Geometric orientation and dg-category Calabi–Yau structure

Fix a characteristic-zero field k and use the derived-stack and cochain conventions of the exact suppliers listed below. Write $C(X_A,E)$ = R $\mathrm{\Gamma }(X_A,E)$ for derived global sections of a perfect complex E on a base change $X_A$ = X × Spec(A), and E dual = R ${\mathrm{Hom}}_{O_{X_A}}(E,O_{X_A})$. A derived stack X is O-compact when, for every connective derived k-algebra A, $O_{X_A}$ is compact in $\mathrm{QCoh}(X_A)$ and $C(X_A,E)$ is a perfect A-module for each perfect E and commutes with derived base change. Compact means that the derived mapping functor out of $O_{X_A}$ preserves arbitrary coproducts.

A degree-d geometric orientation on an O-compact derived stack X is a map [X]: $C(X,O_X)$ -> k[-d] such that for every connective derived k-algebra A and every perfect complex E on $X_A$, the evaluation and cup-product map $C(X_A,E)\to R{\mathrm{Hom}}_A(C(X_A,E\mathrm{d}\mathrm{u}\mathrm{a}\mathrm{l}),A)[-d]$ induced by $[X{]}_A=[X]\otimes i{d}_A$ is a quasi-isomorphism. Use cohomological shifts $K[r{]}^j=K^{j+r}$. The orientation is a datum on a derived stack; it is not the same object as a volume form or as a nondegenerate pairing specified only on individual vector bundles.

For the following projective-volume construction assume the Axiom of Choice, as required by the trace and projective-volume suppliers. If X is a smooth projective pure d-dimensional k-scheme with a chosen trivialization nu: $O_X$ -> ${\omega }_X$ = $\det ({\mathrm{\Omega }}_{X\mathrm{/}k}^1)$, the normalized Serre trace defines a candidate map $C(X,O_X)$ -> k[-d]: compose R Gamma(X,nu) with the derived morphism R $\mathrm{\Gamma }(X,{\omega }_X)$ -> k[-d] corresponding to $t_X$ on $H^d(X,{\omega }_X)$. The local projective-volume lemma proves that X is O-compact and this trace is a geometric orientation in the full sense above: it constructs the cap quasi-isomorphism for every perfect complex after every connective derived base change, using the finite affine Čech complex, the pulled-back coordinate Koszul generator, and exact finite-cone and retract closure. Its original vector-bundle pairing is the restriction at $A=k$. The source-specific strict Calabi–Yau convention remains distinct from this orientation condition.

For a small k-linear cohomological dg category A, let $A^e=A^{op}$ ${\otimes }_k$ A and regard A as its diagonal $A^e$-module. The category is smooth when the diagonal is perfect as an $A^e$-module. Hochschild chains HH(A/k) are the derived trace of the diagonal, equivalently the realization of the cyclic bar construction; they carry the canonical circle action. Negative cyclic chains are HN(A/k) = $HH(A\mathrm{/}k{)}^{h{S}^1}$, the homotopy-fixed-point object for that action. Define the inverse-dualizing bimodule A! = R ${\mathrm{Hom}}_{A^e}(A,A^e)$.

For a smooth A, an absolute left d-Calabi–Yau structure on A is a map k[d] -> HN(A/k) whose image in HH(A/k) induces a quasi-isomorphism A! -> A[-d]. Here smoothness is an explicit hypothesis for the identification HH(A/k) = R ${\mathrm{Hom}}_{A^e}(A!,A)$ used to obtain this bimodule map; defining smoothness alone does not impose it on an arbitrary dg category. For a discrete commutative C-algebra R and a compactly generated smooth R-linear stable category C, use the relative convention HN(C/R) = $HH(C\mathrm{/}R{)}^{h{S}^1}$; a relative left d-Calabi–Yau structure is a map R[d] -> HN(C/R) whose induced natural map $i{d}_C!\to i{d}_C[-d]$ is an equivalence. Here C! denotes the inverse-dualizing endofunctor represented by the dual of the diagonal C-bimodule. Smoothness and finite type for C are the source's categorical finiteness hypotheses; they are not implied by the existence of the negative-cyclic class.

These geometric and categorical Calabi–Yau data are kept separate. The right Calabi–Yau convention uses a properness hypothesis and a map from the diagonal to its k-linear dual; neither it nor a geometric orientation is identified with the left negative-cyclic structure without a separately proved comparison.

Strict Calabi–Yau variety in the Morimura source convention

Derived Artin stack and cotangent complex

Dualizing line bundle and trace datum of a smooth projective variety

Serre duality for locally free sheaves on a smooth projective variety

Derived schemes and the cotangent complex of a morphism

Small dg categories, perfect modules, and Morita deformation

The Axiom of Choice

Projective volume trace duality for perfect complexes under derived base change

The geometric orientation is the PTVV $\mathcal{O}$-orientation, not a Calabi–Yau condition on a dg category. The projective volume-trace construction requires a chosen trivialization of ${\omega }_X$ but does not require strict intermediate-cohomology vanishings. When its input is a $k$-variety in the Morimura convention, connectedness follows from integrality and is not a separate hypothesis; the source's triviality condition itself retains no chosen trivialization.

The full perfect-complex and derived-base-change assertion is established by Projective volume trace duality for perfect complexes under derived base change, proof steps 1.2, 2.2, 3.2, and 4.1. The lemma applies without connectedness or strict intermediate-cohomology vanishing, and makes no comparison with a negative-cyclic Calabi–Yau class.