On the new k-th yau algebras of isolated hypersurface singularities

NAVEED HUSSAIN, Tsinghua University Stephen S.-T. Yau Tsinghua University Huaiqing Zuo Tsinghua University

mathscidoc:1903.01003

Mathematische Zeitschrift, 2019.1
Let V be a hypersurface with an isolated singularity at the origin defined by the holomorphic function f : (Cn , 0) → (C, 0). The Yau algebra L(V ) is defined to be the Lie algebra of derivations of the moduli algebra A(V ) := On/(f, ∂f , · · · , ∂f ), ∂x1 ∂xn i.e., L(V ) = Der(A(V ), A(V )) and plays an important role in singularity theory. It is known that L(V ) is a finite dimensional Lie algebra and its dimension λ(V ) is called Yau number. In this article, we generalize the Yau algebra and introduce a new series of k-th Yau algebras Lk(V) which are defined to be the Lie algebras of derivations of the moduli algebras Ak(V) = On/(f,mkJ(f)),k ≥ 0, i.e., Lk(V) = Der(Ak(V),Ak(V)) and where m is the maximal ideal of On. In particular, it is Yau algebra when k = 0. The dimension of Lk(V ) is denoted by λk(V ). These numbers i.e., k-th Yau numbers λk(V ), are new numerical analytic invariants of an isolated singularity. In this paper we studied these new series of Lie algebras Lk(V ) and also compute the Lie algebras L1(V ) for fewnomial isolated singularities. We also formulate a sharp upper estimate conjecture for the λk(V ) of weighted homogeneous isolated hypersurface singularities and we prove this conjecture in case of k = 1 for large class of singularities.
Isolated hypersurface singularity · Lie algebra · Moduli algebra
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@inproceedings{naveed2019on,
  title={ON THE NEW k-TH YAU ALGEBRAS OF ISOLATED HYPERSURFACE SINGULARITIES},
  author={NAVEED HUSSAIN,, Stephen S.-T. Yau, and Huaiqing Zuo},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20190311162009610824198},
  booktitle={Mathematische Zeitschrift},
  year={2019},
}
NAVEED HUSSAIN,, Stephen S.-T. Yau, and Huaiqing Zuo. ON THE NEW k-TH YAU ALGEBRAS OF ISOLATED HYPERSURFACE SINGULARITIES. 2019. In Mathematische Zeitschrift. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20190311162009610824198.
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