Logarithmic jets and the chiral de Rham complex of a pair
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866918154790240256 |
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| author | Bouaziz, Emile |
| author_facet | Bouaziz, Emile |
| contents | To a smooth variety $X$ with simple normal crossings divisor $D$, we associate a sheaf of vertex algebras on $X$, denoted $Ω^{ch}_{X}(\operatorname{log}D)$, whose conformal weight $0$ subspace is the algebra $Ω_{X}(\operatorname{log}D)$ of forms with log poles along $D$. We prove various basic structural results about $Ω^{ch}_{X}(\operatorname{log}D)$. In particular, if $X^{*}=X\setminus D$ has a volume form then we show that $Ω^{ch}_{X}(\operatorname{log}D)$ admits a topological structure of rank $d=\operatorname{dim}(X)$, which is enhanced to an extended topological structure if $D\sim -K_{X}$ is in fact anticanonical. In this latter case we also show that the resulting $(q,y)$ character $\operatorname{Ell}(X,D)(q,y)$ is a section of the line bundle $Θ^{\otimes d}$ on the elliptic curve $E=\mathbf{C}^{*}/q^{\mathbf{Z}}$. We further show how $Ω^{ch}_{X}(\operatorname{log}D)$ can be understood in terms of a simple birational modification of the space of jets into $X$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_04515 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Logarithmic jets and the chiral de Rham complex of a pair Bouaziz, Emile Algebraic Geometry To a smooth variety $X$ with simple normal crossings divisor $D$, we associate a sheaf of vertex algebras on $X$, denoted $Ω^{ch}_{X}(\operatorname{log}D)$, whose conformal weight $0$ subspace is the algebra $Ω_{X}(\operatorname{log}D)$ of forms with log poles along $D$. We prove various basic structural results about $Ω^{ch}_{X}(\operatorname{log}D)$. In particular, if $X^{*}=X\setminus D$ has a volume form then we show that $Ω^{ch}_{X}(\operatorname{log}D)$ admits a topological structure of rank $d=\operatorname{dim}(X)$, which is enhanced to an extended topological structure if $D\sim -K_{X}$ is in fact anticanonical. In this latter case we also show that the resulting $(q,y)$ character $\operatorname{Ell}(X,D)(q,y)$ is a section of the line bundle $Θ^{\otimes d}$ on the elliptic curve $E=\mathbf{C}^{*}/q^{\mathbf{Z}}$. We further show how $Ω^{ch}_{X}(\operatorname{log}D)$ can be understood in terms of a simple birational modification of the space of jets into $X$. |
| title | Logarithmic jets and the chiral de Rham complex of a pair |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2510.04515 |