Sheaves of AV-modules on quasi-projective varieties
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929486152335360 |
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| author | Billig, Yuly Bouaziz, Emile |
| author_facet | Billig, Yuly Bouaziz, Emile |
| contents | We study sheaves of modules for the Lie algebra of vector fields with the action of the algebra of functions, compatible via the Leibniz rule. A crucial role in this theory is played by the virtual jets of vector fields - jets that evaluate to a zero vector field under the anchor map. Virtual jets of vector fields form a vector bundle $\mathcal{L}_+$ whose fiber is Lie algebra $\widehat{L}_+$ of vanishing at zero derivations of power series. We show that a sheaf of $AV$-modules is characterized by two ingredients - it is a module for $\mathcal{L}_+$ and an $\mathcal{L}_+$-charged $D$-module.
For each rational finite-dimensional representation of $\widehat{L}_+$, we construct a bundle of jet $AV$-modules. We also show that Rudakov modules may be realized as tensor products of jet modules with a $D$-module of delta functions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_02677 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Sheaves of AV-modules on quasi-projective varieties Billig, Yuly Bouaziz, Emile Representation Theory Algebraic Geometry 17B66 (Primary) 14F10, 17B10 (Secondary) We study sheaves of modules for the Lie algebra of vector fields with the action of the algebra of functions, compatible via the Leibniz rule. A crucial role in this theory is played by the virtual jets of vector fields - jets that evaluate to a zero vector field under the anchor map. Virtual jets of vector fields form a vector bundle $\mathcal{L}_+$ whose fiber is Lie algebra $\widehat{L}_+$ of vanishing at zero derivations of power series. We show that a sheaf of $AV$-modules is characterized by two ingredients - it is a module for $\mathcal{L}_+$ and an $\mathcal{L}_+$-charged $D$-module. For each rational finite-dimensional representation of $\widehat{L}_+$, we construct a bundle of jet $AV$-modules. We also show that Rudakov modules may be realized as tensor products of jet modules with a $D$-module of delta functions. |
| title | Sheaves of AV-modules on quasi-projective varieties |
| topic | Representation Theory Algebraic Geometry 17B66 (Primary) 14F10, 17B10 (Secondary) |
| url | https://arxiv.org/abs/2409.02677 |