We extend the lower bounds on the complexity of computing Betti numbers proved in [6] to complex algebraic varieties. More precisely, we first prove that the problem of deciding connectedness of a complex affine or projective variety given as the zero set of integer polynomials is PSPACE-hard. Then we prove PSPACE-hardness for the more general problem of deciding whether the Betti number of fixed order of a complex affine or projective variety is at most some given integer. Key words: connected components, Betti numbers, PSPACE, lower bounds 