kTH YAU NUMBER OF ISOLATED HYPERSURFACE SINGULARITIES AND AN INEQUALITY CONJECTURE

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, partial derivative f/partial derivative x(1), . . . , partial derivative f/partial derivative x(n)), that is, L(V) = Der(A(V); A(V)). It is known that L(V) is finite dimensional and its dimension lambda(V) is called the Yau number. We introduce a new series of Lie algebras, that is, kth Yau algebras L-k(V), k >= 0, which are a generalization of the Yau algebra. The algebra L-k(V) is defined to be the Lie algebra of derivations of the kth moduli algebra A(k)(V) :=O-n/(f, m(k) J(f)), k >= 0, that is, L-k(V) = Der(A(k)(V), A(k)(V)), where m is the maximal ideal of O-n. The kth Yau number is the dimension of L-k(V), which we denote by lambda(k)(V). In particular, L-0(V) is exactly the Yau algebra, that is, L-0(V) = L(V); lambda(0)(V) = lambda(V). These numbers lambda(k)(V) are new numerical analytic invariants of singularities. In this paper we formulate a conjecture that lambda((k+1))(V) > lambda(k)(V), k >= 0. We prove this conjecture for a large class of singularities.