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]: https://faculty.math.illinois.edu/Macaulay2/doc/Macaulay2/share/doc/Macaulay2/Macaulay2Doc/html/___H__H%5E__Z__Z_sp__Coherent__Sheaf.html