A 'Darboux Theorem' for shifted symplectic structures on derived Artin stacks, with applications
This is the fifth in a series arXiv:1304.4508, arXiv:1305,6302, arXiv:1211.3259, arXiv:1305.6428 on the '$k$-shifted symplectic derived algebraic geometry' of Pantev, Toen, Vaquie and Vezzosi, arXiv:1111.3209. This paper extends the previous three from (derived) schemes to (derived) Artin...
| المؤلفون الرئيسيون: | , , , |
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| التنسيق: | Journal article |
| منشور في: |
Mathematical Sciences Publishers
2015
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| _version_ | 1826271259424456704 |
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| author | Ben-Bassat, O Brav, C Bussi, V Joyce, D Bussi, V |
| author_facet | Ben-Bassat, O Brav, C Bussi, V Joyce, D Bussi, V |
| author_sort | Ben-Bassat, O |
| collection | OXFORD |
| description | This is the fifth in a series arXiv:1304.4508, arXiv:1305,6302, arXiv:1211.3259, arXiv:1305.6428 on the '$k$-shifted symplectic derived algebraic geometry' of Pantev, Toen, Vaquie and Vezzosi, arXiv:1111.3209. This paper extends the previous three from (derived) schemes to (derived) Artin stacks. We prove four main results: (a) If $(X,\omega)$ is a $k$-shifted symplectic derived Artin stack for $k<0$ in the sense of arXiv:1111.3209, then near each $x\in X$ we can find a 'minimal' smooth atlas $\varphi:U\to X$ with $U$ an affine derived scheme, such that $(U,\varphi^*(\omega))$ may be written explicitly in coordinates in a standard 'Darboux form'. (b) If $(X,\omega)$ is a $-1$-shifted symplectic derived Artin stack and $X'$ the underlying classical Artin stack, then $X'$ extends naturally to a 'd-critical stack' $(X',s)$ in the sense of arXiv:1304.4508. (c) If $(X,s)$ is an oriented d-critical stack, we can define a natural perverse sheaf $P^\bullet_{X,s}$ on $X$, such that whenever $T$ is a scheme and $t:T\to X$ is smooth of relative dimension $n$, then $T$ is locally modelled on a critical locus Crit$(f:U\to{\mathbb A}^1)$ for $U$ smooth, and $t^*(P^\bullet_{X,s})[n]$ is locally modelled on the perverse sheaf of vanishing cycles $PV_{U,f}^\bullet$ of $f$. (d) If $(X,s)$ is a finite type oriented d-critical stack, we can define a natural motive $MF_{X,s}$ in a ring of motives $\bar{\mathcal M}^{st,\hat\mu}_X$ on $X$, such that whenever $T$ is a finite type scheme and $t:T\to X$ is smooth of dimension $n$, then $T$ is locally modelled on a critical locus Crit$(f:U\to{\mathbb A}^1)$ for $U$ smooth, and ${\mathbb L}^{-n/2}\odot t^*(MF_{X,s})$ is locally modelled on the motivic vanishing cycle $MF^{mot,\phi}_{U,f}$ of $f$ in $\bar{\mathcal M}^{st,\hat\mu}_T$. Our results have applications to categorified and motivic extensions of Donaldson-Thomas theory of Calabi-Yau 3-folds |
| first_indexed | 2024-03-06T21:53:51Z |
| format | Journal article |
| id | oxford-uuid:4c39c93c-ae37-4006-aac5-f71782a8e2ac |
| institution | University of Oxford |
| last_indexed | 2024-03-06T21:53:51Z |
| publishDate | 2015 |
| publisher | Mathematical Sciences Publishers |
| record_format | dspace |
| spelling | oxford-uuid:4c39c93c-ae37-4006-aac5-f71782a8e2ac2022-03-26T15:48:17ZA 'Darboux Theorem' for shifted symplectic structures on derived Artin stacks, with applicationsJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:4c39c93c-ae37-4006-aac5-f71782a8e2acSymplectic Elements at OxfordMathematical Sciences Publishers2015Ben-Bassat, OBrav, CBussi, VJoyce, DBussi, VThis is the fifth in a series arXiv:1304.4508, arXiv:1305,6302, arXiv:1211.3259, arXiv:1305.6428 on the '$k$-shifted symplectic derived algebraic geometry' of Pantev, Toen, Vaquie and Vezzosi, arXiv:1111.3209. This paper extends the previous three from (derived) schemes to (derived) Artin stacks. We prove four main results: (a) If $(X,\omega)$ is a $k$-shifted symplectic derived Artin stack for $k<0$ in the sense of arXiv:1111.3209, then near each $x\in X$ we can find a 'minimal' smooth atlas $\varphi:U\to X$ with $U$ an affine derived scheme, such that $(U,\varphi^*(\omega))$ may be written explicitly in coordinates in a standard 'Darboux form'. (b) If $(X,\omega)$ is a $-1$-shifted symplectic derived Artin stack and $X'$ the underlying classical Artin stack, then $X'$ extends naturally to a 'd-critical stack' $(X',s)$ in the sense of arXiv:1304.4508. (c) If $(X,s)$ is an oriented d-critical stack, we can define a natural perverse sheaf $P^\bullet_{X,s}$ on $X$, such that whenever $T$ is a scheme and $t:T\to X$ is smooth of relative dimension $n$, then $T$ is locally modelled on a critical locus Crit$(f:U\to{\mathbb A}^1)$ for $U$ smooth, and $t^*(P^\bullet_{X,s})[n]$ is locally modelled on the perverse sheaf of vanishing cycles $PV_{U,f}^\bullet$ of $f$. (d) If $(X,s)$ is a finite type oriented d-critical stack, we can define a natural motive $MF_{X,s}$ in a ring of motives $\bar{\mathcal M}^{st,\hat\mu}_X$ on $X$, such that whenever $T$ is a finite type scheme and $t:T\to X$ is smooth of dimension $n$, then $T$ is locally modelled on a critical locus Crit$(f:U\to{\mathbb A}^1)$ for $U$ smooth, and ${\mathbb L}^{-n/2}\odot t^*(MF_{X,s})$ is locally modelled on the motivic vanishing cycle $MF^{mot,\phi}_{U,f}$ of $f$ in $\bar{\mathcal M}^{st,\hat\mu}_T$. Our results have applications to categorified and motivic extensions of Donaldson-Thomas theory of Calabi-Yau 3-folds |
| spellingShingle | Ben-Bassat, O Brav, C Bussi, V Joyce, D Bussi, V A 'Darboux Theorem' for shifted symplectic structures on derived Artin stacks, with applications |
| title | A 'Darboux Theorem' for shifted symplectic structures on derived Artin
stacks, with applications |
| title_full | A 'Darboux Theorem' for shifted symplectic structures on derived Artin
stacks, with applications |
| title_fullStr | A 'Darboux Theorem' for shifted symplectic structures on derived Artin
stacks, with applications |
| title_full_unstemmed | A 'Darboux Theorem' for shifted symplectic structures on derived Artin
stacks, with applications |
| title_short | A 'Darboux Theorem' for shifted symplectic structures on derived Artin
stacks, with applications |
| title_sort | darboux theorem for shifted symplectic structures on derived artin stacks with applications |
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