Let $X$ be a smooth complex quasi-projective variety. We can find good compactification: a smooth proper variety $\bar{X}$ such that ${\bar X} \setminus X$ is a divisor with normal crossing. The variety $\bar{X}$ is then stratified by the singulartities of the divisor. And one can compute the mixed Hodge structure on $H^{\bullet}(X)$ in terms of the pure Hodge structures $H^{\bullet}(S_\alpha)$ of the smooth closed strata using a spectral sequence.

Let's say a variety $Y$ is Hodge-Tate if $h^{p,q}(Y) = 0$ for $p\neq q$.

If all the closed strata of $\bar{X}$ are Hodge-Tate then $X$ is Hodge-Tate.

Question: Let $X$ be a smooth complex quasi-projective variety. Assume $X$ is Hodge-Tate.

- Can one find a good compactification $\bar{X}$ with Hodge-Tate strata?
- Are all good compactifications of $X$ of this type? (Edit: Answer is no, see Torsten's elementary example).