[Paper Review] Algebraic aspects of higher nonabelian Hodge theory

Carlos Simpson|ArXiv.org|Feb 10, 1999

Homotopy and Cohomology in Algebraic Topology49 references13 citations

TL;DR

This paper develops the algebraic foundations of higher nonabelian Hodge theory using n-stacks to formalize the de Rham shape of smooth projective varieties. It establishes key Hodge-theoretic structures—Hodge filtration, Gauss-Manin connection, and Griffiths transversality—within this framework, extending the connection to have regular singularities across singularities, and introduces new categorical tools for n-categories, including a fibrant replacement for nCAT.

ABSTRACT

We look more closely at the higher nonabelian de Rham cohomology of a smooth projective variety or family of varieties that had been defined in some previous papers. We formalize using $n$-stacks the notion of shape underlying this nonabelian cohomology. A generalization of the de Rham construction to any appropriate formal category or family of formal categories, yields various algebraic aspects of Hodge theory for the de Rham shape, such as the Hodge filtration, the Gauss-Manin connection, Griffiths transversality, an extension of the Gauss-Manin connection with regular singularities across singular points, etc. Along the way, we develop a little bit more technology for $n$-categories, such as a canonical fibrant replacement for the $n+1$-category $nCAT$; and we pose the following general type of question: what are the properties of the nonabelian cohomology $n$-stack $Hom(X,T)$ as a function of the properties of the coefficient $n$-stack $T$ and the domain $n$-stack $X$?

Motivation & Objective

• To formalize the de Rham shape of smooth projective varieties using n-stacks as a foundational structure in higher nonabelian Hodge theory.
• To generalize the de Rham construction to formal categories, thereby deriving algebraic aspects of Hodge theory such as the Hodge filtration and Gauss-Manin connection.
• To extend the Gauss-Manin connection to have regular singularities at singular points, ensuring compatibility with the nonabelian cohomological framework.
• To develop categorical tools, particularly a canonical fibrant replacement for the (n+1)-category nCAT, to support higher categorical constructions.
• To investigate the relationship between the nonabelian cohomology n-stack Hom(X,T) and the properties of the domain X and coefficient T, posing a general structural question in higher category theory.

Proposed method

• Utilizes n-stacks to model the nonabelian de Rham cohomology of smooth projective varieties, providing a geometric realization of their shape.
• Applies the de Rham construction in the context of formal categories or families of formal categories to derive algebraic Hodge structures.
• Introduces a canonical fibrant replacement for the (n+1)-category nCAT to enhance the categorical framework for higher nonabelian cohomology.
• Derives the Hodge filtration and Gauss-Manin connection from the universal properties of the de Rham shape and its deformation theory.
• Establishes Griffiths transversality by analyzing the compatibility of the Gauss-Manin connection with the Hodge filtration in the nonabelian setting.
• Extends the Gauss-Manin connection to have regular singularities across singular points by analyzing monodromy and formal neighborhood behavior.

Experimental results

Research questions

• RQ1 How can the de Rham shape of a smooth projective variety be formalized algebraically using n-stacks?
• RQ2 What algebraic structures—such as Hodge filtration and Gauss-Manin connection—arise naturally from the nonabelian de Rham cohomology of varieties?
• RQ3 How can the Gauss-Manin connection be extended to have regular singularities at singular points within the nonabelian framework?
• RQ4 What are the structural properties of the nonabelian cohomology n-stack Hom(X,T) in terms of the domain X and coefficient T?
• RQ5 What categorical tools, such as fibrant replacements, are necessary to support higher nonabelian Hodge theory in n-category theory?

Key findings

• The paper constructs a canonical fibrant replacement for the (n+1)-category nCAT, enabling better control over higher categorical structures in nonabelian cohomology.
• It establishes the Hodge filtration on the nonabelian de Rham cohomology of a variety via the universal properties of the de Rham shape.
• The Gauss-Manin connection is derived and shown to satisfy Griffiths transversality in the nonabelian setting.
• The Gauss-Manin connection is extended to have regular singularities across singular points, preserving compatibility with the Hodge filtration.
• The paper provides a detailed proof of Lemma 6.4.3, strengthening the foundational technical results of the theory.
• It poses a general question on the relationship between Hom(X,T) and the properties of X and T, opening a new direction in higher nonabelian Hodge theory.