2006/10/28 by Jing Zhang, Zhang, Jing
Mathematics · #14J30 #32Q28 #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Complex Variables (math.CV) #FOS: Mathematics #Meromorphic and Entire Functions
paper · pdf · doi:10.48550/arxiv.math/0610881
openalex publication_date 2006/10/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let Y be an algebraic manifold of dimension 3 with Hi(Y, ΩjY)=0 for all j≥ 0, i>0 and h0(Y, OY) > 1. Let X be a smooth completion of Y such that the boundary X-Y is the support of an effective divisor D on X with simple normal crossings. We prove that the D-dimension of X cannot be 2, i.e., either any two nonconstant regular functions are algebraically dependent or there are three algebraically independent nonconstant regular functions on Y. Secondly, if the D-dimension of X is greater than 1, then the associated scheme of Y is isomorphic to SpecΓ(Y, OY). Furthermore, we prove that an algebraic manifold Y of any dimension d≥ 1 is affine if and only if Hi(Y, ΩjY)=0 for all j≥ 0, i>0 and it is regularly separable, i.e., for any two distinct points y1, y2 on Y, there is a regular function f on Y such that f(y1)≠ f(y2).