Harder-Narasimhan Filtration Overview

• Harder–Narasimhan Filtration is a canonical decomposition that breaks bundles into semistable subobjects with strictly decreasing slopes, ensuring uniqueness.
• It extends to decorated, self-dual bundles and quiver representations, employing algorithmic and GIT-based methods to analyze stability and stratify moduli spaces.
• The filtration underpins obstruction theory, categorical frameworks, and game-theoretic models, offering a unified approach to moduli classification and stability analysis.

The Harder–Narasimhan (HN) filtration is a fundamental structural invariant arising in the geometric theory of vector bundles, principal bundles, and representations of quivers. It canonically decomposes complex objects into semistable pieces with strictly decreasing "slopes," providing a unifying tool for moduli theory, geometric invariant theory, and representation theory. The concept, originally appearing in the context of vector bundles on curves, has been extended to principal $G$-bundles, decorated bundles, and other categorical settings, with profound implications for the structure and classification of moduli spaces and algebraic stacks (Roth et al., 21 Nov 2025).

1. Formal Definition and Construction

Let $E$ be a holomorphic vector bundle of rank $r$ over a compact Riemann surface $X$. Its degree is defined as $\deg(E)=c_1(E)[X]$, and its slope as $\mu(E)=\deg(E)/\operatorname{rk}(E)$. $E$ is slope-semistable if every nontrivial proper subbundle $F$ satisfies $\mu(F)\leq\mu(E)$.

Harder–Narasimhan Filtration (Classical case)

Every $E$ admits a unique filtration by subbundles,

$E$0

where each successive quotient $E$1 is slope-semistable and the sequence of slopes is strictly decreasing: $E$2 Uniqueness follows from the "strongly contradicting semistability" condition, and existence is obtained by recursively selecting subbundles of maximal slope and passing to quotients (Roth et al., 21 Nov 2025).

Decorated and Self-dual Bundles

For decorated bundles (e.g., symplectic or special-orthogonal structures) only isotropic (i.e., structure-compatible) subbundles are considered. A symplectic or orthogonal bundle admits a unique isotropic filtration with semistable quotients and decreasing slopes (positive for the nontrivial pieces), followed by a coisotropic semistable quotient. The full HN filtration of $E$3 is recovered by appending the coisotropic piece.

2. Generalization to Principal $E$4-Bundles and Reductions

Semistability is extended to principal $E$5-bundles $E$6 for $E$7 a complex reductive group by considering reductions $E$8 to parabolic subgroups and evaluating the degree of associated vector bundles.

• $E$9 is semistable iff for every maximal parabolic $r$0 and reduction $r$1 the associated "vertical tangent bundle" $r$2 has $r$3.
• The general filtration is constructed via rational reductions to parabolics, leading to a "canonical reduction" as defined by Atiyah–Bott and by Biswas–Holla via GIT-based criteria:
  • The Levi factor bundle is semistable.
  • For every nontrivial dominant character $r$4, $r$5.

The two approaches coincide and characterize existence and uniqueness of canonical reductions up to conjugacy (Roth et al., 21 Nov 2025).

3. Obstruction Theory and HN Type for Principal Bundles

The HN filtration in the principal bundle context is encoded by an obstruction-theoretic invariant:

• The "second obstruction" $r$6, refined to $r$7 (where $r$8 is a maximal torus).
• The HN type $r$9, living in the closed Weyl chamber. For $X$0, if $X$1 has HN steps with slopes $X$2 and ranks $X$3, then

$X$4

corresponding to the graded pieces' slopes.

This encoding allows a complete classification of possible HN types and their stratifications on moduli spaces (Roth et al., 21 Nov 2025).

4. Stratification of Moduli Spaces and Artin Stacks

The HN filtration yields a stratification of parameter schemes and stacks, formalizing the variation of semistability in families.

• For families of principal bundles on a family $X$5 of smooth projective varieties, the parameter scheme $X$6 admits a stratification by locally closed subschemes $X$7—the \textbf{schematic Harder–Narasimhan stratification}.
• Each $X$8 corresponds to points where the fiberwise bundle has HN-type $X$9 and has a universal property: base changes factoring through $\deg(E)=c_1(E)[X]$0 induce families with relative HN filtration of type $\deg(E)=c_1(E)[X]$1.
• On the stack of principal $\deg(E)=c_1(E)[X]$2-bundles, these strata glue to a stratification by locally closed substacks. The closure relations reflect the partial order on HN types in the Weyl chamber (Gurjar et al., 2015).

This framework generalizes to $\deg(E)=c_1(E)[X]$3-modules and Higgs bundles, so that each moduli problem admits an intrinsic HN stratification compatible with the underlying stability theory (Gurjar et al., 2012).

5. Algorithmic and GIT Interpretations

For quiver representations and general objects in finitely generated categories, the HN filtration admits deterministic, polynomial-time algorithms.

• For an acyclic quiver representation $\deg(E)=c_1(E)[X]$4, the filtration is found by iteratively extracting subrepresentations of maximal slope (via matrix discrepancy problems), yielding canonical refinements and supporting effective computation for large classes of moduli problems (Cheng, 2021).
• The HN filtration coincides with the flag associated to Kempf's maximally destabilizing $\deg(E)=c_1(E)[X]$5-PS in GIT: for any unstable object in a parameter space (e.g., a Quot scheme or quiver representation space), the maximally destabilizing $\deg(E)=c_1(E)[X]$6-PS produces a weighted flag whose associated filtration agrees with the HN filtration (Ferino et al., 9 Nov 2025, Gomez et al., 2011).

This correspondence clarifies the link between geometric invariant theory (optimal $\deg(E)=c_1(E)[X]$7-parameter subgroups) and the intrinsic structure of destabilizing filtrations in moduli theory.

6. Extensions, Categorical, and Game-Theoretic Frameworks

The HN filtration generalizes beyond classical settings:

• Chen–Jeannin's order-theoretic and game-theoretic construction reinterprets HN filtrations as minimax profiles in zero-sum games on lattices of subobjects, where convexity replaces degree additivity, and the unique filtration arises from a Nash equilibrium in a bounded lattice (Chen et al., 2023, Yuan, 23 Sep 2025).
• Categorical approaches utilize abstract slope functions satisfying strong "slope inequalities" on proto-abelian categories, allowing existence and uniqueness theorems for HN filtrations in settings lacking classical degree/rank additivity, with broad applicability to isocrystals, normed lattices, and even linear codes (Li, 2021).
• Algebraic frameworks describe HN filtrations as arising from chains of torsion classes and their intersections (slicings), linking the theory to wall-crossing formulas in Hall algebras and providing a unified algebraic structure for stability theory (Treffinger, 2018).

These approaches demonstrate the universality of HN filtrations as organizing structures not only in moduli theory but across broad categorical and representation-theoretic landscapes.

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• "Harder-Narasimhan filtrations of decorated vector bundles" (Roth et al., 21 Nov 2025)
• "Schematic Harder-Narasimhan stratification for families of principal bundles in higher dimensions" (Gurjar et al., 2015)
• "A deterministic algorithm for Harder-Narasimhan filtrations for representations of acyclic quivers" (Cheng, 2021)
• "Harder-Narasimhan games" (Chen et al., 2023)
• "Categorification of Harder-Narasimhan Theory via slope functions" (Li, 2021)
• "An algebraic approach to Harder-Narasimhan filtrations" (Treffinger, 2018)
• "Formalization of Harder-Narasimhan theory" (Yuan, 23 Sep 2025)