AFFINE STRUCTURES ON NILMANIFOLDS

We investigate the existence of affine structures on nilmanifolds Gamm a\G in the case where the Lie algebra g of the Lie group G is filiform nilpotent of dimension less or equal to 11. Here we obtain examples o f nilmanifolds without any affine structure in dimensions 10, 11. Thes e are new counterexamples to the Milnor conjecture. So far examples in dimension 11 were known where the proof is complicated, see [5] and [ 4]. Using certain 2-cocycles we realize the filiform Lie algebras as d eformation algebras from a standard graded filiform algebra. Thus we s tudy the affine algebraic variety of complex filiform nilpotent Lie al gebra structures of a given dimension less than or equal to 11. This a pproach simplifies the calculations, and the counterexamples in dimens ion 10 are less complicated than the known ones. We also obtain result s for the minimal dimension mu(g) of a faithful g-module for these fil iform Lie algebras g.