科研之友
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摘要
Let f : (C-n, 0) -> (C, 0) be a germ of a complex analytic function with an isolated critical point at the origin. Let V = {z is an element of C-n : f (z) = 0}. A beautiful theorem of Saito [1971] gives a necessary and sufficient condition for V to be defined by a weighted homogeneous polynomial. It is a natural and important question to characterize (up to a biholomorphic change of coordinates) a homogeneous polynomial with an isolated critical point at the origin. For a two-dimensional isolated hypersurface singularity V, Xu and Yau [1992; 1993] found a coordinate-free characterization for V to be defined by a homogeneous polynomial. Lin and Yau [2004] and Chen, Lin, Yau, and Zuo [2001] gave necessary and sufficient conditions for 3- and 4-dimensional isolated hypersurface singularities with p(g) >= 0 and p(g) > 0, respectively. However, it is quite difficult to generalize their methods to give characterization of homogeneous polynomials. In 2005, Yau formulated the Yau Conjecture 1.1: (1) Let mu and nu be the Milnor number and multiplicity of (V, 0), respectively. Then mu >= (nu - 1)(n), and the equality holds if and only if f is a semihomogeneous function. (2) If f is a quasihomogeneous function, then mu = (nu - 1)(n) if and only if f is a homogeneous polynomial after change of coordinates. In this paper we solve part (1) of Yau Conjecture 1.1 for general n. We introduce a new method, which allows us to solve the part (2) of Yau Conjecture 1.1 for n = 5 and 6. As a result we have shown that for n D 5 or 6, f is a homogeneous polynomial after a biholomorphic change of coordinates if and only if mu = tau = (nu - 1)(n). As a by-product we have also proved Yau Conjecture 1.2 in some special cases.
- 出版日期2012-11
- 单位清华大学
