Proposition 12.24.3D15FEB2692E7

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

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

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

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. $\square$