Effective zero-cycles and the Bloch-Beilinson filtration
Fuente:
arXiv
Salvato in:
| Autori principali: | , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2022
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866917690257440768 |
|---|---|
| author | Martin, Olivier Vial, Charles |
| author_facet | Martin, Olivier Vial, Charles |
| contents | A conjecture of Voisin states that two points on a smooth projective complex variety whose algebra of holomorphic forms is generated in degree 2 are rationally equivalent to each other if and only if their difference lies in the third step of the Bloch-Beilinson filtration. In this note, we formulate a generalization that allows for rational equivalence of effective zero-cycles of higher degree, at the expense of looking deeper in the Bloch-Beilinson filtration. In the first half, we provide evidence in support of this conjecture in the case of abelian varieties and projective hyper-Kähler manifolds. Notably, we give explicit criteria for rational equivalence of effective zero-cycles on moduli spaces of semistable sheaves on K3 surfaces, generalizing that of Marian-Zhao. In the second half, in an effort to explain our main conjecture, we formulate a second conjecture predicting when the diagonal of a smooth projective variety belongs to a subalgebra of the ring of correspondences generated in low degree. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2208_10026 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Effective zero-cycles and the Bloch-Beilinson filtration Martin, Olivier Vial, Charles Algebraic Geometry 14C15, 14C25, 14C30, 14J42, 14J28, 14K05 A conjecture of Voisin states that two points on a smooth projective complex variety whose algebra of holomorphic forms is generated in degree 2 are rationally equivalent to each other if and only if their difference lies in the third step of the Bloch-Beilinson filtration. In this note, we formulate a generalization that allows for rational equivalence of effective zero-cycles of higher degree, at the expense of looking deeper in the Bloch-Beilinson filtration. In the first half, we provide evidence in support of this conjecture in the case of abelian varieties and projective hyper-Kähler manifolds. Notably, we give explicit criteria for rational equivalence of effective zero-cycles on moduli spaces of semistable sheaves on K3 surfaces, generalizing that of Marian-Zhao. In the second half, in an effort to explain our main conjecture, we formulate a second conjecture predicting when the diagonal of a smooth projective variety belongs to a subalgebra of the ring of correspondences generated in low degree. |
| title | Effective zero-cycles and the Bloch-Beilinson filtration |
| topic | Algebraic Geometry 14C15, 14C25, 14C30, 14J42, 14J28, 14K05 |
| url | https://arxiv.org/abs/2208.10026 |