Can proper-smooth base change be used to show that varieties cannot be lifted to characteristic zero? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T20:25:44Z http://mathoverflow.net/feeds/question/67485 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/67485/can-proper-smooth-base-change-be-used-to-show-that-varieties-cannot-be-lifted-to Can proper-smooth base change be used to show that varieties cannot be lifted to characteristic zero? Akhil Mathew 2011-06-11T03:01:17Z 2011-06-11T13:05:06Z <p>Recall the following corollary to the proper and smooth base change theorems:</p> <blockquote> <p>Let $\pi: X \to S$ be a proper, smooth morphism. Then the direct images $R^i \pi_* \mathcal{F}$ are locally constant constructible for any l.c.c. sheaf $\mathcal{F}$ (with torsion prime to the characteristic of the residue fields) on $X_{et}$.</p> </blockquote> <p>It follows in particular that if $S$ is the Spec of a DVR (say with an algebraically closed residue field), then the cohomology of the generic fiber (at least base-changed to a separable closure) is the same as that of the specific fiber. Consider the case where $S$ has unequal characteristic --- then essentially, this means that the special fiber $X_0$ admits a (smooth) lifting to characteristic zero. Then the above observations says something about what the cohomology of $X_0$ has to be (by comparing to the cohomology of the generic fiber, which is also the complex cohomology).</p> <p>Can this be used to show that schemes in characteristic $p$ aren't liftable to characteristic zero? I don't know if this is easy, because etale cohomology groups seem to satisfy many of the same properties (e.g. Poincare duality, dimensional vanishing) of complex cohomology. Milne's book seems to list this as an application of proper-smooth base change but does not actually give an example.</p> http://mathoverflow.net/questions/67485/can-proper-smooth-base-change-be-used-to-show-that-varieties-cannot-be-lifted-to/67491#67491 Answer by ulrich for Can proper-smooth base change be used to show that varieties cannot be lifted to characteristic zero? ulrich 2011-06-11T06:24:23Z 2011-06-11T06:24:23Z <p>There is an example due to Hirokado of a Calabi-Yau threefold in characteristic 3 with third Betti number zero which implies that it cannot be lifted to characteristic zero. See: Hirokado, Masayuki - A non-liftable Calabi-Yau threefold in characteristic 3. Tohoku Math. J. (2) 51 (1999), no. 4, 479–487. </p>