On the Dimension of a New Class of Derivation Lie Algebras Associated to Singularities

Naveed Hussain, Stephen S.-T. Yau and Huaiqing Zuo
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Naveed Hussain: School of Data Sciences, Guangzhou Huashang College, Guangzhou 511300, China
Stephen S.-T. Yau: Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China
Huaiqing Zuo: Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China

Mathematics, 2021, vol. 9, issue 14, 1-15

Abstract: Let ( V , 0 ) = { ( z 1 , … , z n ) ? C n : f ( z 1 , … , z n ) = 0 } be an isolated hypersurface singularity with m u l t ( f ) = m . Let J k ( f ) be the ideal generated by all k -th order partial derivatives of f . For 1 ? k ? m ? 1 , the new object L k ( V ) is defined to be the Lie algebra of derivations of the new k -th local algebra M k ( V ) , where M k ( V ) : = O n / ( ( f ) + J 1 ( f ) + … + J k ( f ) ) . Its dimension is denoted as ? k ( V ) . This number ? k ( V ) is a new numerical analytic invariant. In this article we compute L 4 ( V ) for fewnomial isolated singularities (binomial, trinomial) and obtain the formulas of ? 4 ( V ) . We also verify a sharp upper estimate conjecture for the ? 4 ( V ) for large class of singularities. Furthermore, we verify another inequality conjecture: ? ( k + 1 ) ( V ) < ? k ( V ) , k = 3 for low-dimensional fewnomial singularities.

Keywords: isolated hypersurface singularity; Lie algebra; local algebra (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
References: View complete reference list from CitEc
Citations: View citations in EconPapers (1)

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