Harder-Narasimhan categories

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.

💡 Research Summary

The paper introduces a substantial generalization of Quillen’s exact categories by defining the notion of an “arithmetic exact category.” In this setting each object is equipped with a real‑valued weight (or degree) together with a rank function, and admissible monomorphisms and epimorphisms are required to preserve these numerical invariants in an additive way. This extra arithmetic data makes it possible to extend the classical Harder–Narasimhan (HN) theory, which traditionally works with integer‑indexed filtrations, to a continuous real‑indexed framework.

The authors first formalize the axioms of an arithmetic exact category. The key requirements are: (i) for any short exact sequence 0→A→B→C→0 the degrees satisfy deg B = deg A + deg C, and (ii) the rank function is additive on split exact sequences. With these in place one can define the slope μ(X)=deg X/rk X for any non‑zero object X. Objects are called μ‑semistable if every admissible subobject has slope ≤ μ(X). This mirrors the usual definition for vector bundles or coherent sheaves but now applies to any arithmetic exact category.

The central technical achievement is the construction of an ℝ‑indexed Harder–Narasimhan filtration. For each real number λ the authors define a subobject Fλ(X) which is the maximal admissible subobject of X with slope ≥ λ. They prove that the assignment λ↦Fλ(X) is monotone, left‑continuous, and that the family {Fλ(X)}λ∈ℝ exhausts X (i.e., Fλ(X)=0 for λ≫∞ and Fλ(X)=X for λ≪−∞). Moreover, the successive quotients Grλ X = Fλ(X)/⋃_{μ>λ}Fμ(X) are μ‑semistable with slope exactly λ. This yields a unique, functorial “Harder–Narasimhan polygon” which is now a piecewise linear, possibly infinitely refined curve in the (rank, degree) plane rather than a finite polygon.

A major novelty is the functoriality theorem: any exact functor between arithmetic exact categories that preserves rank and degree also preserves the entire ℝ‑indexed HN filtration. In other words, for such a functor F one has F(Fλ(X)) ≅ Fλ(F(X)) for all λ. This result goes beyond the classical HN formalism, where functoriality is usually limited to specific situations (e.g., pull‑back of bundles under finite morphisms). The theorem opens the door to systematic transport of stability data across a wide variety of categorical contexts, including derived categories equipped with t‑structures and filtered objects.

The paper also discusses the geometric representation of the filtration via Harder–Narasimhan polygons. By plotting the points (rk Fλ(X), deg Fλ(X)) for all λ one obtains a continuous, convex curve whose slopes record the distribution of semistable pieces. The authors show that the polygon’s break points correspond precisely to the slopes of the semistable factors, and that the curvature encodes how “unstable” an object is. This continuous picture is particularly useful when dealing with families of objects where the slope varies continuously, such as in moduli problems with real‑valued stability parameters.

To illustrate the theory, three families of examples are presented. First, the classical category of finite‑dimensional vector spaces over a field, where the degree is taken to be the dimension; the ℝ‑indexed filtration collapses to the obvious filtration by subspaces of increasing dimension. Second, the category of vector bundles on a smooth projective curve, with degree given by the usual first Chern class; the construction recovers the classical HN filtration and shows that the continuous refinement adds no new information in this case but provides a uniform language. Third, a derived‑category example: objects equipped with a finite filtration (a “filtered derived object”) form an arithmetic exact category where degree is the alternating sum of the degrees of the graded pieces. Here the ℝ‑indexed filtration yields a genuinely new stability notion that interacts with the triangulated structure.

In conclusion, the authors have built a robust categorical framework that enriches exact categories with arithmetic data, enabling a continuous Harder–Narasimhan theory. The key contributions are (1) the definition of arithmetic exact categories, (2) the existence and uniqueness of ℝ‑indexed HN filtrations, (3) a strong functoriality result, and (4) a geometric interpretation via continuous HN polygons. This work not only unifies several existing stability theories but also suggests new directions in areas such as moduli of filtered objects, derived‑category stability conditions, and applications to mathematical physics where real‑valued stability parameters naturally arise.