A remark on lower bound of Milnor number and characterization of homogeneous hypersurface singularities

We consider maximal estimates for the solution to an initial value problem of the linear Schrödinger equation with a singular potential. We show a result about the pointwise convergence of solutions to this special variable coefficient Schrödinger equation with initial data u 0 2 H s ‫ޒ.‬ n / for s > 1 2 or radial initial data u 0 2 H s ‫ޒ.‬ n / for s > 1 and that the solution does not converge when s < 1 4 .

Section: Proof Of Theoremmentioning

confidence: 99%

Untitled

2015

“…Yau Conjecture 1.1 [Lin et al 2006b]. Let f : ‫ރ(‬ n , 0) → ‫,ރ(‬ 0) be a holomorphic germ defining an isolated hypersurface singularity V = { z : f (z) = 0 } at the origin.…”

Section: Based On Above Theorem a Conjecture Was Made By Yau In 2005mentioning

confidence: 99%

“…The Yau conjectures were proved only for very low dimensional singularities. For Yau Conjecture 1.1, Lin, Wu, Yau, and Luk proved the following two theorems: Theorem 1.3 [Lin et al 2006b]. Let f : ‫ރ(‬ 2 , 0) → ‫,ރ(‬ 0) be a germ of a holomorphic function defining an isolated plane curve singularity V = { z ∈ ‫ރ‬ 2 : f (z) = 0 } at the origin.…”

Section: Based On Above Theorem a Conjecture Was Made By Yau In 2005mentioning

confidence: 99%

Lower estimate of Milnor number and characterization of isolated homogeneous hypersurface singularities

Yau¹,

Zuo²

2012

Let f : ‫ރ(‬ n , 0) → ‫,ރ(‬ 0) be a germ of a complex analytic function with an isolated critical point at the origin. Let V = { z ∈ ‫ރ‬ 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. 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 µ and ν be the Milnor number and multiplicity of (V, 0), respectively. Then µ ≥ (ν−1) n , and the equality holds if and only if f is a semihomogeneous function. (2) If f is a quasihomogeneous function, then µ = (ν − 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 = 5 or 6, f is a homogeneous polynomial after a biholomorphic change of coordinates if and only if µ = τ = (ν − 1) n . As a by-product we have also proved Yau Conjecture 1.2 in some special cases.

Section: Introductionmentioning

confidence: 99%

Complete characterization of isolated homogeneous hypersurface singularities

Yau

Zuo

2015

Dedicated to Professor Michael Artin on the occasion of his 80th birthday Let X be a nonsingular projective variety in ‫ސރ‬ n−1. Then the cone over X in ‫ރ‬ n is an affine variety V 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 an 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 1971 paper by Saito. However, the latter question had remained open for 40 years until Xu and 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 are hard to generalize. 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 µ = τ = (ν − 1) n , where µ, τ and ν are the Milnor number, Tjurina number and multiplicity of the singularity respectively. We also prove that there are at most µ 1/n + 1 multiplicities within the same topological type of the isolated hypersurface singularity, while the famous Zariski multiplicity problem asserts that there is only one multiplicity.