On the monodromy conjecture for determinantal varieties
Yifan Chen, Huaiqing Zuo
TL;DR
This work proves the monodromy conjecture for determinantal varieties by adapting a generalized framework based on jet schemes, arc spaces, and motivic zeta functions, and by deriving a detailed A'Campo-type formula for monodromy in the ideal setting via Verdier monodromy. Central to the argument is a careful analysis of multiple contact loci and a complete orbit classification in the arc space of determinantal varieties, which yields explicit expressions for the monodromy zeta function at points on the exceptional divisor. The main results show that the poles of the motivic zeta function correspond to monodromy eigenvalues for determinantal varieties (Theorem A) and provide an explicit description of the monodromy zeta function values (Theorem B); as applications, the Holomorphy conjecture is established (Theorem C) and Brill-Noether loci follow locally from the determinantal structure. Together, these results extend the monodromy framework to a broad class of singularities and connect intricate geometric invariants with arithmetic zeta data, with implications for Brill-Noether theory and beyond.
Abstract
This paper presents a proof of the monodromy conjecture for determinantal varieties. Our strategy centers on an in-depth analysis of monodromy zeta functions, leveraging a generalized A'Campo formula, an examination of multiple contact loci, and the exploitation of the intrinsic symmetric structures inherent to these varieties. Furthermore, we prove the holomorphy conjecture for determinantal varieties and the monodromy conjecture for Brill-Noether loci of generic curves. Keywords. monodromy conjecture, determinantal varieties, monodromy zeta function.