As the title suggest it seems standard conjectures mean different things depending on the context. I had the impression that in characteristic 0 they are a list of conjectures about varieties over an arbitrary field of characteristic zero and any Weil cohomology theory (not necessarily the usual suspects) just like the way it is explained in the Wikipedia page.

After reading this paper that claims the standard conjecture over any field of char $0$ follows from Suslin's Lawson homology conjecture I am confused.

- First, the base field $k$ admits embeddings into $\mathbb{C}$ as mentioned in the first page third paragraph, so $k$ cannot be any field of char $0$.
- Second, in the proof of the Proposition $2.2$ page $5$ the cohomology theory that is used is the singular cohomology and it is not just any Weil cohomology theory.

Are the conjectures stated in the generality of the Wikipedia page obviously not correct or is it possible somehow to go from singular cohomology and $\mathbb{C}$ to any Weil cohomology theory and field of characteristic $0$?