## Cones of special cycles of codimension 2 on orthogonal Shimura varieties

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## Abstract:

Let $X$ be an orthogonal Shimura variety associated to a unimodular lattice. We investigate the polyhedrality of the cone $\mathcal {C}_X$ of special cycles of codimension 2 on $X$. We show that the rays generated by such cycles accumulate towards infinitely many rays, the latter generating a non-trivial cone. We also prove that such an accumulation cone is polyhedral. The proof relies on analogous properties satisfied by the cones of Fourier coefficients of Siegel modular forms. We show that the accumulation rays of $\mathcal {C}_X$ are generated by combinations of Heegner divisors intersected with the Hodge class of $X$. As a result of the classification of the accumulation rays, we implement an algorithm in SageMath to certify the polyhedrality of $\mathcal {C}_X$ in some cases.## References

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## Additional Information

**Riccardo Zuffetti**- Affiliation: Institut für Mathematik, Goethe–Universität Frankfurt, Robert-Mayer-Str. 6–8, 60325 Frankfurt am Main, Germany
- ORCID: 0000-0001-9702-0785
- Email: zuffetti@math.uni-frankfurt.de, riccardo.zuffetti@gmail.com
- Received by editor(s): March 8, 2022
- Received by editor(s) in revised form: May 1, 2022
- Published electronically: August 16, 2022
- Additional Notes: This research was founded by the LOEWE research unit “Uniformized Structures in Arithmetic and Geometry”, and by the Collaborative Research Centre TRR 326 “Geometry and Arithmetic of Uniformized Structures”, project number 444845124.
- © Copyright 2022 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**375**(2022), 7385-7441 - MSC (2020): Primary 14C25; Secondary 11F30
- DOI: https://doi.org/10.1090/tran/8757