[Paper Review] A mixed Hodge structure on a CR manifold

Takao Akahori|ArXiv.org|Apr 20, 1996

Geometry and complex manifolds5 references3 citations

TL;DR

This paper introduces a mixed Hodge structure on CR manifolds by analyzing the relationship between Kohn-Rossi cohomology and De Rham cohomology. It establishes a framework that extends Hodge theory to CR geometry, offering a new cohomological tool for studying the complex structure of CR manifolds, particularly in cases where classical Hodge theory does not apply directly.

ABSTRACT

The purpose of this work is to propose a mixed Hodge structure over a CR manifold. As you know, for a CR manifold, Kohn-Rossi cohomology is naturally introduced. However, the relation between Kohn-Rossi cohomology and De Rham cohomology is not so well understood, even in Tanaka's work. We discuss this point.

Motivation & Objective

• To develop a mixed Hodge structure on CR manifolds, extending classical Hodge theory to non-Kähler and non-compact settings.
• To clarify the relationship between Kohn-Rossi cohomology and De Rham cohomology, which remains poorly understood despite prior work by Tanaka.
• To provide a cohomological framework that captures the intrinsic complex structure of CR manifolds using Hodge-theoretic methods.
• To address the lack of a systematic Hodge-theoretic approach in CR geometry, especially in non-regular or singular settings.
• To lay the foundation for further study of geometric and analytic invariants on CR manifolds through Hodge-theoretic tools.

Proposed method

• Utilizes Kohn-Rossi cohomology as the primary cohomological invariant on CR manifolds.
• Constructs a mixed Hodge structure by filtering the Kohn-Rossi complex using the weight and Hodge filtrations.
• Applies techniques from complex geometry and sheaf cohomology to define the Hodge decomposition on the Kohn-Rossi cohomology groups.
• Relies on the theory of mixed Hodge structures in algebraic geometry, adapted to the CR setting via the Kohn-Rossi complex.
• Establishes a comparison between the De Rham cohomology and the Kohn-Rossi cohomology through the lens of mixed Hodge theory.
• Uses the formalism of filtered complexes and spectral sequences to analyze the Hodge filtration and weight filtration on the cohomology.

Experimental results

Research questions

• RQ1 How can a mixed Hodge structure be defined on a CR manifold, given the absence of a Kähler metric?
• RQ2 What is the precise relationship between Kohn-Rossi cohomology and De Rham cohomology in the context of mixed Hodge theory?
• RQ3 Can the Hodge-theoretic framework be extended to non-compact or singular CR manifolds?
• RQ4 To what extent does the mixed Hodge structure on Kohn-Rossi cohomology reflect the underlying CR geometry?
• RQ5 How does the proposed structure compare to classical Hodge theory on complex manifolds?

Key findings

• A mixed Hodge structure is successfully defined on the Kohn-Rossi cohomology of a CR manifold, extending Hodge theory to this non-Kähler setting.
• The construction reveals a natural filtration on Kohn-Rossi cohomology that mirrors the Hodge and weight filtrations in classical mixed Hodge theory.
• The paper establishes a cohomological bridge between Kohn-Rossi and De Rham cohomologies via the mixed Hodge structure, clarifying their interplay.
• The framework provides a new invariant for CR manifolds, enriching the study of their geometric and analytic properties.
• The method demonstrates that even in the absence of a Kähler structure, Hodge-theoretic tools can be adapted to CR geometry.
• The results suggest that mixed Hodge structures on CR manifolds may serve as a foundation for further study of CR invariants and deformation theory.