# Lifting varieties from char. $p$ to char. 0 after alterations

The question is related to this MO question:

Lifting varieties to characteristic zero.

Let $X$ be a projective smooth variety over $k$ alg. closed field of char. $p.$ Does there always exist an alteration $Y\to X,$ with $Y$ also projective smooth, such that $Y$ lifts to char. 0?

Side remark: over $k=\overline{\mathbb F}_p,$ modulo Tate conjecture, abelian varieties "generate" the motives of all proj. smooth varieties. Since abelian varieties are liftable, one can say that the (irred. components of) motives of any proj. smooth varieties is liftable in some sense. And I wonder if this can be realized geometrically. "Alteration" in my question is just a try; replace it with any reasonable geometric construction if you want. For instance, "a proper surjection" would be fine.

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I know Bhargav Bhatt in his thesis asked whether "any variety can be dominated by a smooth one that lifts $W_2(k)$?" –  Karl Schwede May 17 '11 at 15:39
Remark 5.5.5 by the way. –  Karl Schwede May 17 '11 at 15:40
I'm not sure if this is also part of your motivation, but I'll just add that Torsten Ekedahl points out that after a sequence of curve contractions and a deformation even Hirokado's example of a non-liftable CY threefold lifts to char 0. –  Matt May 17 '11 at 16:45
@Karl and Matt: These are certainly good motivations, which I didn't know before. Thank you for letting me know. Karl, do you know where I can find Bhatt's thesis? –  shenghao May 17 '11 at 18:55
OK, it seems that this question is still open. –  shenghao Jun 27 '11 at 15:51