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.
