A Darboux theorem for derived schemes with shifted symplectic structure
We prove a 'Darboux theorem' for derived schemes with symplectic forms of degree $k<0$, in the sense of Pantev, Toen, Vaquie and Vezzosi arXiv:1111.3209. More precisely, we show that a derived scheme $X$ with symplectic form $\omega$ of degree $k$ is locally equivalent to (Spec $A,\...
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| Format: | Journal article |
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American Medical Society
2018
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| _version_ | 1826269803978948608 |
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| author | Brav, C Bussi, V Joyce, D |
| author_facet | Brav, C Bussi, V Joyce, D |
| author_sort | Brav, C |
| collection | OXFORD |
| description | We prove a 'Darboux theorem' for derived schemes with symplectic forms of degree $k<0$, in the sense of Pantev, Toen, Vaquie and Vezzosi arXiv:1111.3209. More precisely, we show that a derived scheme $X$ with symplectic form $\omega$ of degree $k$ is locally equivalent to (Spec $A,\omega'$) for Spec $A$ an affine derived scheme whose cdga $A$ has Darboux-like coordinates in which the symplectic form $\omega'$ is standard, and the differential in $A$ is given by Poisson bracket with a Hamiltonian function $H$ in $A$ of degree $k+1$. When $k=-1$, this implies that a $-1$-shifted symplectic derived scheme $(X,\omega)$ is Zariski locally equivalent to the derived critical locus Crit$(H)$ of a regular function $H:U\to{\mathbb A}^1$ on a smooth scheme $U$. We use this to show that the underlying classical scheme of $X$ has the structure of an 'algebraic d-critical locus', in the sense of Joyce arXiv:1304.4508. In the sequels arXiv:1211.3259, arXiv:1305.6428, arXiv:1312.0090, arXiv:1504.00690, 1506.04024 we will discuss applications of these results to categorified and motivic Donaldson-Thomas theory of Calabi-Yau 3-folds, and to defining new Donaldson-Thomas type invariants of Calabi-Yau 4-folds, and to defining 'Fukaya categories' of Lagrangians in algebraic symplectic manifolds using perverse sheaves, and we will extend the results of this paper and arXiv:1211.3259, arXiv:1305.6428 from (derived) schemes to (derived) Artin stacks, and to give local descriptions of Lagrangians in $k$-shifted symplectic derived schemes. Bouaziz and Grojnowski arXiv:1309.2197 independently prove a similar 'Darboux Theorem'. |
| first_indexed | 2024-03-06T21:30:52Z |
| format | Journal article |
| id | oxford-uuid:44a257f5-31ef-45ae-9edb-4305c9dcf3b8 |
| institution | University of Oxford |
| last_indexed | 2024-03-06T21:30:52Z |
| publishDate | 2018 |
| publisher | American Medical Society |
| record_format | dspace |
| spelling | oxford-uuid:44a257f5-31ef-45ae-9edb-4305c9dcf3b82022-03-26T15:02:53ZA Darboux theorem for derived schemes with shifted symplectic structureJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:44a257f5-31ef-45ae-9edb-4305c9dcf3b8Symplectic Elements at OxfordAmerican Medical Society2018Brav, CBussi, VJoyce, DWe prove a 'Darboux theorem' for derived schemes with symplectic forms of degree $k<0$, in the sense of Pantev, Toen, Vaquie and Vezzosi arXiv:1111.3209. More precisely, we show that a derived scheme $X$ with symplectic form $\omega$ of degree $k$ is locally equivalent to (Spec $A,\omega'$) for Spec $A$ an affine derived scheme whose cdga $A$ has Darboux-like coordinates in which the symplectic form $\omega'$ is standard, and the differential in $A$ is given by Poisson bracket with a Hamiltonian function $H$ in $A$ of degree $k+1$. When $k=-1$, this implies that a $-1$-shifted symplectic derived scheme $(X,\omega)$ is Zariski locally equivalent to the derived critical locus Crit$(H)$ of a regular function $H:U\to{\mathbb A}^1$ on a smooth scheme $U$. We use this to show that the underlying classical scheme of $X$ has the structure of an 'algebraic d-critical locus', in the sense of Joyce arXiv:1304.4508. In the sequels arXiv:1211.3259, arXiv:1305.6428, arXiv:1312.0090, arXiv:1504.00690, 1506.04024 we will discuss applications of these results to categorified and motivic Donaldson-Thomas theory of Calabi-Yau 3-folds, and to defining new Donaldson-Thomas type invariants of Calabi-Yau 4-folds, and to defining 'Fukaya categories' of Lagrangians in algebraic symplectic manifolds using perverse sheaves, and we will extend the results of this paper and arXiv:1211.3259, arXiv:1305.6428 from (derived) schemes to (derived) Artin stacks, and to give local descriptions of Lagrangians in $k$-shifted symplectic derived schemes. Bouaziz and Grojnowski arXiv:1309.2197 independently prove a similar 'Darboux Theorem'. |
| spellingShingle | Brav, C Bussi, V Joyce, D A Darboux theorem for derived schemes with shifted symplectic structure |
| title | A Darboux theorem for derived schemes with shifted symplectic structure |
| title_full | A Darboux theorem for derived schemes with shifted symplectic structure |
| title_fullStr | A Darboux theorem for derived schemes with shifted symplectic structure |
| title_full_unstemmed | A Darboux theorem for derived schemes with shifted symplectic structure |
| title_short | A Darboux theorem for derived schemes with shifted symplectic structure |
| title_sort | darboux theorem for derived schemes with shifted symplectic structure |
| work_keys_str_mv | AT bravc adarbouxtheoremforderivedschemeswithshiftedsymplecticstructure AT bussiv adarbouxtheoremforderivedschemeswithshiftedsymplecticstructure AT joyced adarbouxtheoremforderivedschemeswithshiftedsymplecticstructure AT bravc darbouxtheoremforderivedschemeswithshiftedsymplecticstructure AT bussiv darbouxtheoremforderivedschemeswithshiftedsymplecticstructure AT joyced darbouxtheoremforderivedschemeswithshiftedsymplecticstructure |