On the D-dimension of a Certain Type of Threefolds

Let Y be an algebraic manifold of dimension 3 with H(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 H(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) 6= f(y2).