Reference for symplectic structures on schemes?

My original goal was to read the PTVV paper Shifted Symplectic Structures https://arxiv.org/pdf/1111.3209v4.pdf. I was quickly humbled!

Being told the theory ought to generalize symplectic structures on algebraic varieties and schemes I was unable to find a clear reference for these structures. I could get a hold of a paper or talk here and there giving some definition, but nothing futher.

• 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 25, 2016 at 13:21
• 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
– user21574
Nov 19, 2017 at 11:54
• 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
– user21574
Nov 19, 2017 at 12:10
• 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
– user21574
Nov 19, 2017 at 12:29
• 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.
– user21574
Nov 19, 2017 at 12:38

Find a specific geometric problem that this stuff solves, which you think is interesting, and then you will find yourself magically learning it. For you, this problem was extending Donaldson-Thomas theory to Calabi-Yau 4-folds. Even in the Calabi-Yau 3-fold case, shifted symplectic structures show Behrend's notion of symmetric obstruction theory comes from a secret $-1$-shifted symplectic structure (this amazed you).