On two inequality conjectures for the k-th Yau numbers of isolated hypersurface singularities

Abstract

Let (V, 0) be an isolated hypersurface singularity defined by the holomorphic function $$f: ({\mathbb {C}}^n, 0)\rightarrow ({\mathbb {C}}, 0)$$ . In our previous work, we introduced a series of novel Lie algebras associated to (V, 0), i.e., k-th Yau algebra $$L^k(V), k\ge 0$$ . It was defined to be the Lie algebra of derivations of the k-th moduli algebras $$A^k(V)= {\mathcal {O}}_n/(f, m^k J(f)), k\ge 0$$ , where m is the maximal ideal of $${\mathcal {O}}_n$$ . I.e., $$L^k(V):=\text {Der}(A^k(V), A^k(V))$$ . The dimension of $$L^k(V)$$ was denoted by $$\lambda ^k(V)$$ . The number $$\lambda ^k(V)$$ , which was called k-th Yau number, is a subtle numerical analytic invariant of (V, 0). Furthermore, we formulated two conjectures for these k-th Yau number invariants: a sharp upper estimate conjecture of $$\lambda ^k(V)$$ for weighted homogeneous isolated hypersurface singularities (see Conjecture 1.2) and an inequality conjecture $$\lambda ^{(k+1)}(V) > \lambda ^k(V), k\ge 0$$ (see Conjecture 1.1). In this article, we verify these two conjectures when k is small for large class of singularities.