Proposition 18.4.3`9AE7825754C3`

Proposition 18.4.3.9AE7825754C3 Let $\mu :Y\to X$ be a modification of connected compact complex manifolds. For every smooth real $(1,1)$-form $\theta $ on $X$ and every $u\in \mathrm{PSH}(X,\theta )$,

\[ P_{\mu ^*\theta }[u\circ \mu ]_{\mathcal I} =P_\theta [u]_{\mathcal I}\circ \mu . \]

Proof. By proposition 5.43.18, pullback identifies the potentials in the two backgrounds. Surjectivity of $\mu $ identifies their nonpositivity constraints, and lemma 5.43.24 identifies their all-weight multiplier-ideal classes. Hence pullback gives a bijection of the complete competitor families. The ideal envelope is itself a competitor and is their greatest member by theorem 18.4.2. Its pullback is therefore the greatest competitor upstairs, which proves the identity. $\square$

Cite this page

BibTeX citation

@misc{pome-tag-9AE7825754C3,
  author = {{Pome contributors}},
  title = {Proposition 18.4.3},
  howpublished = {Pome},
  url = {https://mingchenxia.github.io/Pome/tag/9AE7825754C3/}
}

Comments

Corrections, questions, and mathematical discussion are welcome. Sign in with GitHub to comment. Comments are public.

Read this page’s discussion on GitHub Post a comment on GitHub

Proposition 18.4.39AE7825754C3

Comments

Proposition 18.4.3`9AE7825754C3`