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PreprintarXiv licencearXiv

A Fan Algorithm for Chow-Witt Rings of Smooth Projective Toric Varieties

  • Haoyang Liu
  • Tianle Liu
  • Guorui Xu
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Abstract

Let R be a real closed field and let XΣ be a smooth projective split toric variety. For an explicitly chosen basis rigidification r of its Picard grading, we prove that the pair (Σ,r) determines the resulting total Chow--Witt ring and give a terminating finite algorithm for all groups, products, twists, and forgetful maps. The ring is identified with an explicit fibre product of the Picard-graded diagonal I-cohomology ring and the integral inverse images of twisted Bockstein kernels over the mod-$2$ Chow ring. Using real cycle classes, we identify the first factor with the cohomology of the real toric variety with all sign local systems. We construct the mod-$2$ cycle map on invariant divisors as explicit deck-transition cocycles and obtain finite signed boundary and product matrices directly from the fan.

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Cite

BibTeXRIS
BibTeX
@misc{liu2026algorithm,
  title = {{A Fan Algorithm for Chow-Witt Rings of Smooth Projective Toric Varieties}},
  author = {Haoyang Liu and Tianle Liu and Guorui Xu},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.26082},
  eprint = {2609.26082},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.26082},
}
RIS
TY  - UNPB
TI  - A Fan Algorithm for Chow-Witt Rings of Smooth Projective Toric Varieties
AU  - Haoyang Liu
AU  - Tianle Liu
AU  - Guorui Xu
PY  - 2026
DA  - 2026/09/01
PB  - arXiv
DO  - 10.48550/arxiv.2609.26082
UR  - https://arxiv.org/abs/2609.26082
AB  - Let R be a real closed field and let X_Σ be a smooth projective split toric variety. For an explicitly chosen basis rigidification r of its Picard grading, we prove that the pair (Σ,r) determines the resulting total Chow--Witt ring and give a terminating finite algorithm for all groups, products, twists, and forgetful maps. The ring is identified with an explicit fibre product of the Picard-graded diagonal I-cohomology ring and the integral inverse images of twisted Bockstein kernels over the mod-$2$ Chow ring. Using real cycle classes, we identify the first factor with the cohomology of the real toric variety with all sign local systems. We construct the mod-$2$ cycle map on invariant divisors as explicit deck-transition cocycles and obtain finite signed boundary and product matrices directly from the fan.
ER  -