# Holomorphic tubular neighborhood of divisors at infinity

For the discussion of holomorphic tubular neighborhoods and some criteria for their existence see this question.

Let $$X$$ be a smooth quasi-projective variety over $$\mathbb{C}$$. Hironaka tells us that there exists a smooth projective compactification $$X \subset \bar{X}$$, such that $$D =\bar{X}\backslash X$$ is a strictly normal crossing divisor. Let $$D=\cup_{i}^N D_i$$ be its decomposition into irreducible smooth divisors. What is the obstruction to finding such $$\bar{X}$$, such that each $$D_i$$ has a holomorphic tubular neighborhood in $$\bar{X}$$?

Example 1: Let $$S$$ be a smooth projective variety, $$E$$ a vector bundle on $$S$$. The space $$X=\text{Tot}(E)$$ is compactified to $$\bar{X}=\mathbb{P}_S(E\oplus \mathcal{O}_S)$$ with the divisor at infinity $$D =\mathbb{P}_S(E)\subset\bar{X}$$. Its normal bundle should be $$\mathcal{O}_{\mathbb{P}(E)}(1)$$ and one should be able to find a holomorphic tubular neighborhood by constructing one in each fiber and gluing together.

Example 2: Using the answer by Joey, I have convinced myself that if the divisor $$D_i$$ is $$\mathbb{P}^n$$ for some $$n\geq 2$$ and $$n+1=\text{dim}(X)$$, then it also has such a neighborhood. If $$n=1$$, then one needs additionally that $$(D_i)^2<0$$.

The above examples make me believe that it could be always achievable, as each blow up at a smooth point introduces a new $$\mathbb{P}^{n}$$ with normal bundle $$\mathcal{O}(-1)$$.In particular, blowing up seems to only help with the existence of these neighborhoods.

Edit: What I am asking is: Does there exists such a procedure of blowing up the divisor at infinity, that leaves me at the end with $$D_i$$ having holomorphic tubular neighborhoods? I don't expect to understand all possible compactifications.