Apparently you can compute the h^{p,q}'s of smooth things in, for example, Macaulay. Here's an example: [computing the h^{p,q}'s of a quintic hypersurface in P^4][1].


  [1]: http://www.math.uiuc.edu/Macaulay2/doc/Macaulay2-1.3.1/share/doc/Macaulay2/Macaulay2Doc/html/___H__H%5e__Z__Z_sp__Coherent__Sheaf.html