Pome

Example 12.26.3698A2684C823

Example 12.26.3.698A2684C823 Let $X=\mathbb P^1\times \mathbb P^ q$, $q\geq 1$, with projections $p_1,p_2$. Choose a smooth nonnegative function $b\not\equiv 0$ on $\mathbb P^1$ that vanishes on a nonempty open set, and a smooth strictly positive nonconstant function $f$ on $\mathbb P^ q$. With $\omega _1$ the normalized Fubini–Study form on $\mathbb P^1$, set

\[ B=p_1^*(b\omega _1),\qquad \theta =(f\circ p_2)B. \]

The smooth nef background $\theta $ is not closed, vanishes on a nonempty open set, and has common order $q$, with two-sided bounds by multiples of $\epsilon ^ q$.

Proof. The form $b\omega _1$ is closed, because the first factor is a curve, and semipositive. Its pullback has rank one where $b>0$ and rank zero where $b=0$, so its maximal rank is one. Since $n=q+1$, corollary 12.26.2 gives the order $q$. A nonconstant smooth function $f$ on the connected second factor has $\mathrm df\ne 0$ somewhere. At a point of the product where $b>0$ and $\mathrm df\neq 0$, the horizontal two-form $B$ wedges nontrivially with the vertical one-form $p_2^*\mathrm df$. Thus $\mathrm d\theta =p_2^*\mathrm df\wedge B\ne 0$. $\square$

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