On the D-dimension of a certain type of threefolds

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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).