[Paper Review] Constructibility of tempered solutions of holonomic D-modules

Giovanni Morando|arXiv (Cornell University)|Nov 26, 2013

Holomorphic and Operator Theory7 references3 citations

TL;DR

This paper proves the constructibility of the sheaf of tempered holomorphic solutions of holonomic 𝒟-modules on complex analytic manifolds with respect to the subanalytic site, confirming a conjecture by Kashiwara and Schapira. The proof proceeds by induction on dimension, using resolution of singularities and properties of the tempered De Rham complex, establishing that the cohomology of the tempered De Rham complex is ℝ-sa-constructible for holonomic 𝒟-modules.

ABSTRACT

In this paper we prove the constructibility on the subanalytic sites of the sheaves of tempered holomorphic solutions of holonomic D-modules on complex analytic manifolds. Such a result solves a conjecture of M. Kashiwara and P. Schapira (2003).

Motivation & Objective

• To resolve a long-standing conjecture by Kashiwara and Schapira on the ℝ-sa-constructibility of tempered holomorphic solutions of holonomic 𝒟-modules.
• To extend the Riemann–Hilbert correspondence to the irregular case by providing a topological characterization of the image category of the enhanced tempered solution functor.
• To establish that the complex of tempered holomorphic solutions of any holonomic 𝒟-module is constructible on the subanalytic site, thereby clarifying the structure of solutions in the presence of irregular singularities.

Proposed method

• The proof uses induction on the dimension of the underlying complex manifold, reducing the problem to lower-dimensional cases.
• Resolution of singularities is applied to decompose the singular locus into regular and singular parts, allowing the use of proper morphisms to transfer the problem to smooth manifolds.
• The tempered De Rham complex is analyzed via the functor $\mathcal{TH}\mathrm{om}$, which relates sheaves on the subanalytic site to 𝒟-modules.
• The key isomorphisms are derived using derived categories, base change, and duality, particularly involving $R\mathcal{H}\mathrm{om}$ and $\mathbb{D}$-functors.
• The argument treats the regular and singular parts of the singular locus separately, using the strict transform under a resolution to reduce to the regular case.
• The inductive hypothesis is applied to $\mathbb{D}\pi_{Z'}^*R\Gamma_{[Z]}\mathcal{M}$ and $R\Gamma_{[Z_s]}\mathcal{M}$, leveraging the fact that their singular supports have lower dimension.

Experimental results

Research questions

• RQ1 Is the complex of tempered holomorphic solutions of a holonomic 𝒟-module on a complex analytic manifold constructible on the subanalytic site?
• RQ2 Does the sheaf of tempered solutions of a holonomic 𝒟-module satisfy ℝ-sa-constructibility, as conjectured by Kashiwara and Schapira?
• RQ3 Can the enhanced tempered solution functor be fully characterized topologically via constructibility in the irregular case?
• RQ4 How does the structure of the singular locus affect the constructibility of tempered solutions?
• RQ5 Can the Riemann–Hilbert correspondence be extended to irregular holonomic 𝒟-modules using tempered solutions?

Key findings

• The sheaf of tempered holomorphic solutions of any holonomic 𝒟-module on a complex analytic manifold is ℝ-sa-constructible on the subanalytic site.
• The result confirms the conjecture of Kashiwara and Schapira regarding the constructibility of tempered solutions in the irregular case.
• The proof establishes that the cohomology of the tempered De Rham complex is constructible by reducing to lower-dimensional cases via resolution of singularities.
• The complex $\mathrm{D\!R}^t_{Z'}(\mathbb{D}\pi_{Z'}^*R\Gamma_{[Z]}\mathcal{M})$ is shown to be ℝ-sa-constructible by induction on dimension.
• The argument successfully treats both the regular and singular parts of the singular locus by using the strict transform and proper pushforward.
• The result provides a foundational step toward a global Riemann–Hilbert correspondence for irregular holonomic 𝒟-modules via tempered solutions.