Regular homomorphisms and mixed motives
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909797724454912 |
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| author | Hernandez, Ivan Pelaez, Pablo |
| author_facet | Hernandez, Ivan Pelaez, Pablo |
| contents | Let $X$ be a smooth projective variety of dimension $d$ over an algebraically closed field $k$. The main goal of this paper is to study, in the context of Voevodsky's triangulated category of motives $DM_k$, the group $CH^n_{\mathrm{alg}}(X)$ of codimension $n$ algebraic cycles of $X$, algebraically equivalent to zero, modulo rational equivalence, $1\leq n \leq d$. Namely, for any regular homomorphism $ψ$ (in the sense of Samuel) defined on $CH^n_{\mathrm{alg}}(X)$, we construct $M^n_ψ(X)\in DM_k$, which is a reasonable approximation, with respect to the slice filtration in $DM_k$, of the motive of $X$, $M(X)$; and a map $z_ψ: M^n_ψ(X)\rightarrow M(X)$ in $DM_k$, which computes the kernel of $ψ$. We construct as well a map, $z_{\mathrm{ab}}^n: M^n_{\mathrm{ab}}(X) \rightarrow M(X)$ having analogue properties but which instead computes the subgroup $CH^n_{\mathrm{ab}}(X)\subseteq CH^n_{\mathrm{alg}}(X)$ of algebraic cycles abelian equivalent to zero (in the sense of Samuel). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_15920 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Regular homomorphisms and mixed motives Hernandez, Ivan Pelaez, Pablo Algebraic Geometry 14C15, 14C25, 19E15 Let $X$ be a smooth projective variety of dimension $d$ over an algebraically closed field $k$. The main goal of this paper is to study, in the context of Voevodsky's triangulated category of motives $DM_k$, the group $CH^n_{\mathrm{alg}}(X)$ of codimension $n$ algebraic cycles of $X$, algebraically equivalent to zero, modulo rational equivalence, $1\leq n \leq d$. Namely, for any regular homomorphism $ψ$ (in the sense of Samuel) defined on $CH^n_{\mathrm{alg}}(X)$, we construct $M^n_ψ(X)\in DM_k$, which is a reasonable approximation, with respect to the slice filtration in $DM_k$, of the motive of $X$, $M(X)$; and a map $z_ψ: M^n_ψ(X)\rightarrow M(X)$ in $DM_k$, which computes the kernel of $ψ$. We construct as well a map, $z_{\mathrm{ab}}^n: M^n_{\mathrm{ab}}(X) \rightarrow M(X)$ having analogue properties but which instead computes the subgroup $CH^n_{\mathrm{ab}}(X)\subseteq CH^n_{\mathrm{alg}}(X)$ of algebraic cycles abelian equivalent to zero (in the sense of Samuel). |
| title | Regular homomorphisms and mixed motives |
| topic | Algebraic Geometry 14C15, 14C25, 19E15 |
| url | https://arxiv.org/abs/2509.15920 |