ncku-talk2013-08-01

NCTS(South)/ NCKU Math Colloquium

DATE 2013-08-01　16:00-17:00

PLACE R203, 2F, NCTS, NCKU

SPEAKER Professor Stephen Yau (丘成棟教授)（北京清華大學數學科學系）

TITLE Complete Characterization of Isolated Homogeneous Hypersurface

ABSTRACT Let X be a nonsingular projective variety in CP^(n-1). Then the cone over X in C^n is an affine variety with an isolated singularity at the origin. It is a very natural and important question to ask when an affine variety with an isolated singularity at the origin is a cone over nonsingular projective variety. This problem is very hard in general. In this paper we shall treat the hypersurface case. Given a function f with a isolated singularity at the origin. We can ask whether f is a weighted homogeneous polynomial or a homogeneous polynomial after a biholomorphic change of coordinates. The former question was answered in a celebrated paper by Saito in 1971. However, the latter question has remained open for 40 years until Xu-Yau solved it for f with three variables. Recently, Yau and Zuo solved it for f with up to six variables. However the methods they used is hard to be generalized. In this paper, we solve the latter question for general n completely, i.e., we show that f is a homogeneous polynomial after a biholomorphic change of coordinates if and only if Milnor number = Tjurina number = (multiplicity - 1)^n. We also prove that there are at most one plus nth root of Milnor number of multiplicities within the same topological type of the isolated hypersurface singularity, while the famous Zariski multiplicity problem asserts that there is only one multiplicity.

NCTS(South)/ NCKU Math Colloquium

DATE	2013-08-01　16:00-17:00

PLACE	R203, 2F, NCTS, NCKU

SPEAKER	Professor Stephen Yau (丘成棟教授)（北京清華大學數學科學系）

TITLE	Complete Characterization of Isolated Homogeneous Hypersurface

ABSTRACT	Let X be a nonsingular projective variety in CP^(n-1). Then the cone over X in C^n is an affine variety with an isolated singularity at the origin. It is a very natural and important question to ask when an affine variety with an isolated singularity at the origin is a cone over nonsingular projective variety. This problem is very hard in general. In this paper we shall treat the hypersurface case. Given a function f with a isolated singularity at the origin. We can ask whether f is a weighted homogeneous polynomial or a homogeneous polynomial after a biholomorphic change of coordinates. The former question was answered in a celebrated paper by Saito in 1971. However, the latter question has remained open for 40 years until Xu-Yau solved it for f with three variables. Recently, Yau and Zuo solved it for f with up to six variables. However the methods they used is hard to be generalized. In this paper, we solve the latter question for general n completely, i.e., we show that f is a homogeneous polynomial after a biholomorphic change of coordinates if and only if Milnor number = Tjurina number = (multiplicity - 1)^n. We also prove that there are at most one plus nth root of Milnor number of multiplicities within the same topological type of the isolated hypersurface singularity, while the famous Zariski multiplicity problem asserts that there is only one multiplicity.