Is there an example of a variety over the complex numbers with no embedding into a smooth variety?



Is there an example of a variety over the complex numbers with no embedding into a smooth variety?
Yes, there are even toric varieties. In fact, there are complete toric varieties with trivial picard group (see e.g., Eickelberg "Picard groups of compact toric varieties..." 1993). We have the following simple observations:


1) Any variety which can be embedded into a smooth variety has an ample family of line bundles.


2) A proper variety with trivial picard group cannot have an ample family of line bundles.


