Let $X$ be a projective integral Cohen-Macaulay variety (over $\mathbb{C}$, if that makes things easier). The Cohen-Macaulay condition says that the dualizing complex (see this answer) is concentrated in degree $-\dim X$, and so we define the dualizing sheaf to be this cohomology sheaf.
My question is simple: are there any situations where this dualizing sheaf is known to be globally generated?
For smooth or Gorenstein curves of arithmetic genus at least one, this is true, but I am specifically interested when the dualizing sheaf is not invertible (such as Cohen-Macaulay curves or suitable surfaces).
Thanks in advance.
