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Special cycles and Picard groups of Baily--Borel compactifications
Abstract
In this paper, we answer two basic questions concerning algebraic cycles on orthogonal Shimura varieties attached to lattices. First, we prove that Kudla's special cycles and Heegner cycles have the same rational span in every codimension. This fills a gap in the codimension-one argument of \cite[Remark 2.4]{BLMM17}, pointed out by Bruinier--Zuffetti \cite{BZ26}. Second, in dimension at least ten, we prove that the rational Picard group of the Baily--Borel compactification is generated by the Hodge class. The proof of the second result reduces, via isotropic induction on the discriminant form, to the K3-type case treated in our previous work.
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BibTeX
@misc{li2026special,
title = {{Special cycles and Picard groups of Baily--Borel compactifications}},
author = {Zhiyuan Li and Zelin Ye},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.38187},
eprint = {2610.38187},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.38187},
}
RIS
TY - UNPB
TI - Special cycles and Picard groups of Baily--Borel compactifications
AU - Zhiyuan Li
AU - Zelin Ye
PY - 2026
DA - 2026/10/28
PB - arXiv
DO - 10.48550/arxiv.2610.38187
UR - https://arxiv.org/abs/2610.38187
AB - In this paper, we answer two basic questions concerning algebraic cycles on orthogonal Shimura varieties attached to lattices. First, we prove that Kudla's special cycles and Heegner cycles have the same rational span in every codimension. This fills a gap in the codimension-one argument of \cite[Remark 2.4]{BLMM17}, pointed out by Bruinier--Zuffetti \cite{BZ26}. Second, in dimension at least ten, we prove that the rational Picard group of the Baily--Borel compactification is generated by the Hodge class. The proof of the second result reduces, via isotropic induction on the discriminant form, to the K3-type case treated in our previous work.
ER -