Moderate Growth and Rapid Decay Nearby Cycles via Enhanced Ind-Sheaves

TL;DR

This paper develops a sheaf-theoretic framework for moderate growth and rapid decay objects related to holomorphic functions, connecting enhanced ind-sheaves with classical de Rham complexes and proving a conjecture on duality in irregular Riemann--Hilbert theory.

Contribution

It introduces a new sheaf-theoretic approach to moderate growth and rapid decay objects, proving Sabbah's duality conjecture for arbitrary divisors within the irregular Riemann--Hilbert correspondence.

Findings

Reconstruction of classical de Rham complexes via enhanced ind-sheaves.

Proof of Sabbah's duality conjecture for arbitrary divisors.

Recovery of the perfect pairing between algebraic de Rham cohomology and rapid decay homology.

Abstract

For any holomorphic function $f\colon X\to \mathbb{C}$ on a complex manifold $X$, we define and study moderate growth and rapid decay objects associated to an enhanced ind-sheaf on $X$. These will be sheaves on the real oriented blow-up space of $X$ along $f$. We show that, in the context of the irregular Riemann--Hilbert correspondence of D'Agnolo--Kashiwara, these objects recover the classical de Rham complexes with moderate growth and rapid decay associated to a holonomic $\mathcal{D}_X$-module. In order to prove the latter, we resolve a recent conjectural duality of Sabbah between these de Rham complexes of holonomic $\mathcal{D}_X$-modules with growth conditions along a normal crossing divisor by making the connection with a classic duality result of Kashiwara--Schapira between certain topological vector spaces. Via a standard d\'evissage argument, we then prove Sabbah's conjecture for arbitrary divisors. As a corollary, we then recover the well-known perfect pairing between the algebraic de Rham cohomology and rapid decay homology associated to integrable connections on smooth varieties due to Bloch--Esnault and Hien.