Proposition 12.34.12.83512AFDC859 The current $[Z]/2$ belongs to $\mathscr G_{\omega _0}$ and annihilates $\alpha _\tau $. Every $R\in \mathscr G_{\omega _0}$ with $\int _ X\alpha _\tau \wedge R=0$ has its trace probability $\sigma _ R=\omega _0\wedge R$ supported on $Z$. In the coefficient-matrix convention of section 12.7, its matrix is
\[ P_ R=n\Pi _{\mathrm{vert}}\qquad \sigma _ R\text{-almost everywhere}, \]
where $\Pi _{\mathrm{vert}}$ is the $\omega _0$-orthogonal projection onto the tangent line of the fiber $\{ p\} \times \mathbb P^1$. In particular, $P_ R$ has rank one, and the determinant-weighted measure of $R$ vanishes for every smooth positive reference density.
Proof.
Only smooth forms need to be restricted to $Z$, and their restrictions satisfy $\omega _0|_ Z=2w$ and $\alpha _\tau |_ Z=0$. This proves that $[Z]/2$ is normalized and annihilates $\alpha _\tau $.
Let $R$ be a general annihilator. Since $\alpha _\tau $ and $\theta $ differ by $\mathrm{dd}^{\mathrm{c}}$ of a smooth function, Stokes' theorem gives $\int _ X\theta \wedge R=0$. The measure $\theta \wedge R$ is nonnegative, so it vanishes. On every compact set $K\subset X\setminus Z$, the explicit form $\theta $ satisfies $\theta \geq c_ K\omega _0$ for some $c_ K>0$. Thus $\sigma _ R(K)=0$, and a countable exhaustion by compact sets shows that $\sigma _ R$ is supported on $Z$.
Relative to $\omega _0$, the matrix of $R$ is characterized, for smooth test forms $\beta $ with matrix $B_\beta $, by
\[ \beta \wedge R=\frac1n\operatorname {tr}(B_\beta P_ R)\sigma _ R, \qquad P_ R\geq 0,\qquad \operatorname {tr}P_ R=n. \]
At points of $Z$, the form $\theta $ is positive on the horizontal $m$-plane and vanishes on the vertical line. The vanishing of $\theta \wedge R$ therefore forces all horizontal diagonal entries of $P_ R$ to vanish, and the inequality $|(P_ R)_{ij}|^2\leq (P_ R)_{ii}(P_ R)_{jj}$ for positive matrices removes their rows and columns. The remaining vertical entry is $n$, which proves the formula and the vanishing of the determinant.
$\square$
Pome
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