This is something I just heard about yesterday. In this article
Nodal quintics in P^4.
B. van Geemen and J. Werner
In: Arithmetic of Complex Manifolds; W.-P. Barth, H. Lange Eds. Springer LNM 1399, (1989) 48-59.
the authors compute the Betti numbers of certain smooth complex Calabi-Yau manifolds using a combination of techniques from algebraic topology and number theory. If I understand they first reduce the equations to some finite field, then combining the use of etale cohomology, the Lefschetz fixed point theorem and other things they reduce the problem to counting a finite set of points.