# defining the canonical divisor

I am just starting to learn some basic algebraic geometry, but I am very confused with some definitions. So I hope I am not asking something that is completely trivial.

Suppose $X$ is a normal variety. I understand (I think!) that the canonical sheaf is simply defined to be the top wedge of the sheaf of differentials. My question is - can the canonical divisor (as a Weil divisor) be defined in general?

Since the variety is normal, as I understand, it is enough to look at the regular part, $X_0 \subset X$. On the regular part, the sheaf is invertible and is the sheaf of sections of the canonical line bundle. But in order for this to induce a Weil divisor, does it not need to have a meromorphic section? So how can one be sure (without any assumptions that $X$ is projective) that there does exist a meromorphic section of $K_{X_0}$.

Also, there are many examples of even smooth varieties with no irreducible sub-varieties. So in particular there will be no non-trivial Weil divisor. In such cases, the canonical divisor would be forced to be trivial. Are there any example of such varieties whose canonical bundle is still non-trivial as a line bundle?

• You should read this (in particular the accepted answer) even though it doesn't answer all of your questions. – Ben Feb 19 '15 at 16:59