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On the singularities and the Kodaira dimension of unitary Shimura varieties

  • Shuji Horinaga
  • Yota Maeda
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Abstract

We study the geometry of Shimura varieties associated with a Hermitian form of signature (p,q) over an imaginary quadratic field E, where 2≤ p≤ q. We prove that when p≥ wE and (p,q)≠(2,2),(2,3), where wE:=#𝒪E×, there exists a toroidal compactification with at worst canonical singularities. As an application, combining this singularity analysis with Arthur's multiplicity formula, we prove that only finitely many pairs (p,q) satisfying the above conditions and p+q≡1pmodwE give rise to unitary Shimura varieties that are not of general type. Our method also improves the singularity bound of Gritsenko--Hulek--Sankaran (Invent.\ Math., 2007) for O+(2,n) to the range n≥6 and shows that this bound is sharp.

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Cite

BibTeXRIS
BibTeX
@misc{horinaga2026singularities,
  title = {{On the singularities and the Kodaira dimension of unitary Shimura varieties}},
  author = {Shuji Horinaga and Yota Maeda},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.18146},
  eprint = {2609.18146},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.18146},
}
RIS
TY  - UNPB
TI  - On the singularities and the Kodaira dimension of unitary Shimura varieties
AU  - Shuji Horinaga
AU  - Yota Maeda
PY  - 2026
DA  - 2026/09/16
PB  - arXiv
DO  - 10.48550/arxiv.2609.18146
UR  - https://arxiv.org/abs/2609.18146
AB  - We study the geometry of Shimura varieties associated with a Hermitian form of signature (p,q) over an imaginary quadratic field E, where 2≤ p≤ q. We prove that when p≥ w_E and (p,q)≠(2,2),(2,3), where w_E:=#𝒪_E^×, there exists a toroidal compactification with at worst canonical singularities. As an application, combining this singularity analysis with Arthur's multiplicity formula, we prove that only finitely many pairs (p,q) satisfying the above conditions and p+q≡1pmodw_E give rise to unitary Shimura varieties that are not of general type. Our method also improves the singularity bound of Gritsenko--Hulek--Sankaran (Invent.\ Math., 2007) for O^+(2,n) to the range n≥6 and shows that this bound is sharp.
ER  -