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 $M$): 


1. it is generated in degree $1$ (by logarithmic differentials),
2. the algebra is formal
3. 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 $1$. 



Cohomology of submanifold complements
If one moves from complements of hyperplanes arrangements to complements of hypersurfaces in $\mathbb{C}^n$, some of the above properties may fail. For instance, if $C$ is a plane curve, then the rational cohomology ring of its complement is not necessarily generated in degree $1$, see for instance José Ignacio Cogolludo-Agustín, *Topological invariants of the complement to arrangements of rational plane curves*, Memoirs of the Amer. Math. Soc. 159 (2002), no. 756, [MR 1921584](http://www.ams.org/mathscinet-getitem?mr=1921584), as well as Cogolludo and Daniel Matei, *Cohomology algebra of plane curves, weak combinatorial type, and formality*, Trans. Amer Math Soc. 364 (2012), no. 11, 5765–5790, [MR 2946931](http://www.ams.org/mathscinet-getitem?mr=2946931). I would have to think whether this can be done while having all the singularities to be arrangement-like, as you ask, but my hunch is that it can happen even then.


