Timeline for Reference for symplectic structures on schemes?
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17 events
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| Feb 4, 2018 at 17:00 | comment | added | user100272 | @HassanJolany Why does one expect moduli spaces of sheaves on an (algebraic) Kahler-Einstein manifold to have shifted symplectic structure? | |
| Nov 19, 2017 at 15:06 | vote | accept | CommunityBot | ||
| Nov 19, 2017 at 12:57 | comment | added | user21574 | Tyurin, generalized the result of S. Mukai on vector bundles on $K3$ surfaces to the case of regular algebraic surfaces $S$ with $p_g>0$ to construct a symplectic structure. See iopscience.iop.org/article/10.1070/IM1989v033n01ABEH000818 | |
| Nov 19, 2017 at 12:38 | comment | added | user21574 | The old question was that the moduli space $ \mathcal M_{X,P}$ of torsion free sheaves on $X$ with Hilbert polynomial $P$ which are generically simple $\mathcal A$-modules( as a sheaf of Azumaya algebras.) is symplectic? Ulrich confirmed it in the case when $X$ is $K3$-surface or Abelian variety. | |
| Nov 19, 2017 at 12:29 | comment | added | user21574 | The existence of a holomorphic symplectic structure on Fano scheme $F(Y) $ of lines on a cubic fourfold $ Y,$ confirmed by Beauville and studied by Dimitri Markushevich also see iopscience.iop.org/article/10.1070/IM2003v067n01ABEH000421 | |
| Nov 19, 2017 at 12:21 | comment | added | user21574 | The main difficulty is about the existence of shifted symplectic structures on derived stacks. There is few result of it. | |
| Nov 19, 2017 at 12:13 | history | edited | Martin Sleziak |
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| Nov 19, 2017 at 12:10 | comment | added | user21574 | Second definition:The $p$-forms on the derived stack $X$ are then naturally defined as sections of $Λ^pL_{X/k}$, and more generally, elements in $H^n(X,Λ^pL_{X/k})$ are called $p$-forms of degree $n$ on $X$. The notion of closed $p$-forms on $X$ is highly non-trivial. | |
| Nov 19, 2017 at 12:10 | comment | added | user21574 | Definition: A symplectic form on a smooth scheme over some base ring $k$ of characteristic zero is the datum of a closed 2-form $ω$ on $X$, which is required to be non-degenerate, i.e. it induces an isomorphism $Θ_ω:T_{X/k}→Ω^1_{X/k}$ between the tangent and cotangent sheaves on $X$. In the context of derived Artin stacks, the cotangent sheaf is replaced by the cotangent complex $L_{X/k}$; due to L. Illusie, idea | |
| Nov 19, 2017 at 12:08 | history | rollback | S. Carnahan♦ |
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| Nov 19, 2017 at 12:06 | history | edited | user21574 | CC BY-SA 3.0 |
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| Nov 19, 2017 at 11:58 | history | edited | user21574 | CC BY-SA 3.0 |
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| Nov 19, 2017 at 11:54 | comment | added | user21574 | From Pantev result: moduli spaces of sheaves on Calabi-Yau manifolds admit shifted symplectic structures. So the question is that moduli spaces of sheaves on Kahler-Einstein manifolds admit shifted symplectic structures. ? link.springer.com/article/10.1007/s10240-013-0054-1 | |
| Nov 19, 2017 at 11:46 | vote | accept | CommunityBot | ||
| Nov 19, 2017 at 11:46 | |||||
| Nov 19, 2017 at 11:44 | answer | added | user100272 | timeline score: 18 | |
| Nov 25, 2016 at 13:21 | comment | added | user40276 | There are expositions about deformation quantization in the contexts of schemes arxiv.org/pdf/math/0106006v1.pdf and arxiv.org/pdf/math/0310399.pdf . But instead I would suggest studying a little bit of Lie $\infty$-algebroids, symplectic groupoids and all this in the DG context (or A_{\infty} DG context). Maybe studying the basics of noncommutative geometry in the DG context will also help (for instance, arxiv.org/pdf/math/0506603v1.pdf). The base field is not so important. Once you know for R, you know for C and, then, you know for every field of characteristic zero. | |
| Nov 24, 2016 at 16:54 | history | asked | user100272 | CC BY-SA 3.0 |