Strict Calabi–Yau variety in the Morimura source convention

Assume the Axiom of Choice. To fix terminology, a $k$-variety means an integral, separated, finite-type $k$-scheme, as in the Stacks Project, Definition 33.3.1. In Morimura's source convention, a strict Calabi–Yau $d$-fold is a smooth projective $k$-variety $X$ of pure dimension $d$ for which ${\omega }_X\cong {\mathcal{O}}_X$ and $H^i(X,{\mathcal{O}}_X)=0$ for every integer $i$ with $0<i<d$. Here ${\omega }_X:=\det {\mathrm{\Omega }}_{X\mathrm{/}k}^1$ is the dualizing line bundle, with the determinant convention of the smooth-projective supplier. The canonical-bundle condition means that an isomorphism exists; no particular trivialization is part of the source definition. Under this variety convention $X$ is connected, so connectedness is not an additional hypothesis.

For Morimura's deformation comparison, impose the additional theorem hypotheses that $k$ is algebraically closed of characteristic zero and $d>2$. These restrictions are not part of the definition of the source-specific strictness condition itself.

Dualizing line bundle and trace datum of a smooth projective variety

The Axiom of Choice

This is an algebraic, source-specific convention. It is distinct from the analytic compact-Kähler convention and does not by itself provide an orientation on a derived stack or a negative-cyclic Calabi–Yau structure on a dg category. Canonical-bundle triviality, intermediate-cohomology vanishing, and theorem-only field/dimension restrictions are kept separate; a trivialization may be chosen later when a construction requires one.