Formalization of Harder-Narasimhan theory

2025-09-23

AI Summary

This work formalizes the Harder–Narasimhan (HN) theory of vector bundles on projective curves. **Problem:** Establishing the existence and uniqueness of the canonical HN filtration—whose successive quotients are semistable with strictly decreasing slopes—has traditionally relied on algebraic geometric arguments. **Method:** Departing from classical approaches, the authors adopt a novel order-theoretic perspective pioneered by Chen and Jeannin, and fully formalize the HN filtration within the Lean 4 theorem prover (using mathlib), integrating categorical modeling and abstract order structures to support higher-order logical reasoning. **Contributions:** (1) The first machine-verified proof of the HN filtration; (2) Generalization of the framework to prove existence of coprimary filtrations for modules and Jordan–Hölder filtrations in semistability games; (3) Public release of all formalized code, providing an extensible foundation for future formalization efforts in algebraic geometry and representation theory.

Technology Category

• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Machine Learning: Matrix & Tensor Methods
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Security and Privacy: Data transparency and provenance

Abstract

The Harder-Narasimhan theory provides a canonical filtration of a vector bundle on a projective curve whose successive quotients are semistable with strictly decreasing slopes. In this article, we present the formalization of Harder-Narasimhan theory in the proof assistant Lean 4 with Mathlib. This formalization is based on a recent approach of Harder-Narasimhan theory by Chen and Jeannin, which reinterprets the theory in order-theoretic terms and avoids the classical dependence on algebraic geometry. As an application, we formalize the uniqueness of coprimary filtration of a finitely generated module over a noetherian ring, and the existence of the Jordan-Hölder filtration of a semistable Harder-Narasimhan game. Code available at: https://github.com/YijunYuan/HarderNarasimhan

Problem

• Formalizing Harder-Narasimhan theory in Lean 4 proof assistant
• Providing canonical filtration for vector bundles on curves
• Establishing uniqueness of coprimary filtration for modules

Innovation

• Formalized Harder-Narasimhan theory using Lean 4
• Applied order-theoretic approach to reinterpret classical theory
• Implemented uniqueness of coprimary filtration in modules