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# FundamentalComputingfundamental groups and singular cohomology of projective varieties

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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 is should be gotten by slowly changing the coefficients of the polynomialsor . Is something like that. this true? Does some kind of argument like this work?

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Are there any general methods for computing fundamental group or singular cohomology 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 is gotten by changing the coefficients of the polynomials or something like that. Does some kind of argument like this work?

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