Micro-local characterization of quasi-homogeneous singularities. (English) Zbl 0927.32022

Xu, Yi-Jing; Yau, Stephen S.-T.

Am. J. Math. 118, No. 2, 389-399 (1996).

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

Keywords:

quasi-homogeneous singularities; moduli algebra; hypersurface singularity