Editing Arrangement of hyperplanes (section)

==Complex arrangements==

In [[complex number|complex]] affine space (which is hard to visualize because even the complex affine plane has four real dimensions), the complement is connected (all one piece) with holes where the hyperplanes were removed.

A typical problem about an arrangement in complex space is to describe the holes.

The basic theorem about complex arrangements is that the [[cohomology]] of the complement ''M''(''A'') is completely determined by the intersection semilattice.  To be precise, the cohomology ring of ''M''(''A'') (with integer coefficients) is [[isomorphic]] to the Orlik–Solomon algebra on '''Z'''.

The isomorphism can be described explicitly and gives a presentation of the cohomology in terms of generators and relations, where generators are represented (in the [[de Rham cohomology]]) as logarithmic [[differential form]]s

:<math>\frac{1}{2\pi i}\frac{d\alpha}{\alpha}.</math>

with <math>\alpha</math> any linear form defining the generic hyperplane of the arrangement.