  By considering Voronoi tessellations of the configurations of a fluid, we
propose two new conserved fields, which provide structural information not
fully accounted for by the usual 2-point density field fluctuations (structure
factor). One of these fields is scalar and associated to the Voronoi cell
volumes, whereas the other one, termed the "geometrical polarisation", is
vectorial, related to the very local anisotropy of the configurations. We study
the static and dynamical properties of these fields in the supercooled regime
of a model glass-forming liquid. We show in particular that the geometrical
polarisation is both statically correlated to the force field and contrary to
it develops a plateau regime when the temperature is lowered. We attribute this
behaviour to the microsopic disorder of the underlying inherent structures (IS)
which dictate the dynamics on time scales larger than the true microscopic
time, in the strong supercooled regime. In this respect, this work raises the
issue of to what extent the inter IS dynamics, intrinsically anisotropic and
collective (cf. T.B. Schr{\o}der et al. {\it J. of Chem. Phys.}, {\bf 112},
9834 (2000)), could be related to their polarisation field.
