The monodromy conjecture for coherent ideal sheaves

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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

e2π−1s0∈E(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.

References

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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