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

Naveed Hussain Huashang College Guangdong University of Finance and Economics Stephen Yau Tsinghua University Huaiqing Zuo Tsinghua University

Algebraic Geometry mathscidoc:2005.45001

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 $L^k(V), k\geq 0$. It was defined to be the Lie algebra of derivations of the $k$-th moduli algebras $A^k(V)= \mathcal{O}_n/(f, m^k J(f)), k\geq 0$, 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 \ref{conj2}) and an inequality conjecture $\lambda^{(k+1)}(V) > \lambda^k(V), k\geq0$ (see Conjecture \ref{conj1}). In this article, we verify these two conjectures when $k$ is small for large class of singularities.
Derivation, Lie algebra, isolated singularity, Yau algebra.
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@inproceedings{naveedon,
  title={On  two inequality conjectures for the $k$-th Yau numbers of  isolated hypersurface singularities},
  author={Naveed Hussain, Stephen Yau, and Huaiqing Zuo},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20200520184830405133687},
}
Naveed Hussain, Stephen Yau, and Huaiqing Zuo. On two inequality conjectures for the $k$-th Yau numbers of isolated hypersurface singularities. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20200520184830405133687.
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