kTH YAU NUMBER OF ISOLATED HYPERSURFACE SINGULARITIES AND AN INEQUALITY CONJECTURE. (30th April 2019) - HUSSAIN, NAVEED; YAU, STEPHEN S.-T.; ZUO, HUAIQING
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, [STIX]x2202f/[STIX]x2202x_1, …, [STIX]x2202f/[STIX]x2202x_n), that is, L(V)=Der(A(V), A(V)) . It is known that L(V) is finite dimensional and its dimension [STIX]x1D706(V) is called the Yau number. We introduce a new series of Lie algebras, that is, k th 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 k th moduli algebra A^k(V):=O_n/(f, m^kJ(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 k th Yau number is the dimension of L^k(V), which we denote by [STIX]x1D706^k(V) . In particular, L^0(V) is exactly the Yau algebra, that is, L^0(V)=L(V), [STIX]x1D706^0(V)=[STIX]x1D706(V) . These numbers [STIX]x1D706^k(V) are new numerical analytic invariants of singularities. In this paper we formulate a conjecture that [STIX]x1D706^(k+1)(V)>[STIX]x1D706^k(V), k≥ 0. We prove this conjecture for a large class of singularities.