Abstract
Let V be a hypersurface with an isolated singularity at the origin in ℂn+1. It is a natural question to ask when V is defined by weighted homogeneous polynomial or homogeneous polynomial up to biholomorphic change of coordinates. In 1971, a beautiful theorem of Saito gives a necessary and sufficient condition for V to be defined by a weighted homogeneous polynomial. For a two-dimensional isolated hypersurface signularity V, Xu and Yau found a coordinate free characterization for V to be defined by a homogeneous polynomial. Recently Lin and Yau gave necessary and sufficient conditions for a 3-dimensional isolated hypersurface singularity with geometric genus bigger than zero to be defined by a homogeneous polynomial. The purpose of this paper is to prove that Lin-Yau’s theorem remains true for singularities with geometric genus equal to zero.
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Dedicated to Professor Sheng GONG on the occasion of his 75th birthday
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Lin, K., Tu, Z. & Yau, S.S.T. Characterization of isolated homogeneous hypersurface singularities in ℂ4 . SCI CHINA SER A 49, 1576–1592 (2006). https://doi.org/10.1007/s11425-006-2062-9
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DOI: https://doi.org/10.1007/s11425-006-2062-9