The $μ$-Trace System
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866914616216387584 |
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| author | Barlet, Daniel |
| author_facet | Barlet, Daniel |
| contents | 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 $k$, $P_σ(z) := z^k + \sum_{h=1}^k (-1)^h σ_h z^{k-h}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_31019 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The $μ$-Trace System Barlet, Daniel Algebraic Geometry Complex Variables 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 $k$, $P_σ(z) := z^k + \sum_{h=1}^k (-1)^h σ_h z^{k-h}$. |
| title | The $μ$-Trace System |
| topic | Algebraic Geometry Complex Variables |
| url | https://arxiv.org/abs/2605.31019 |