  We start from the Maurer-Cartan (MC) equations of the Osp(N|4) superalgebras
satisfied by the left-invariant super-forms realized on supercoset manifolds of
the corresponding supergroups and we derive some new pure spinor constraints.
They are obtained by "ghostifying" the MC forms and extending the differential
d to a BRST differential. From the superalgebras G =Osp(N|4) we single out
different subalgebras H contained in G associated with the different cosets
G/H: each choice of H leads to a different weakening of the pure spinor
constraints. In each case, the number of parameter is counted and we show that
in the cases of Osp(6|4)/U(3) x SO(1,3), Osp(4|4)/SO(3) x SO(1,3) and finally
Osp(4|4) U(2)} x SO(1,3) the bosonic and fermionic degrees of freedom match in
order to provide a c=0 superconformal field theory. We construct both the
Green-Schwarz and the pure spinor sigma model for the case Osp(6|4)/U(3)x
SO(1,3) corresponding to AdS_4 x P^3. The pure spinor sigma model can be
consistently quantized.
