Tautological projection for cycles on the moduli space of abelian varieties
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866910954090921984 |
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| author | Canning, Samir Molcho, Sam Oprea, Dragos Pandharipande, Rahul |
| author_facet | Canning, Samir Molcho, Sam Oprea, Dragos Pandharipande, Rahul |
| contents | We define a tautological projection operator for algebraic cycle classes on the moduli space of principally polarized abelian varieties $\mathcal{A}_g$: every cycle class decomposes canonically as a sum of a tautological and a non-tautological part. The main new result required for the definition of the projection operator is the vanishing of the top Chern class of the Hodge bundle over the boundary $\bar{\mathcal{A}}_g\smallsetminus \mathcal{A}_g$ of any toroidal compactification $\bar{\mathcal{A}}_g$ of the moduli space $\mathcal{A}_g$. We prove the vanishing by a careful study of residues in the boundary geometry.
The existence of the projection operator raises many natural questions about cycles on $\mathcal{A}_g$. We calculate the projections of all product cycles $\mathcal{A}_{g_1}\times \ldots \times \mathcal{A}_{g_\ell}$ in terms of Schur determinants, discuss Faber's earlier calculations related to the Torelli locus, and state several open questions. The Appendix contains a conjecture about the projection of the locus of abelian varieties with real multiplication. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_15768 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Tautological projection for cycles on the moduli space of abelian varieties Canning, Samir Molcho, Sam Oprea, Dragos Pandharipande, Rahul Algebraic Geometry 14C15, 14C17, 14K10 We define a tautological projection operator for algebraic cycle classes on the moduli space of principally polarized abelian varieties $\mathcal{A}_g$: every cycle class decomposes canonically as a sum of a tautological and a non-tautological part. The main new result required for the definition of the projection operator is the vanishing of the top Chern class of the Hodge bundle over the boundary $\bar{\mathcal{A}}_g\smallsetminus \mathcal{A}_g$ of any toroidal compactification $\bar{\mathcal{A}}_g$ of the moduli space $\mathcal{A}_g$. We prove the vanishing by a careful study of residues in the boundary geometry. The existence of the projection operator raises many natural questions about cycles on $\mathcal{A}_g$. We calculate the projections of all product cycles $\mathcal{A}_{g_1}\times \ldots \times \mathcal{A}_{g_\ell}$ in terms of Schur determinants, discuss Faber's earlier calculations related to the Torelli locus, and state several open questions. The Appendix contains a conjecture about the projection of the locus of abelian varieties with real multiplication. |
| title | Tautological projection for cycles on the moduli space of abelian varieties |
| topic | Algebraic Geometry 14C15, 14C17, 14K10 |
| url | https://arxiv.org/abs/2401.15768 |