  We compactify the pure spinor formalism on a K3 surface. The pure spinor
splits into a six-dimensional pure spinor, a projective superspace harmonic,
and 6 non-covariant variables. A homological algebra argument reduces the
calculation of the cohomology of the Berkovits differential to a "small"
Hilbert space which is the string-theoretic analogue of projective superspace.
The description of the physical state conditions is facilitated by lifting to
the full harmonic superspace, which is accomplished by the introduction of the
missing harmonics as non-minimal variables. Finally, contact with the hybrid
formalism is made by returning to the small Hilbert space and fermionizing the
projective parameter.
