New $k$-th Yau algebras of isolated hypersurface singularities and weak Torelli-type theorem

Abstract

Let V be a hypersurface with an isolated singularity at the origin defined by the holomorphic function f : ( C n , 0) → ( C , 0). The Yau algebra L ( V ) is defined to be the Lie algebra of derivations of the moduli algebra A ( V ) := O n / ( f, ∂f∂x 1 , · · · , ∂f∂x n ), i.e., L ( V ) = Der( A ( V ) , A ( V )). It is known that L ( V ) is a finite dimensional Lie algebra and its dimension λ ( V ) is called Yau number. In this paper, we introduce a new series of Lie algebras, i.e., k -th Yau algebras L k ( V ), k ≥ 0, which are a generalization of Yau algebra. L k ( V ) is defined to be the Lie algebra of derivations of the k th moduli algebra A k ( V ) := O n / ( f, m k J ( f )) , k ≥ 0, i.e., L k ( V ) = Der( A k ( V ) , A k ( V )), where m is the maximal ideal of O n . The k -th Yau number is the dimension of L k ( V ) which we denote as λ k ( V ). In particular, L 0 ( V ) is exactly the Yau algebra, i.e., L 0 ( V ) = L ( V ) , λ 0 ( V ) = λ ( V ). These numbers λ k ( V ) are new numerical analytic invariants of singularities. In this paper we obtain the weak Torelli-type theorems of simple elliptic singularities using Lie algebras L 1 ( V ) and L 2 ( V ). We shall also characterize the simple singularities completely using L 1 ( V ).