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Example 12.25.15EB3698E8A068

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

\[ \beta _ s=(t-s)H+(g+s)Q,\qquad \theta _ t+\mathrm{dd}^{\mathrm{c}}(s\psi _ E)=s[E]+\beta _ s. \]

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 $m-2$. $\square$

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