PomeCorollary 12.10.39.1E2D5DAE5C6C Under the hypotheses of theorem 12.10.17, let $u$ be a threshold solution and let $D_1,\ldots ,D_ k$ be the prime divisors with $\nu (u,D_ j)>0$. Then the classes $\{ D_ j\} $ are linearly independent in $H^{1,1}_{\mathrm{BC}}(X,\mathbb R)$. In particular their number is at most $\dim _{\mathbb R}H^{1,1}_{\mathrm{BC}}(X,\mathbb R)$.
Corollary 12.10.391E2D5DAE5C6C
Proof. Since $u$ is smooth off $Z$, these divisors are among the finitely many components of $Z$ of codimension one. Put $a_ j=\nu (u,D_ j)>0$. By lemma 5.38.1,
\[ R=\omega +\mathrm{dd}^{\mathrm{c}}u-\sum _ j a_ j[D_ j]\geq 0. \]
Suppose that real numbers $c_1,\ldots ,c_ k$ satisfy $\sum _ j c_ j\{ D_ j\} =0$. Fix a Gauduchon metric $\eta $ and apply lemma 20.4.20 to the currents of integration $[D_ j]$, with the integrable potentials $v_\eta ([D_ j])$ given there. Since the smooth representative in that lemma depends linearly on the Bott–Chern class, we obtain
\[ h=\sum _ j c_ j v_\eta ([D_ j])\in L^1(X),\qquad \mathrm{dd}^{\mathrm{c}}h=\sum _ j c_ j[D_ j]. \]
Choose $t\ne 0$ sufficiently small that $a_ j+tc_ j\geq 0$ for every $j$. The distribution $u_ t=u+th$ satisfies
\[ \omega +\mathrm{dd}^{\mathrm{c}}u_ t=R+\sum _ j(a_ j+tc_ j)[D_ j]\geq 0, \]
so it is represented by a function $u_ t\in \mathrm{PSH}(X,\omega )$. On $X\setminus Z$, the function $h$ is smooth and pluriharmonic, so $u_ t$ has the same Monge–Ampère measure as $u$ there. Since neither extended measure charges the pluripolar set $Z$, $u_ t$ is another threshold solution. Uniqueness makes $th=u_ t-u$ constant. Hence $\sum _ jc_ j[D_ j]=0$, and testing near a generic point of each of the distinct divisors $D_ j$ gives $c_ j=0$. $\square$
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