Gluing invariants of Donaldson--Thomas type -- Part I: the Darboux stack
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arXiv
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| Natura: | Preprint |
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2024
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| author | Hennion, Benjamin Holstein, Julian Robalo, Marco |
| author_facet | Hennion, Benjamin Holstein, Julian Robalo, Marco |
| contents | Let $X$ be a (-1)-shifted symplectic derived Deligne--Mumford stack. In this paper we introduce the Darboux stack of $X$, parametrizing local presentations of $X$ as a derived critical locus of a function $f$ on a smooth formal scheme $U$. Local invariants such as the Milnor number $μ_f$, the perverse sheaf of vanishing cycles $\mathsf{P}_{U,f}$ and the category of matrix factorizations $\mathsf{MF}(U,f)$ are naturally defined on the Darboux stack, without ambiguity. The stack of non-degenerate flat quadratic bundles acts on the Darboux stack and our main theorem is the contractibility of the quotient stack when taking a further homotopy quotient identifying isotopic automorphisms. As a corollary we recover the gluing results for vanishing cycles by Brav--Bussi--Dupont--Joyce--Szendr\H oi. In a second part (to appear), we will apply this general mechanism to glue the motives of the locally defined categories of matrix factorizations $\mathsf{MF}(U,f)$ under the prescription of additional orientation data, thus answering positively conjectures by Kontsevich--Soibelman and Toda in motivic Donaldson--Thomas theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_08471 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Gluing invariants of Donaldson--Thomas type -- Part I: the Darboux stack Hennion, Benjamin Holstein, Julian Robalo, Marco Algebraic Geometry Mathematical Physics Representation Theory Let $X$ be a (-1)-shifted symplectic derived Deligne--Mumford stack. In this paper we introduce the Darboux stack of $X$, parametrizing local presentations of $X$ as a derived critical locus of a function $f$ on a smooth formal scheme $U$. Local invariants such as the Milnor number $μ_f$, the perverse sheaf of vanishing cycles $\mathsf{P}_{U,f}$ and the category of matrix factorizations $\mathsf{MF}(U,f)$ are naturally defined on the Darboux stack, without ambiguity. The stack of non-degenerate flat quadratic bundles acts on the Darboux stack and our main theorem is the contractibility of the quotient stack when taking a further homotopy quotient identifying isotopic automorphisms. As a corollary we recover the gluing results for vanishing cycles by Brav--Bussi--Dupont--Joyce--Szendr\H oi. In a second part (to appear), we will apply this general mechanism to glue the motives of the locally defined categories of matrix factorizations $\mathsf{MF}(U,f)$ under the prescription of additional orientation data, thus answering positively conjectures by Kontsevich--Soibelman and Toda in motivic Donaldson--Thomas theory. |
| title | Gluing invariants of Donaldson--Thomas type -- Part I: the Darboux stack |
| topic | Algebraic Geometry Mathematical Physics Representation Theory |
| url | https://arxiv.org/abs/2407.08471 |