My question arises from the proof of theorem 5.11 page 18 ~ 19 of the notes by Mihnea Popa:
https://sites.math.northwestern.edu/~mpopa/483-3/notes/notes.pdf#page16
We wish to prove Kleinman’s theorem: A $\mathbb{Q}$ divisor $D$ on a proper scheme $X$ is nef iff $D^{\dim V}\cdot V \ge 0$ for all irreducible subvariety $V\subset X$.
In the third paragraph, he wrote that since $H$ is ample, then “$H^{n-k}$ is represented by an effective cycle class of dimension $k$“. My question is how does one show this? He seem to use this argument multiple times thereafter in several proofs as well. I tried looking up but I can’t seem to find any literature detailing this.
Any help rendered would be greatly appreciated!