Topologies and sheaves on causal manifolds
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866915584338296832 |
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| author | Schapira, Pierre |
| author_facet | Schapira, Pierre |
| contents | A causal manifold $(M,γ)$ is a manifold $M$ endowed with a closed proper cone $γ$ in the tangent bundle $TM$ such that the projection $TM\to M$ is surjective when restricted to the interior of $γ$. Let $λ$ be the antipodal of the polar cone of $γ$.
An open set $U$ of $M$ is called $γ$-open if its Whitney normal cone contains the interior of $γ$. Similarly, $U$ is called $λ$-open if the micro-support of the constant sheaf on $U$ is contained in $λ$. We begin by proving that the two notions coincide.
Next, we prove that if $(M,γ)$ admits a ``future time function'' the functor of direct images establishes an equivalence of triangulated categories between the derived category of sheaves on $M$ micro-supported by $λ$ and the derived category of sheaves on the manifold $M$ endowed with the $γ$-topology. This generalizes a result of~\cite{KS90} which dealt with the case of a constant cone in a vector space. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_10364 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Topologies and sheaves on causal manifolds Schapira, Pierre Algebraic Geometry 35A27, 81T20 A causal manifold $(M,γ)$ is a manifold $M$ endowed with a closed proper cone $γ$ in the tangent bundle $TM$ such that the projection $TM\to M$ is surjective when restricted to the interior of $γ$. Let $λ$ be the antipodal of the polar cone of $γ$. An open set $U$ of $M$ is called $γ$-open if its Whitney normal cone contains the interior of $γ$. Similarly, $U$ is called $λ$-open if the micro-support of the constant sheaf on $U$ is contained in $λ$. We begin by proving that the two notions coincide. Next, we prove that if $(M,γ)$ admits a ``future time function'' the functor of direct images establishes an equivalence of triangulated categories between the derived category of sheaves on $M$ micro-supported by $λ$ and the derived category of sheaves on the manifold $M$ endowed with the $γ$-topology. This generalizes a result of~\cite{KS90} which dealt with the case of a constant cone in a vector space. |
| title | Topologies and sheaves on causal manifolds |
| topic | Algebraic Geometry 35A27, 81T20 |
| url | https://arxiv.org/abs/2505.10364 |