Deformation of Singularities via L∞-Algebras

This is an addendum to the paper “Deformation of L∞-Algebras” [9]. We explain in which way the deformation theory of L∞-algebras extends the deformation theory of singularities. We show that the construction of semi-universal deformations of L∞-algebras gives explicit formal semiuniversal deformations of isolated singularities. Introduction In this paper, we apply the following general idea for the construction of moduli spaces to isolated singularities: Take the differential graded Lie algebra L describing a deformation problem (for isolated singularities, this is the tangent complex) and find a minimal representative M of L in the class of formal L∞algebras (see [9]). In geometric terms, M is a formal DG-manifold, containing the moduli space as analytic substructure. This general concept is also sketched in [7]. We define a functor F from the category of complex analytic space germs to the localization of the category of L∞-algebras by L∞-equivalence. For a singularity X , we take the semi-universal L∞-deformation (V,Q V ) of F (X) constructed in [9]. For isolated singularities, the components V i are of finite dimension. The restriction of the vectorfield Q defines a formal map (Kuranishimap) V 0 −→ V 1 whose zero locus gives the formal moduli space. 1 Definitions and reminders In the whole paper, we work over a ground field k of characteristic zero. Denote the category of formal (resp. convergent) complex analytic space germs by Anf (resp. An). Denote the category of isomorphism classes of formal DG manifolds by DG-Manf. We use the following superscripts to denote full subcategories of DG-Manf: L (“local”): the subcategory of all (M,Q ) in DG-Manf such that Q0 = 0; M (“minimal”): the subcategory of all (M,Q ) in DG-Manf such that Q1 = 0; G (“g-finite”): the subcategory of all (M,Q ) in DG-Manf such thatH(M,Q1 ) ∗Supported by: Doktorandenstipendium des Deutschen Akademischen Austauschdienstes im Rahmen des gemeinsamen Hochschulsonderprogramms III des Bundes und der Länder