Microlocal homology
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arXiv
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| Format: | Preprint |
| Published: |
2022
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| _version_ | 1866918145090912256 |
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| author | Schefers, Kendric |
| author_facet | Schefers, Kendric |
| contents | Let $Z$ be an l.c.i. scheme over $\mathbb{C}$. In this paper, we introduce a Kashiwara--Schapira-style functor of derived microlocalization, which we use to define a perverse sheaf $μ_{Z}$ on the $-1$-shifted cotangent bundle, $T^*[-1]Z$. The sheaf $μ_{Z}$ is designed to be a refinement of the microlocal homology of $Z$: a family of invariants introduced by Nadler that interpolates between the singular cohomology and Borel--Moore homology of $Z$.
Our main result is an equivalence between $μ_{Z}$ and the DT sheaf $φ_{T^*[-1]Z}$ on $T^*[-1]Z$. This provides an alternative construction for the DT sheaf in the case of a shifted cotangent bundle. The main step of our argument, which may be of independent interest, is a local computation -- closely related to one obtained recently by Kinjo using different methods -- providing a description of the classical microlocalization functor in terms of vanishing cycles. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2205_12436 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Microlocal homology Schefers, Kendric Algebraic Geometry Representation Theory Let $Z$ be an l.c.i. scheme over $\mathbb{C}$. In this paper, we introduce a Kashiwara--Schapira-style functor of derived microlocalization, which we use to define a perverse sheaf $μ_{Z}$ on the $-1$-shifted cotangent bundle, $T^*[-1]Z$. The sheaf $μ_{Z}$ is designed to be a refinement of the microlocal homology of $Z$: a family of invariants introduced by Nadler that interpolates between the singular cohomology and Borel--Moore homology of $Z$. Our main result is an equivalence between $μ_{Z}$ and the DT sheaf $φ_{T^*[-1]Z}$ on $T^*[-1]Z$. This provides an alternative construction for the DT sheaf in the case of a shifted cotangent bundle. The main step of our argument, which may be of independent interest, is a local computation -- closely related to one obtained recently by Kinjo using different methods -- providing a description of the classical microlocalization functor in terms of vanishing cycles. |
| title | Microlocal homology |
| topic | Algebraic Geometry Representation Theory |
| url | https://arxiv.org/abs/2205.12436 |