The Topological Monodromy Conjecture and its strong form

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Let ff be a divisor on a smooth complex variety, let Zf(s)Z_f(s) be its topological zeta function, and let bf(s)b_f(s) be the Bernstein–Sato polynomial. A pole of Zf(s)Z_f(s) is a complex number at which the rational function Zf(s)Z_f(s) has a pole.

Topological Monodromy Conjecture. Any pole of Zf(s)Z_f(s) yields under exponentiation an eigenvalue of the monodromy operator at some p∈Var⁡(f)p\in\operatorname{Var}(f). The strong form asserts that any pole of Zf(s)Z_f(s) is a root of bf(s)b_f(s).

These are the monodromy and strong monodromy conjectures for the topological zeta function. The source states them as conjectural claims, without supplying resolution evidence.

References

Primary source

Anton Leykin and Uli Walther, “Survey on the D-module f^s”, arXiv:1504.07516 (2015).

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