[Paper Review] Sato hyperfunctions via relative Dolbeault cohomology

Naofumi Honda, Takeshi IZAWA+1 more|arXiv (Cornell University)|Jul 5, 2018

Homotopy and Cohomology in Algebraic Topology23 references3 citations

TL;DR

This paper establishes Sato hyperfunctions using relative Dolbeault cohomology, providing a concrete, computable framework that replaces abstract local cohomology with smooth forms. It yields explicit representations of hyperfunctions, simplifies operations like integration and duality, and constructs an explicit embedding of Schwartz distributions into the space of hyperfunctions via a canonical morphism.

ABSTRACT

The relative Dolbeault cohomology which naturally comes up in the theory of Cech-Dolbeault cohomology turns out to be canonically isomorphic with the local (relative) cohomology of A. Grothendieck and M. Sato so that it provides a handy way of representing the latter. In this paper we use this cohomology to give simple explicit expressions of Sato hyperfunctions, some fundamental operations on them and related local duality theorems. This approach also yields a new insight into the theory of hyperfunctions and leads to a number of further results and applications. As one of such, we give an explicit embedding morphism of Schwartz distributions into the space of hyperfunctions.

Motivation & Objective

• To reformulate Sato hyperfunctions using relative Dolbeault cohomology to avoid reliance on Stein coverings and Čech cohomology.
• To provide explicit, computable representations of hyperfunctions using smooth forms on neighborhoods and complements of the real manifold.
• To simplify fundamental operations such as integration, duality, and boundary value morphisms in hyperfunction theory.
• To construct an explicit embedding of Schwartz distributions into the space of hyperfunctions.
• To lay a foundation for further developments in algebraic analysis, including Laplace hyperfunctions and pseudodifferential operators.

Proposed method

• Utilize relative Dolbeault cohomology as a canonical model for local cohomology, replacing sheaf-theoretic constructions with smooth forms.
• Represent a hyperfunction by a pair (τ₁, τ₀₁), where τ₁ is a C∞ (0,n)-form near M and τ₀₁ is a C∞ (0,n−1)-form on the complement, satisfying a cocycle condition.
• Apply partitions of unity and fine resolutions to handle global constructions without requiring Stein coverings.
• Construct the boundary value morphism explicitly via the Thom class and integration over infinitesimal wedges.
• Use the quasi-isomorphism between the complex of sheaves and the Čech-Dolbeault resolution to relate cohomology classes to hyperfunctions.
• Apply the functor RHom to relate the derived category of sheaves to hyperfunction cohomology, enabling explicit computation of morphisms.

Experimental results

Research questions

• RQ1 Can relative Dolbeault cohomology provide a more concrete and computable representation of Sato hyperfunctions than traditional local cohomology?
• RQ2 How can the boundary value morphism and integration of hyperfunctions be explicitly described in this framework?
• RQ3 Can the space of Schwartz distributions be explicitly embedded into the space of hyperfunctions using this cohomological model?
• RQ4 What are the implications of this approach for the theory of Laplace hyperfunctions and pseudodifferential operators?
• RQ5 Does this method simplify the proof of fundamental properties such as purity of codimension or flabbiness of the hyperfunction sheaf?

Key findings

• The relative Dolbeault cohomology is canonically isomorphic to Grothendieck–Sato local cohomology, providing a concrete model for hyperfunctions.
• Hyperfunctions admit explicit representatives as pairs of smooth forms (τ₁, τ₀₁), enabling direct computation and compact support representations.
• The integration of a hyperfunction is explicitly computable via the pairing of smooth forms in the relative Dolbeault complex.
• The boundary value morphism bΩ is shown to be a morphism of D-modules, confirming its compatibility with differential operators.
• An explicit embedding morphism of Schwartz distributions into the space of hyperfunctions is constructed via the cohomological residue map.
• The framework enables new results in algebraic analysis, including applications to Laplace hyperfunctions and symbol theory of pseudodifferential operators.