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# Communications in Analysis and Geometry

## Volume 19 (2011)

### Number 4

### Coordinate-free characterization of homogeneous polynomials with isolated singularities

Pages: 661 – 704

DOI: http://dx.doi.org/10.4310/CAG.2011.v19.n4.a2

#### Authors

#### Abstract

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.