The monodromy conjecture for coherent ideal sheaves

Let XX be a smooth irreducible complex variety and let I\mathcal I be a coherent ideal sheaf of XX with a non-empty zero locus. Let ZImot(T)Z_{\mathcal I}^{\mathrm{mot}}(T) be its motivic zeta function, and let E(I)E(\mathcal I) denote the set of Verdier monodromy eigenvalues of I\mathcal I. Monodromy conjecture. If s0s_0 is a pole of ZImot(T)Z_{\mathcal I}^{\mathrm{mot}}(T), then

e2π1s0E(I).e^{2\pi \sqrt{-1}s_0} \in E(\mathcal I).

This conjecture relates poles of motivic zeta functions to Verdier monodromy. The source presents it as an established conjecture; its resolution status is not specified in the supplied text.

Sources & referencesView supporting material

Primary source

Yifan Chen, Quan Shi, Yongxin Xu and Huaiqing Zuo, “On the monodromy conjecture, holomorphy conjecture, and embedded Nash problem for Pfaffian ideals”, arXiv:2511.01262 (2025).

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