Variation of complex structures and variation of Lie algebras (Q1116993)

The simple-elliptic singularities \(\tilde E_ 7\) and \(\tilde E_ 8\) are studied by means of the derivation algebra of the moduli algebra. These singularities occur in 1-parameter families, which constitute (\(\mu\),\(\tau)\)-constant deformations. The authors introduce the notion of a derivation liftable to a vector field on the parameter space. For bad parameter values the liftable subalgebra is a proper subalgebra. For all parameter values the nilradical is liftable. This is conjectured to be true for arbitrary weighted-homogeneous isolated hypersurface singularities. Unlike the example of the \(\tilde E_ 6\) singularities, the isomorphism class of the nilradical is an analytic invariant of the singularity. This is shown by computing the cross-ratio of four invariant lines in a 2-dimensional subquotient of the nilradical.