Fourier transform on hyperplane arrangements

TL;DR

This paper explores the application of Fourier transform to hyperplane arrangements by describing perverse sheaves via hyperbolic sheaves and computing key sheaf operations in this framework.

Contribution

It provides a detailed calculation of sheaf-theoretic operations like vanishing cycles and Fourier-Sato transform using hyperbolic sheaves for hyperplane arrangements.

Findings

Explicit formulas for vanishing cycles and specialization.

Calculation of Fourier-Sato transform in hyperbolic sheaf terms.

Enhanced understanding of sheaf operations on hyperplane arrangements.

Abstract

We consider the category of perverse sheaves on a complex vector space smooth with respect to a stratification given by an arrangement of hyperplanes with real equations. As shown in an earlier wotk of two of the authors, this category can be described in terms of certain diagrams of vector spaces labelled by all the faces of the real arrangement (we call such diagrams hyperbolic sheaves). In this paper we calculate, in these terms, several fundamental operations of sheaf theory such as forming the space of vanishing cycles, specialization and the Fourier-Sato transform.