  It has recently been shown that the ten-dimensional superstring can be
quantized using the BRST operator $Q=\oint\lambda^\alpha d_\alpha$ where
$\lambda^\alpha$ is a pure spinor satisfying $\lambda \gamma^m \lambda=0$ and
$d_\alpha$ is the fermionic supersymmetric derivative. In this paper, the pure
spinor version of superstring theory is formulated in a curved supergravity
background and it is shown that nilpotency and holomorphicity of the pure
spinor BRST operator imply the on-shell superspace constraints of the
supergravity background. This is shown to lowest order in $\alpha'$ for the
heterotic and Type II superstrings, thus providing a compact pure spinor
version of the ten-dimensional superspace constraints for N=1, Type IIA and
Type IIB supergravities. Since quantization is straightforward using the pure
spinor version of the superstring, it is expected that these methods can also
be used to compute higher-order $\alpha'$ corrections to the ten-dimensional
superspace constraints.
