# Cohomology of the complement of the resonance hyperplane arrangement

Here was a question about resonance arrangement. It is defined as follows.

Let $$x_i$$ be the standard coordinates on $$\mathbb{C}^n$$. For each nonempty $$I\subseteq\{1,\dots,n\}$$, define the hyperplane $$H_I$$ to be the hyperplane given by $$l_I:=\sum _{i∈I}x_i=0$$.

My question is about the top degree (co)homology of the complement to this arrangement.

I have the following conjecture: the space of top degree forms, which are wedge products of forms $$d \log l_I$$, is generated by tree-forms.

A tree form is defined by a rooted tree with $$n$$ edges labeled by $$x_i$$'s. Any (non-root) vertex gives a linear function on $$\mathbb{C}^n$$, which is the sum of all labels on edges, lying on the shortest path from the root to the vertex. The wedge product of $$d\log$$'s of these linear functions for all vertices is the tree form. By the way, is it a known object?

I would deeply appreciate any comments.

• The resonance arrangement is notoriously difficult to understand in any nice way. One of the most recent papers about it I know of is arxiv.org/abs/1903.06595. (I don't think it talks about cohomology of complement of complex arrangement at all- but of course # of regions of complement of real arrangement is a closely related thing, which again is not well understood.) Sep 23, 2020 at 16:57
• For something even more recent (and in line with Sam Hopkins's comment), see arxiv.org/pdf/2008.10553.pdf
– user35313
Sep 24, 2020 at 2:07
• @SamHopkins Thank you for your comment and the reference. Sep 24, 2020 at 16:56
• @user61318 Thank you for the reference. Sep 24, 2020 at 16:57
• The cohomology of hyperplane arrangements is described by the Orlik-Solomon algebra which has exactly the dlog forms as basis. Section 3 of the book of Orlik and Terao "Arrangements of hyperplanes" has a description of the "no broken circuits" basis of the top degree. I suppose that means the question is about expressing the "no broken circuits" basis in terms of your "tree form" basis. Oct 11, 2020 at 19:40