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Another way to change the order of degeneration is to subtract a positive current from the background. If $\theta -\beta $ differs from a positive current by a $\mathrm{dd}^{\mathrm{c}}$-exact term, then every Gauduchon pairing of $\theta $ dominates the corresponding pairing of $\beta $. When the residual $\beta $ is still nef, its perturbed volumes are therefore smaller, and its orders are at least those of $\theta $. The main example is the subtraction of the curvature form of an effective divisor. Such a subtraction can preserve the order, raise it by an integer or by a fraction, or produce unequal lower and upper orders; and the residual need not be nef.
Proposition 12.24.1.7CE9EDC18591 Let $\theta ,\beta $ be smooth nef real $(1,1)$-forms on a compact complex manifold $X$ of dimension $n$, and suppose that
\[ S=(\theta -\beta )+\mathrm{dd}^{\mathrm{c}}\varphi \geq 0 \]
is a positive $(1,1)$-current for some integrable function $\varphi $. For every Hermitian form $h$, every smooth strictly positive probability form $\mu $ and every $\epsilon >0$,
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12.24.2
\begin{equation} \mathcal V_\mu (\beta +\epsilon h)\leq \mathcal V_\mu (\theta +\epsilon h). \end{equation}
Consequently $\underline a(\beta )\geq \underline a(\theta )$ and $\overline a(\beta )\geq \overline a(\theta )$.
Proof.
For every Gauduchon metric $\gamma $, distributional integration by parts gives
\[ \int _ X(\theta -\beta )\wedge \gamma ^{n-1} =\langle S,\gamma ^{n-1}\rangle \geq 0. \]
Hence, for each test metric, the quotient for $\theta +\epsilon h$ is at least the quotient for $\beta +\epsilon h$, and taking infima gives $\lambda _\mu (\beta +\epsilon h)\leq \lambda _\mu (\theta +\epsilon h)$. Both sides are positive because $\beta $ and $\theta $ are nef, so raising to the $n$th power gives (12.24.2). Dividing the logarithms by the negative number $\log \epsilon $ reverses the inequality and proves both comparisons of orders.
$\square$
For an effective real divisor $D$ with curvature form $\eta _ D$, the Poincaré–Lelong formula gives $[D]=\eta _ D+\mathrm{dd}^{\mathrm{c}}\varphi $ for a quasi-plurisubharmonic function $\varphi $. Thus proposition 12.24.1 applies to $\beta =\theta -\eta _ D$, with $S=[D]$, whenever $\beta $ is nef. For a convergent countable positive combination of divisors, the same argument applies only when its curvature representative is smooth and a global potential realizes the current difference. If $T=\theta +\mathrm{dd}^{\mathrm{c}}u$ is a positive current with $T-S\geq 0$, then $T-S=\beta +\mathrm{dd}^{\mathrm{c}}(u-\varphi )$, so $\beta $ is the background of the positive current that remains after $S$ is removed from $T$.
Proposition 12.24.3.D15FEB2692E7 Subtracting the curvature form of an effective divisor from a smooth nef background, with a smooth nef residual, can preserve the common order, raise it by any integer between $1$ and the dimension or by a fraction, or produce a residual whose lower and upper orders differ. The residual can also fail to be nef.
Proof.
Integer jumps. On $E^ n$, where $E$ is an elliptic curve, let $\omega _ i$ be the pullback of a flat area form of integral one from the $i$th factor, and set $h=\sum _ i\omega _ i$ and $\mu =\omega _1\wedge \cdots \wedge \omega _ n$. The divisor $F_ i$ obtained by fixing a point in the $i$th factor has $\omega _ i$ as a smooth curvature form: on that elliptic curve, any curvature form of the corresponding line bundle differs from $\omega _ i$ by $\mathrm{dd}^{\mathrm{c}}$ of a solution of a zero-mean Poisson equation, so the metric can be adjusted, and then pulled back. For $\theta =h$ and $D=\sum _{i=1}^ kF_ i$, the residual $\beta =\sum _{i=k+1}^ n\omega _ i$ is smooth and nef. The perturbed forms are closed, positive and of constant density with respect to $\mu $, so the zero potential solves their equations, and theorem 12.2.3 gives
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12.24.4
\begin{equation} \mathcal V_\mu (\theta +\epsilon h)=n!(1+\epsilon )^ n,\qquad \mathcal V_\mu (\beta +\epsilon h)=n!\epsilon ^ k(1+\epsilon )^{n-k}. \end{equation}
For $n=1$ the same formula follows from $\lambda _\mu (\xi )=\int _ X\xi $. These examples realize each integer jump $1,\ldots ,n$. A nonzero divisor with jump zero is given by $\theta =2h$ and $D=F_1$, since then the residual is still strictly positive.
A fractional jump. Use the background $\beta =b\omega _1+q\omega _2$ of proposition 12.21.4, where $q=|\zeta |^{2m}$ near its unique zero and $m>1$. Choose $c>0$ and put $\theta =\beta +c\omega _2>0$ and $D=c(E\times \{ w_0\} )$, whose smooth curvature form is $c\omega _2$. The original background is strictly positive, so both its orders are zero, whereas the residual has common order $1-1/m$.
Unequal orders. Let $\beta _{\rm res}$ be the oscillatory rank-one background of proposition 12.21.5, and put
\[ \theta =\beta _{\rm res}+ch>0,\qquad D=c(\{ z_0\} \times E)+c(E\times \{ w_0\} ),\qquad \eta _ D=ch. \]
The background $\theta $ has both orders zero: since $\theta \geq ch$, every test quotient is bounded below by a positive constant, while one fixed test gives a finite upper bound. Its smooth nef residual is exactly $\beta _{\rm res}$, whose lower and upper orders differ. In each of these examples, the residual plus $[D]$ is a positive current in the smooth $\mathrm{dd}^{\mathrm{c}}$-class of $\theta $, and its divisorial part is exactly $[D]$ because the residual is smooth. We do not need this current to have minimal singularities; that case is treated in corollary 12.24.5.
A residual that is not nef. Blow up $\mathbb P^2$ at two distinct points. Let $H$ be the pullback of the class of the normalized Fubini–Study form, let $E_1,E_2$ be the exceptional curves, and let $L$ be the strict transform of the line through the two points. The pullback $\theta $ of the Fubini–Study form is smooth, semipositive and nef. Subtract curvature forms of $D=E_1+E_2$. The residual $\beta $ has class $H-E_1-E_2$, which is the class of $L$, and the Poincaré–Lelong formula for the total transform of the line shows that it contains the positive current $[L]$. The intersection numbers $H^2=1$, $H\cdot E_ i=0$, $E_ i^2=-1$ and $E_1\cdot E_2=0$ give
\[ \int _ L\beta =(H-E_1-E_2)^2=-1. \]
If $\beta $ were nef, smooth representatives $\beta +\epsilon h+\mathrm{dd}^{\mathrm{c}}u_\epsilon >0$ would give $-1+\epsilon \int _ Lh\geq 0$ for all $\epsilon >0$, which is impossible.
$\square$
For a prime divisor $D$ and $\delta >0$, consider the infimum of the generic Lelong numbers along $D$ of the positive currents $\theta +\delta h+\mathrm{dd}^{\mathrm{c}}u$; its limit as $\delta \downarrow 0$ is the minimal multiplicity of $\theta $ along $D$. For a nef background all minimal multiplicities vanish, since every $\theta +\delta h$ has a smooth positive representative. Removing the divisor defined by these multiplicities therefore leaves a nef background unchanged. Removing the divisorial part of a particular positive current in the class of $\theta $ is a different operation, even when that current has minimal singularities.
Corollary 12.24.5.9860E4D04E6C On the ruled surface $X$ of lemma 16.29.16, keep its nef curvature form $\theta $, its section $D$ at infinity and its Kähler form $\kappa =\theta +\pi ^*\sigma _0$. For every smooth positive probability form $\mu $,
\[ \mathcal V_\mu (\theta +\epsilon \kappa )=2\epsilon (1+\epsilon ), \qquad \mathcal V_\mu (\epsilon \kappa )=2\epsilon ^2. \]
The only positive current of the form $\theta +\mathrm{dd}^{\mathrm{c}}u$ is $[D]$. It has minimal singularities, and removing its whole divisor leaves the zero background and changes the common order from one to two.
Proof.
Lemma 16.29.16 gives the uniqueness of $[D]$ and the nefness of $\theta $, and the calculation in the proof of proposition 16.29.18 gives $\int _ X\theta ^2=0$. Put $B=\pi ^*\sigma _0$. Then $B^2=0$, and
\[ \int _ X\theta \wedge B=\int _ D B=1, \]
by the Poincaré–Lelong formula, Stokes' theorem and the fact that the section $D$ maps isomorphically onto the elliptic base. The closed nef formula following theorem 12.11.1 now computes both volumes by expanding the squares, for every $\mu $. Every $\theta $-plurisubharmonic function $u$ satisfies $\theta +\mathrm{dd}^{\mathrm{c}}u=[D]$, so any two such functions differ by a pluriharmonic function, hence by a constant; in particular each has minimal singularities. Since $\theta $ is itself a curvature form of $\mathcal O_ X(D)$, removing the whole divisor leaves the zero background. The two volume polynomials give the orders one and two.
$\square$
Pome
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