Communications in Analysis and Geometry

Volume 19 (2011)

Number 4

Coordinate-free characterization of homogeneous polynomials with isolated singularities

Pages: 661 – 704



Irene Chen (Massachusetts Institute of Technology, Cambridge, Massachusetts)

Ke-Pao Lin (Department of Information Management, Chang Gung University of Science and Technology, Taiwan)

Stephen Yau (Department of Mathematical Science, Tsinghua University, Beijing, China)

Huaiqing Zuo (Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago)


The Durfee conjecture, proposed in 1978, relates twoimportant invariants of isolated hypersurface singularitiesby a famous\break inequality; however, the inequality in thisconjecture is not sharp. In 1995, Yau announced hisconjecture which proposed a sharp inequality. The Yauconjecture characterizes the conditions under which anaffine hypersurface with an isolated singularity at theorigin is a cone over a nonsingular projectivehypersurface; in other words, the conjecture gives acoordinate-free characterization of when a convergent powerseries is a homogeneous polynomial after a biholomorphicchange of variables. In this paper, we have proved that if$p_{g}>0$, then $5!p_{g}\leq\mu-p(v)$,$p(v)=(v-1)^{5}-v(v-1)\dots(v-4)$ and $p_{g}$, $\mu$ and$v$ are, respectively, the geometric genus, the Milnornumber, and the multiplicity of the isolated singularity atthe origin of a weighted homogeneous polynomial. As aconsequence, we prove that the Yau conjecture holds for$n=5$ if $p_{g}>0$. In the process, we have also definedyet another sharp upper bound for the number of positiveintegral points within a five-dimensional simplex.

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