Closed n-shifted symplectic form

Fix a characteristic-zero field $k$, the connective derived-affine model of Derived schemes and the cotangent complex of a morphism, and a derived Artin stack $F$ with cotangent complex $L_{F\mathrm{/}k}$. Assume AC for the cotangent-complex construction inherited from Derived schemes and the cotangent complex of a morphism. In the affine calculation, a nonpositive commutative dg $k$-algebra $A={⨁}_{i\le 0}{A}^i$ is a unital associative cochain algebra with degree-$+1$ differential satisfying $d_A^2=0$, $ab=(-1{)}^{\mathrm{\mid }a\mathrm{\mid }\mathrm{\mid }b\mathrm{\mid }}ba$, and $d_A(ab)=d_A(a)b+(-1{)}^{\mathrm{\mid }a\mathrm{\mid }}a{d}_A(b)$. A morphism preserves unit, multiplication, grading, and differential. This is the graded-commutative convention used in $\mathrm{D}\mathrm{R}$. On a cofibrant such algebra form the derived de Rham algebra $\mathrm{D}\mathrm{R}(A\mathrm{/}k)={\mathrm{Sym}}_A^{\mathbf{L}}(L_{A\mathrm{/}k}[1])\mathrm{.}$ Its weight-$p$ component is ${\bigwedge }_A^p{L}_{A\mathrm{/}k}[p]$. It has the internal cochain differential of degree $+1$ and a mixed de Rham differential $ϵ$ of degree $-1$ and weight $+1$, with $dϵ+ϵd=0$ and ${ϵ}^2=0$. For a derived stack, $\mathrm{D}\mathrm{R}(F\mathrm{/}k)$ is the homotopy limit of these graded mixed complexes over all derived affine maps into $F$; the construction uses derived cotangent complexes and is independent of cofibrant resolution.

For a graded mixed complex $E$, its Hodge-completed weighted negative-cyclic complex in weight $p$ has cochain degree $j$ term ${\mathrm{N}\mathrm{C}}^w(E)(p{)}^j={\prod }_{r\ge 0}{E}^{j-2r}(p+r)\mathrm{.}$ For $m=(m_r{)}_{r\ge 0}$ of degree $j$, its differential is $(Dm{)}_r=d{m}_r+ϵm_{r-1},m_{-1}=0.$ Both summands belong to $E^{j+1-2r}(p+r)$, and $D^2=0$ follows from $d^2={ϵ}^2=dϵ+ϵd=0$. This is the total differential in ${\mathrm{RHom}}_{k[ϵ]}(k(p),E)$; the completion is a product over all higher weights, not a direct sum. For any cochain complex $E$ of $k$-vector spaces, let ${\tau }_{\le 0}E$ be its good truncation: it agrees with $E$ in degrees below $0$, has $\ker (d:E^0\to E^1)$ in degree $0$, and is zero in positive degrees. Define $\mathrm{\mid }E\mathrm{\mid }$ to be the underlying simplicial set of the simplicial $k$-vector space obtained by applying the Dold–Kan correspondence to the nonnegative chain complex $C_n=({\tau }_{\le 0}E{)}^{-n}$, $n\ge 0$, with differential inherited from $E$ (Dold-Kan equivalence for simplicial modules with explicit inverse). Define the space of degree-$n$ $p$-forms and the space of closed degree-$n$ $p$-forms by ${\mathcal{A}}^p(F,n)={\mathrm{Map}}_{\mathrm{Q}\mathrm{C}\mathrm{o}\mathrm{h}(F)}({\mathcal{O}}_F,{\bigwedge }^p{L}_{F\mathrm{/}k}[n]),{\mathcal{A}}^{p,\mathrm{c}\mathrm{l}}(F,n)=\mid {\mathrm{N}\mathrm{C}}^w(\mathrm{D}\mathrm{R}(F\mathrm{/}k))(p)[n-p]\mid \mathrm{.}$ The natural projection ${\mathrm{N}\mathrm{C}}^w(E)(p)\to E(p)$ gives the underlying form map ${\mathcal{A}}^{p,\mathrm{c}\mathrm{l}}(F,n)\to {\mathcal{A}}^p(F,n)$.

For $p=2$, assume explicitly that $L_{F\mathrm{/}k}$ is perfect in $\mathrm{Q}\mathrm{C}\mathrm{o}\mathrm{h}(F)$. This hypothesis makes $T_{F\mathrm{/}k}:=R{\mathrm{Hom}}_{{\mathcal{O}}_F}(L_{F\mathrm{/}k},{\mathcal{O}}_F)$ its dual and identifies a 2-form $\omega$ of degree $n$ with its adjoint morphism ${\mathrm{\Theta }}_{\omega }:T_{F\mathrm{/}k}⟶L_{F\mathrm{/}k}[n]\mathrm{.}$ It is nondegenerate when this map is a quasi-isomorphism. An $n$-shifted symplectic structure is a closed 2-form of degree $n$ whose underlying form is nondegenerate. A pointwise or merely internally closed pairing without a lift to ${\mathcal{A}}^{2,\mathrm{c}\mathrm{l}}(F,n)$ is not an $n$-shifted symplectic structure.

Derived Artin stack and cotangent complex

Derived schemes and the cotangent complex of a morphism

Dold-Kan equivalence for simplicial modules with explicit inverse