[Paper Review] Holonomic D-modules with Betti structure

Takuro Mochizuki|arXiv (Cornell University)|Jan 13, 2010

Finite Group Theory Research18 references15 citations

TL;DR

This paper introduces the concept of Betti structure for holonomic $\mathcal{D}$-modules, even in the non-regular singular case, establishing foundational functorial properties. It develops an auxiliary analysis of holomorphic functions on the real blow-up, extending the theory of Stokes structures to non-regular holonomic $\mathcal{D}$-modules and providing a framework for studying their topological and analytic behavior via Betti structures.

ABSTRACT

We define the notion of Betti structure for holonomic D-modules which are not necessarily regular singular. We establish the fundamental functorial properties. We also give auxiliary analysis of holomorphic functions of various types on the real blow up.

Motivation & Objective

• To extend the notion of Betti structures beyond regular singular holonomic $\mathcal{D}$-modules to include non-regular cases.
• To establish fundamental functorial properties of these Betti structures in the non-regular setting.
• To provide a detailed analysis of holomorphic functions of various types on the real blow-up space.
• To unify the study of holonomic $\mathcal{D}$-modules with Betti and Stokes structures in the non-regular context.

Proposed method

• Defining Betti structures for holonomic $\mathcal{D}$-modules via the real blow-up of the complex manifold along singularities.
• Using the real blow-up to analyze the asymptotic behavior of solutions and holomorphic functions near singularities.
• Applying techniques from $\mathcal{D}$-module theory and sheaf cohomology to study the topological and analytic structure of solutions.
• Establishing functoriality of Betti structures under pullbacks and pushforwards via the real blow-up construction.
• Integrating Stokes data into the Betti structure framework to handle irregular singularities.
• Analyzing the interplay between holomorphic functions, monodromy, and the real blow-up to characterize the Betti structure.

Experimental results

Research questions

• RQ1 How can Betti structures be defined for holonomic $\mathcal{D}$-modules that are not necessarily regular singular?
• RQ2 What are the functorial properties of Betti structures in the non-regular case?
• RQ3 How do holomorphic functions behave on the real blow-up of a complex manifold near singularities?
• RQ4 What role does the real blow-up play in capturing the topological and analytic data of non-regular holonomic $\mathcal{D}$-modules?
• RQ5 How can Stokes structures be incorporated into the Betti structure framework for irregular holonomic $\mathcal{D}$-modules?

Key findings

• The paper successfully defines Betti structures for holonomic $\mathcal{D}$-modules without requiring regular singularities.
• It establishes that Betti structures satisfy key functorial properties such as compatibility with pullbacks and pushforwards.
• The analysis of holomorphic functions on the real blow-up reveals refined asymptotic behavior near singularities, crucial for understanding solution spaces.
• The framework unifies Betti and Stokes structures, allowing a coherent treatment of irregular holonomic $\mathcal{D}$-modules.
• The real blow-up construction provides a geometric setting where the topological and analytic data of $\mathcal{D}$-modules become accessible and computable.