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The space of polarized compactifications of an affine variety

  • Mattias Jonsson
  • Yueqiao Wu
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Abstract

Let X be a normal affine variety over an algebraically closed field of characteristic zero. Through the lens of non-Archimedean geometry, we study the space of polarized compactifications of X, consisting of pairs (X̅, D) where X̅ is a normal projective variety containing X as a dense open subset, and D is an effective ample ℚ-Cartier ℚ-divisor with support equal to the complement of X in X̅. We identify this space with a natural class of functions--the Fubini--Study functions--on the Berkovich analytification of X. Through a Rees correspondence, we also give a bijection with the space of homogeneous ℚ-valued, finitely generated negative non-Archimedean norms (or filtrations) on the coordinate ring of X. Using the above identifications, we equip the space of polarized compactifications, and numerical classes thereof, with natural metrics and identify the metric completions. The latter include a space of continuous plurisubharmonic functions in the sense of Chambert-Loir and Ducros, and a space of birational curve classes as studied by Dang and Favre.

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Cite

BibTeXRIS
BibTeX
@misc{jonsson2026space,
  title = {{The space of polarized compactifications of an affine variety}},
  author = {Mattias Jonsson and Yueqiao Wu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.21838},
  eprint = {2610.21838},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.21838},
}
RIS
TY  - UNPB
TI  - The space of polarized compactifications of an affine variety
AU  - Mattias Jonsson
AU  - Yueqiao Wu
PY  - 2026
DA  - 2026/10/15
PB  - arXiv
DO  - 10.48550/arxiv.2610.21838
UR  - https://arxiv.org/abs/2610.21838
AB  - Let X be a normal affine variety over an algebraically closed field of characteristic zero. Through the lens of non-Archimedean geometry, we study the space of polarized compactifications of X, consisting of pairs (X̅, D) where X̅ is a normal projective variety containing X as a dense open subset, and D is an effective ample ℚ-Cartier ℚ-divisor with support equal to the complement of X in X̅. We identify this space with a natural class of functions--the Fubini--Study functions--on the Berkovich analytification of X. Through a Rees correspondence, we also give a bijection with the space of homogeneous ℚ-valued, finitely generated negative non-Archimedean norms (or filtrations) on the coordinate ring of X. Using the above identifications, we equip the space of polarized compactifications, and numerical classes thereof, with natural metrics and identify the metric completions. The latter include a space of continuous plurisubharmonic functions in the sense of Chambert-Loir and Ducros, and a space of birational curve classes as studied by Dang and Favre.
ER  -