Moderate and rapid-decay nearby cycles for holonomic D-modules

Abstract

We introduce the notion of moderate and rapid decay nearby cycles relative to a holomorphic function $f$ for an arbitrary holonomic $\mathcal{D}$-module. They are proved to be $\mathbb{R}$-constructible complexes on the product of the special fiber of the function and the circle $S^1$ parametrizing the values of $f/|f|$. Duality properties are proved by B.\,Hepler and A.\,Hohl [HH25], and we also prove them in special cases. Relations with the irregularity complexes as defined by Z.\,Mebkhout are given. Most of the arguments rely on the results of T.\,Mochizuki [Moc14].