The Orlik-Terao algebra and 2-formality

The Orlik-Solomon algebra is the cohomology ring of the complement of a hyperplane arrangement $\A \subseteq \C^n$; it is the quotient of an exterior algebra $\Lambda(V)$ on $|\A|$ generators. In \cite{ot1}, Orlik and Terao introduced a commutative analog $Sym(V^*)/I$ of the Orlik-Solomon algebra to answer a question of Aomoto and showed the Hilbert series depends only on the intersection lattice $L(\A)$. In \cite{fr}, Falk and Randell define the property of 2-formality; in this note we study the relation between 2-formality and the Orlik-Terao algebra. Our main result is a necessary and sufficient condition for 2-formality in terms of the quadratic component $I_2$ of the Orlik-Terao ideal $I$. The key is that 2-formality is determined by the tangent space $T_p(V(I_2))$ at a generic point $p$.

Full Text (PDF format)

Published 1 January 2009