Contents Online
Mathematical Research Letters
Volume 29 (2022)
Number 2
New $k$-th Yau algebras of isolated hypersurface singularities and weak Torelli-type theorem
Pages: 455 – 486
DOI: https://dx.doi.org/10.4310/MRL.2022.v29.n2.a7
Authors
Abstract
Let $V$ be a hypersurface with an isolated singularity at the origin defined by the holomorphic function $f : (\mathbb{C}^n, 0) \to (\mathbb{C}, 0)$. The Yau algebra $L(V)$ is defined to be the Lie algebra of derivations of the moduli algebra $A(V) := \mathcal{O}_n / (f, \frac{\partial f}{\partial x_1}, \dotsc, \frac{\partial f}{\partial x_n})$, i.e., $L(V)=\operatorname{Der}(A(V),A(V))$. It is known that $L(V)$ is a finite dimensional Lie algebra and its dimension $\lambda (V)$ is called Yau number. In this paper, we introduce a new series of Lie algebras, i.e., k-th Yau algebras $L^k(V), k \geq 0$, which are a generalization of Yau algebra. $L^k(V)$ is defined to be the Lie algebra of derivations of the k-th moduli algebra $A^k(V) := \mathcal{O}_n / (f, m^k J(f)), k \geq 0$, i.e., $L^k(V) =\operatorname{Der}(A^k(V), A^k(V))$, where $m$ is the maximal ideal of $\mathcal{O}_n$. The k-th Yau number is the dimension of $L^k(V)$ which we denote as $\lambda^k(V)$. In particular, $L^0(V)$ is exactly the Yau algebra, i.e., $L^0(V) = L(V), \lambda^0 (V) = \lambda (V)$. These numbers $\lambda^k(V)$ are new numerical analytic invariants of singularities. In this paper we obtain the weak Torelli-type theorems of simple elliptic singularities using Lie algebras $L^1(V)$ and $L^2(V)$. We shall also characterize the simple singularities completely using $L^1(V)$.
Both Yau and Zuo are supported by NSFC Grants 11961141005, 11531007. Zuo is supported by NSFC Grant 12271280 and Tsinghua University Initiative Scientific Research Program. Yau is supported by Tsinghua University Education Foundation fund(042202008) and the start-up fund from Tsinghua University.
Received 24 February 2020
Accepted 29 July 2020
Published 29 September 2022