Corollary 5.38.6`F5277854ECB3`

Corollary 5.38.6.F5277854ECB3 Let $Z$ be a connected compact complex manifold, let $(D_ j)$ be a finite or countable family of distinct prime divisors on $Z$, and let $c_ j>0$. Suppose that $S=\sum _ jc_ j[D_ j]$ is a closed positive current of finite mass. Then there are quasi-plurisubharmonic functions $v$ on $Z$ satisfying

\[ \mathrm{dd}^{\mathrm{c}}v=S-\beta \]

for a smooth closed real $(1,1)$-form $\beta $, and any two such functions differ by a smooth function. In particular, they determine a canonical ordinary singularity type, which depends only on $S$.

Proof. Lemma 5.38.4, applied to $T=S$, gives the decomposition, with $v$ quasi-plurisubharmonic since $S\geq 0$, and shows that any two such potentials differ by a smooth function. By proposition 5.33.1, they have the same ordinary singularity type. $\square$

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@misc{pome-tag-F5277854ECB3,
  author = {{Pome contributors}},
  title = {Corollary 5.38.6},
  howpublished = {Pome},
  url = {https://mingchenxia.github.io/Pome/tag/F5277854ECB3/}
}

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Corollary 5.38.6F5277854ECB3

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Corollary 5.38.6`F5277854ECB3`