  It is shown that the BRST resolution of the spaces of physical states of the
systems with anomalies can be consistently defined. The appropriate anomalous
complexes are obtained by canonical restrictions of the ghost extended spaces
to the kernel of anomaly operator without any modifications of the "matter"
sector. The cohomologies of the anomalous complex for the case of constarints
constituting a centrally extended simple Lie algebra of compact type are
calculated and analyzed in details within the framework of Hodge - deRham -
Kaehler theory: the vanishing theorem of the relative cohomologies is proved
and the absolute cohomologies are reconstructed.
