[Paper Review] A Category of Motivic Sheaves

Donu Arapura|arXiv (Cornell University)|Jan 1, 2008

Algebraic Geometry and Number Theory4 citations

TL;DR

This paper constructs an abelian category of motivic sheaves on algebraic varieties over subfields of ℂ using Nori’s method, establishing faithful exact realization functors to classical and étale constructible sheaves. It further identifies a tannakian subcategory of motivic local systems that realizes into variations of mixed Hodge structures, with all standard geometric examples of the latter arising from this construction.

ABSTRACT

The goal of this paper is to construct a category of motivic on an algebraic variety defined over a subfield of C, using Nori's method. This categoryis abelian and it possesses faithful exact realization functors to the categoriesof constructible sheaves for the classical and etale topologies. Moreover, there is a tannakian subcategory of motivic local systems with a realization functor into the category of variations of mixed Hodge structures. Conversely, all basic geometric examples of the latter come from this motivic category.

Motivation & Objective

• To define a category of motivic sheaves on algebraic varieties over subfields of ℂ using Nori’s method.
• To ensure the category is abelian and admits faithful exact realization functors to constructible sheaves in the classical and étale topologies.
• To identify a tannakian subcategory of motivic local systems that maps faithfully to variations of mixed Hodge structures.
• To show that all basic geometric examples of variations of mixed Hodge structures arise from this motivic construction.
• To unify motivic sheaves with classical and Hodge-theoretic realizations in a single categorical framework.

Proposed method

• Employing Nori’s diagrammatic method to construct the category of motivic sheaves from a diagram of algebraic varieties and morphisms.
• Defining the category as the abelian envelope of a diagram of constructible sheaves under the classical and étale topologies.
• Constructing faithful exact realization functors from the motivic category to the categories of constructible sheaves for the classical and étale topologies.
• Identifying a tannakian subcategory of motivic local systems within the motivic sheaf category.
• Establishing a realization functor from this tannakian subcategory to the category of variations of mixed Hodge structures.
• Using the properties of tannakian categories to ensure the realization is both faithful and exact.

Experimental results

Research questions

• RQ1 Can a well-behaved abelian category of motivic sheaves be constructed on algebraic varieties over subfields of ℂ using Nori’s method?
• RQ2 Do the motivic sheaves admit exact and faithful realization functors to classical and étale constructible sheaves?
• RQ3 Is there a tannakian subcategory of motivic local systems that realizes into variations of mixed Hodge structures?
• RQ4 Can all standard geometric examples of variations of mixed Hodge structures be recovered from this motivic construction?
• RQ5 How does the motivic sheaf category unify different realizations (classical, étale, Hodge-theoretic) in algebraic geometry?

Key findings

• The constructed category of motivic sheaves is abelian, ensuring good homological properties for further study.
• Faithful exact realization functors exist from the motivic sheaf category to both the classical and étale categories of constructible sheaves.
• A tannakian subcategory of motivic local systems is identified, which admits a realization into the category of variations of mixed Hodge structures.
• All basic geometric examples of variations of mixed Hodge structures arise as realizations of objects in this motivic category.
• The construction provides a unifying framework linking motivic sheaves with classical, étale, and Hodge-theoretic realizations.
• The realization functors preserve exactness and faithfulness, ensuring that the motivic structure reflects the geometric and cohomological data of the varieties.