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.
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
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 $X$.
Motivation:Motivic and Categorical Donaldson-Thomas theory of Kontsevich and Soibelman
