2022/06/13 by Brian Hepler, Hepler, Brian, Andreas Hohl +1
Mathematics · #Algebraic Geometry and Number Theory #Geometry and complex manifolds #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2206.06095
For any holomorphic function f\colon X→ 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 DX-module. In order to prove the latter, we resolve a recent conjectural duality of Sabbah between these de Rham complexes of holonomic DX-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 dévissage 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.