[Paper Review] Harder-Narasimhan categories

Huayi Chen|arXiv (Cornell University)|Jun 18, 2007

Algebraic structures and combinatorial models11 references4 citations

TL;DR

This paper introduces arithmetic exact categories as a generalization of Quillen's exact categories, enabling the construction of Harder-Narasimhan filtrations indexed by the real numbers (R). It establishes functoriality of these filtrations and proves that categories of Hermitian vector bundles, filtered (ϕ,N)-modules, and torsion-free sheaves on polarized varieties are Harder-Narasimhan categories under appropriate degree and rank functions.

ABSTRACT

We propose a generalization of Quillen's exact category -- arithmetic exact category and we discuss conditions on such categories under which one can establish the notion of Harder-Narasimhan filtrations and Harder-Narsimhan polygons. Furthermore, we show the functoriality of Harder-Narasimhan filtrations (indexed by $\mathbb R$), which can not be stated in the classical setting of Harder and Narasimhan's formalism.

Motivation & Objective

• To generalize Quillen's exact categories to arithmetic exact categories for studying semistability in arithmetic and geometric contexts.
• To define and establish the existence of Harder-Narasimhan filtrations indexed by the real numbers R, overcoming the non-functoriality of classical flags.
• To prove that such filtrations are functorial, a property not available in classical formalism.
• To verify that key arithmetic and geometric categories—such as Hermitian vector bundles, filtered (ϕ,N)-modules, and torsion-free sheaves—satisfy the conditions of a Harder-Narasimhan category.
• To provide a categorical framework for studying Harder-Narasimhan polygons and associated probability measures via degree and rank functions.

Proposed method

• Introduces the notion of I-filtrations in a category, with emphasis on left-continuous filtrations indexed by a totally ordered set I.
• Defines arithmetic exact categories as a generalization of exact categories, incorporating arithmetic structures such as degree and rank functions.
• Establishes conditions under which a category admits semistable objects and Harder-Narasimhan filtrations, using finite sets of degrees and slopes.
• Constructs a filtration indexed by R using successive minimal slopes, ensuring functoriality across morphisms.
• Applies the formalism to three key categories: (1) Hermitian vector bundles over Spec(OK), (2) filtered (ϕ,N)-modules, and (3) torsion-free sheaves on polarized projective varieties.
• Shows that the normalized Arakelov degree and rank functions extend to group homomorphisms on K0, enabling the definition of Harder-Narasimhan categories.

Experimental results

Research questions

• RQ1 Can the classical Harder-Narasimhan flag construction be generalized to a functorial filtration indexed by the real numbers R?
• RQ2 What categorical conditions ensure the existence of Harder-Narasimhan filtrations in arithmetic contexts?
• RQ3 How can the functoriality of Harder-Narasimhan filtrations be established in settings where classical flags fail to be functorial?
• RQ4 Which classical arithmetic and geometric categories—such as Hermitian vector bundles or filtered (ϕ,N)-modules—satisfy the axioms of a Harder-Narasimhan category?
• RQ5 What is the role of the normalized Arakelov degree and rank in defining Harder-Narasimhan filtrations and associated polygons?

Key findings

• The paper constructs a Harder-Narasimhan filtration indexed by R, which is functorial, resolving a key limitation of classical flags.
• The category of Hermitian vector bundles over Spec(OK) equipped with normalized Arakelov degree and rank functions forms a Harder-Narasimhan category.
• Filtered (ϕ,N)-modules with degree and rank functions also form a Harder-Narasimhan category, with semistable objects of slope 0 corresponding to admissible modules.
• Torsion-free coherent sheaves on a polarized projective variety over a field form a Harder-Narasimhan category under the usual degree and rank functions.
• The normalized Arakelov degree satisfies d̃deg(E) = d̃deg(E′) + d̃deg(E′′) for short exact sequences, ensuring compatibility with K0-group homomorphism.
• The construction yields a Borel probability measure on R as a linear combination of Dirac masses, which is instrumental in studying Harder-Narasimhan polygons in subsequent work.