Let $X$ be a finite-dimensional complex manifold (possibly non-compact). Let $\mathcal{H}$ be a union of codimension-$1$ submanifolds such that the local picture is that of intersecting hyperplanes. I am interested in $M$ the complement of $\mathcal{H}$ inside $X$.

The well-known case, that I am aware of, is when $M$ is the complement of hyperplanes inside $\mathbb{C}^n$. The following is true for $H^*(M)$ (the cohomology of $M$):

- it is generated in degree $1$ (by logarithmic differentials),
- the algebra is formal
- and the integral cohomology is torsion free.

I would like to know of other examples that are similar to hyperplane complements in the above respect. Counterexamples would also help; in particular, I am interested in knowing the cases in which cohomology is not generated in degree $1$.