Are there any general methods for computing fundamental group or singular cohomology (including the ring structure, hopefully) of a projective variety (over C of course), if given the equations defining the variety?

I seem to recall that, if the variety is smooth, we can compute the H^{p,q}'s by computer -- and thus the H^n's by Hodge decomposition -- is this correct? However this won't work if the variety is not smooth -- are there any techniques that work even for non-smooth things?

Also I seem to recall some argument that, at least if we restrict our attention to smooth things only, all varieties defined by polynomials of the same degrees will be homotopy equivalent. The homotopy should be gotten by slowly changing the coefficients of the polynomials. Is something like this true? Does some kind of argument like this work?

complexdimension, yet a projective variery has non vanishing up to therealdimension. You need to compute the hypercohomology of the algebraic de Rham complex to get something useful. But there is very little gain in using HKR when you can start with the algebraic de Rham complex directly! $\endgroup$ – Mariano Suárez-Álvarez Feb 12 '10 at 14:18