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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 \(L(V)\).

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