LECHS CONJECTURE ON DEFORMATIONS OF SINGULARITIES AND 2ND HARRISON COHOMOLOGY

Let B-0 be a local singularity of dimension d. Then we consider the pr oblem of Lech, whether for every deformation (A, m) --> (B, n) of B-0 the inequality H-A(d+1) less than or equal to H-B(1) between the Hilbe rt functions is true, and give a positive answer in the case that the formal versal deformation of B-0 is a base change of an algebraic fami ly (R, M) --> (S, N), where R is regular and dim S = dim R+d. So one w ould hope to lift versal deformations in that way. There are obstructi ons against this in certain second Harrison cohomology groups.