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?
Best Answer
[I've incorporated or addressed comments of Dan Petersen and Daniel Litt into this. My thanks to them.]
One sometimes says that $X$ admits a cellular decomposition if it admits a stratification by affine spaces. The isomorphism of the type you mention was known before Totaro, cf Fulton's Intersection Theory 19.1.11 (in my edition). This applies to flag varieties. Also such an isomorphism holds for toric varieties for similar reasons.
A surface with $p_g=q=0$ i.e. $\dim H^0(X,\Omega^2)=\dim H^0(X,\Omega^1)=0$, and $A_0(X)\cong \mathbb{Z}$ satisfies the condition of $A(X)\cong H^*(X)$ by the Lefschetz $(1,1)$ theorem. Note that Bloch has conjectured that the second condition follows from the first. One has examples of surfaces of Kodaira dimension zero $\ge 0$, such as Enriques surfaces (Bloch-Kas-Lieberman) and Godeaux surfaces (Voisin) where these conditions hold. These do not have cellular decompositions.
Finally, assuming the Hodge and Bloch-Beilinson conjectures [thanks Dan], an isomorphism $A(X)\otimes \mathbb{Q}\cong H^{2*}(X,\mathbb{Q})$ holds if and only if the Hodge numbers $h^{pq}=0$ for $p\not= q$.