TITLE: Hodge numbers of compactifications
QUESTION [9 upvotes]: 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).

REPLY [7 votes]: I asked the question to Claire Voisin and she immediatly gave me the following counter example:
Consider Fermat's cubic of dimension 3 $X$. There are 5 cones on elliptic curves $E_i$ inside $X$ and one can verify that $Y=X\setminus U_iE_i$ is Hodge-Tate. But $Y$ doesn't admit a Hodge-Tate compactification as that would imply that $X$ is birational to a Tate variety and this would contradict Clemens-Griffiths' theorem saying that the intermediate jacobian of $X$ is not a direct sum of jacobians of curves as a polarized variety.