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 $\dim _{\mathbb R}H^{1,1}_{\mathrm{BC}}(X,\mathbb R)$.
12.10.38 The classes of the divisorial components622292D98711
Uniqueness constrains the divisors along which the threshold solution can have positive Lelong numbers. A linear relation between their Bott–Chern classes would produce a global potential that changes the coefficients of these divisors while preserving the equation off the degeneracy set, contradicting uniqueness. We write $\{ D\} \in H^{1,1}_{\mathrm{BC}}(X,\mathbb R)$ for the Bott–Chern class of the current of integration $[D]$ along a prime divisor $D$, and $\nu (u,D)$ for the generic Lelong number of $u$ along $D$.
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 $c_ j=0$. $\square$
The conclusion concerns only the divisors along which the solution has positive generic Lelong number. It imposes no independence condition on the other components of codimension one of the degeneracy set.
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