My apologies if this question is too naive.

Let $X$ be a smooth projective complex variety. There is a natural map $A^{\bullet}(X) \to H^{2\bullet}(X)$ of graded rings from the Chow ring of $X$ to the integral cohomology of $X$ given by taking Poincaré duals of fundamental classes. (Is there a convenient name for it?) For some particularly nice varieties, e.g. projective spaces and more generally Grassmannians, this map is an isomorphism.

What are some more general $X$ for which $A^{\bullet}(X) \to H^{2\bullet}(X)$ is an isomorphism?

I think it suffices that $X$ have a stratification by affine spaces. (Is there a convenient name for such spaces?) In this case there is apparently a difficult theorem of Totaro asserting that $A(X)$ is free abelian on the strata, which I think is also the case for the integral cohomology via cellular cohomology, and degrees and intersections ought to match as well. Are there interesting families of examples where $X$ doesn't admit such a stratification?

**Edit:** A somewhat more general sufficient condition, if I've understood my reading correctly, is that the Chow motive of $X$ is a polynomial in the Lefschetz motive. Guletskii and Pedrini showed that this is true (edit: rationally) of the Godeaux surface so this is more general than admitting a stratification by affine spaces. Are there interesting families of examples where this doesn't hold either?