The $μ$-Trace System

Abstract

We study a simple 1-parameter perturbation of the regular holonomic Trace System satisfied by a complex power of the root of the universal polynomial of degree k as a holomorphic function of the coefficients. We prove that these systems have many analogous properties than the Trace System studied in [4] and we prove that they are, in general, minimal extensions of a simple pole meromorphic connection on a rank $k$ trivial bundle on $\mathbb{C}^k$. We also examine the structure of these $D$-modules for the special values of the parameters. This explicites many examples of perverse sheaves associated to representations of the $π_1$ of the complement of the hyper-surface $\{σ_kΔ(σ) = 0\}$ in the affine space with coordinates $σ_1,\ldots,σ_k$, where $Δ(σ)$ is the discriminant of the universal monic polynomial of degree $k$, $P_σ(z) := z^k + \sum_{h=1}^k (-1)^h σ_h z^{k-h}$.