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Micro-local characterization of quasi-homogeneous singularities. (English) Zbl 0927.32022
Summary: The moduli algebra $$A(V)$$ of a hypersurface singularity $$(V,0)$$ is a finite-dimensional $$\mathbb{C}$$-algebra. In 1982, Mather and Yau proved that two germs of complex-analytic hypersurfaces of the same dimension with isolated singularities are biholomorphically equivalent if and only if their moduli algebras are isomorphic. It is a natural question to ask for a necessary and sufficient condition for a complex-analytic isolated hypersurface singularity to be quasi-homogeneous in terms of its moduli algebra.
In this paper, we prove that $$(V,0)$$ admits a quasi-homogeneous structure if and only if its moduli algebra is isomorphic to a finite-dimensional nonnegatively graded algebra.
In 1983, Yau introduced the finite-dimensional Lie algebra $$L(V)$$ of an isolated hypersurface singularity $$(V,0)$$. $$L(V)$$ is defined to be the algebra of derivations of the moduli algebra $$A(V)$$ and is finite-dimensional.
We prove that $$(V,0)$$ is a quasi-homogeneous singularity if (1) $$L(V)$$ is isomorphic to a nonnegatively graded Lie algebra without center, (2) there exists $$E$$ in $$L(V)$$ of degree zero such that $$[E,D_i]= iD_i$$ for any $$D_i$$ in $$L(V)$$ of degree $$i$$, and (3) for any element $$a\in m-m^2$$, where $$m$$ is the maximal ideal of $$A(V)$$, $$aE$$ is not in the degree zero part of $$L(V)$$.

##### MSC:
 32S25 Complex surface and hypersurface singularities 17B70 Graded Lie (super)algebras 14J17 Singularities of surfaces or higher-dimensional varieties
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