Upon reading K. Costello's paper on Witten genus, I wonder when, on a smooth (quasi-)projective variety $X$, the canonical bundle $\omega\_X$ admits a left $D$-module structure (other than the Calabi-Yau case).



Projective volume form
If and only if it admits an integrable connection. In the projective case, this means that some power of $\omega \_X$ is trivial, hence a finite covering of $X$ has trivial canonical bundle. In the quasi-projective case it just means that $\omega \_X$ is flat.


