  We investigate the complex geometry of D=10 pure spinor space. The K\"ahler
structure and the corresponding metric giving rise to the desired Calabi-Yau
property are determined, and an explicit covariant expression for the Laplacian
is given. The metric is not that of a cone obtained by embedding pure spinor
space in a flat space of unconstrained spinors. Some directions for future
studies, concerning regularisation and generalisation to eleven dimensions, are
briefly discussed.
