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

For any holomorphic function \( f \) on a complex manifold \( X \), we define and study moderate growth and rapid decay objects associated to an enhanced ind-sheaf on \( X \). These are 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 holonomic \( \mathcal{D} \)-modules. To prove this, we resolve a conjectural duality by Sabbah concerning these de Rham complexes with growth conditions along a normal crossing divisor. This is achieved via a connection to a classic duality result by Kashiwara–Schapira between certain topological vector spaces. By a dévissage argument, we extend this duality to arbitrary divisors, thereby proving Sabbah’s conjecture. As a corollary, we recover the perfect pairing between algebraic de Rham cohomology and rapid decay homology for integrable connections, previously established by Bloch–Esnault and Hien.

Publications of the Research Institute for Mathematical Sciences, Vol. 61, No. 1 (To appear)