Cohomology of submanifold complements

Question

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 $1$.

Answer 1

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, 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. 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.

Answer 2

Regarding counterexamples, it's easy enough to construct them by taking a smooth quasiprojective variety $M$ where 1, 2 or 3 fails. Choose projective compactification $X'$ of $X$, then by resolution of singularities, we can replace it by a nonsingular compactification $X$ so that $X-M$ is divisor with simple normal crossings.

For an explicit counterexample to 2, let $M$ be a nontrivial $\mathbb{C}^*$-bundle over an elliptic curve. The fundamental group is the Heisenberg group, so it won't be ($1$-)formal. For the others, you can even assume that $M$ is projective, so take an Enriques surface. Here $H^1=0$, but $H^2$ has $2$-torsion, so 1 and 3 both fail.