )$ and $a(\theta )$ satisfy
Equality holds if and only if, for every factor,
the second condition concerns only blown-up factors.
Proof. The residual parameters $(a_ i-\delta _ i,b_ i,c_ i-\rho _ i,d_ i)$ are nonnegative, so $\beta $ lies in the fixed cone; in particular it is smooth, semipositive and nef. Also
which is the positive-current comparison of proposition 12.24.1. Apply theorem 12.25.8 to $\theta $ and to $\beta $. If $H_ i$ occurs in $\theta $ but not in $\beta $, the rank $R_ i$ drops from $m_ i$ to $r_ i<m_ i$ or to zero. If $H_ i$ does not occur in $\theta $, then $a_ i=b_ i=0$, and the range of lemma 12.25.13 together with $a_ i\geq \delta _ i$ forces $\delta _ i=0$ and hence $\rho _ i\geq 0$; thus $Q_ i$ cannot appear in $\beta $ if it was absent from $\theta $. Therefore no rank $R_ i$ increases, each difference $R_ i(\theta )-R_ i(\beta )$ is nonnegative, and the sum vanishes exactly when the generator that determines $R_ i(\theta )$ still occurs in $\beta $ for every $i$. This is the stated pair of conditions. $\square$
The parameter inequalities guarantee that the residual stays in the fixed cone; they do not describe all divisors whose residual is nef.
Example 12.25.15.EB3698E8A068 On $Z=\operatorname {Bl}_{\mathbb P^\ell }\mathbb P^ m$, choose a smooth function $g>0$ with $\mathrm dg\wedge Q\ne 0$. For $t>0$, the nonclosed positive background $\theta _ t=tH+gQ$ has order zero. Removing $sE$, with its fixed curvature form $s(H-Q)$ and $0\leq s\leq t$, gives
Every residual $\beta _ s$ is nef and not closed. Its order equals that of $\theta _ t$ exactly for $s<t$, and at $s=t$ the order jumps by $\ell +1$. These examples realize each integer jump $1,\ldots ,m-1$.
Proof. The lower bound $\theta _ t\geq \min (t,\min g)\kappa >0$ shows that $\theta _ t$ is positive. For $s<t$ the form $H$ occurs in $\beta _ s$ and the rank is $m$; at $s=t$ only $Q$ occurs, with rank $r=m-\ell -1$. The order formula gives zero and $m-r=\ell +1$, respectively. The exterior derivative of every residual is $\mathrm dg\wedge Q\ne 0$. Finally, let $\ell $ range from zero to $m-2$. $\square$
At the endpoint $s=t$, the current $t[E]+\beta _ t$ in the class of $\theta _ t$ has a divisorial part, although the minimal multiplicities of $\theta _ t$ vanish because $\theta _ t$ is nef, as explained before corollary 12.24.5. The construction uses three hypotheses in an essential way: the coefficient functions are strictly positive, the subtracted divisor keeps the residual in the fixed cone, and, for general modifications, the source manifold is Kähler. Without such hypotheses the question in 12.33.4 is open.
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