Let $X$ be a simply connected projective manifold of dimension $n$ over $\mathbb{C}$ and $D = \cup D_i$ be a divisor with normal crossings such that its all components $D_i$ are smooth and ample.

I am trying to follow the proof of Theorem 2.1 of http://arxiv.org/pdf/math/0404341v1.pdf, where the author claims the following.

"surjectivity of $π_i(D_i − \cup_{j\neq i} D_j) \to π_i(X − D)$ ... follows from ampleness of the components $D_i$."

Based upon the statement of theorem 2.1, I presume that the author means in the range $2 \leq i \leq n-1$.

There are a few things I don't understand about this statement.

1) The first is what is the map on $\pi_i$ that the author is proposing?

2) Second, what version of the Lefschetz hyperplane is the author invoking ? I know that for experts there are many Lefschetz hyperplane theorems (e.g. affine Lefschetz hyperplane theorem due to Hamm, etc.). Perhaps if one has a deep enough understanding of the theorem this result is trivial. For me, who just understands the classical version in say Milnor's book, this is not so clear. Can anyone please explain a bit more of what the author is asserting and from what results in the literature it follows?