Arrangements and Frobenius like structures

Varchenko, Alexander

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 24 (2015) no. 1, pp. 133-204.

On considère une famille d’arrangements pondérés génériques de $n$ hyperplans dans $C^k$ et montre que la connexion de Gauss - Manin pour les intégrales hypergéométriques associées, la forme contravariante sur l’espace des vecteurs singuliers et l’algébre de fonctions sur l’ensemble des points critiques définissent une structure du type Frobenius sur la base de la famille. Comme un résultat de cette construction nous montrons que les éléments matriciels des opérateurs linéaires de la connexion de Gauss - Manin sont donnés par les $(2k+1)$-mes dérivées d’une seule fonction sur la base de la famille, cf. la formule (6.46).

We consider a family of generic weighted arrangements of $n$ hyperplanes in ${ℂ}^k$ and show that the Gauss-Manin connection for the associated hypergeometric integrals, the contravariant form on the space of singular vectors, and the algebra of functions on the critical set of the master function define a Frobenius like structure on the base of the family. As a result of this construction we show that the matrix elements of the linear operators of the Gauss-Manin connection are given by the $2k+1$-st derivatives of a single function on the base of the family, the function called the potential of second kind, see formula (6.46).

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