Harder-Narasimhan theory

A Harder-Narasimhan stratification of the $B_{dR}^+$-Grassmannian

We establish a Harder-Narasimhan formalism for modifications of $G$-bundles on the Fargues-Fontaine curve. The semi-stable stratum of the associated stratification of the $B_{dR}^+$-Grassmannian coincides with the variant of the weakly admissible locus defined by Viehmann, and its classical points agree with those of the basic Newton stratum. When restricted to minuscule affine Schubert cells, the stratification corresponds to the Harder-Narasimhan stratification of Dat, Orlik and Rapoport. We also study basic geometric properties of the strata, and the relation to the Hodge-Newton decompositi…

On Harder-Narasimhan strata in flag manifolds

This paper deals with a question of Fontaine and Rapoport which was posed in math.NT/0204293. There they asked for the determination of the index set of the Harder-Narasimhan vectors of the filtered isocrystals with fixed Newton- and Hodge vector. The aim of this paper is to give an answer to their question.

Convergence of Harder-Narasimhan polygons

We establish in this article convergence results of normalized Harder-Narasimhan polygons both in geometric and in arithmetic frameworks by introducing the Harder-Narasimhan filtration indexed by $\mathbb R$ and the associated Borel probability measure.

Schematic Harder-Narasimhan Stratification

For any flat family of pure-dimensional coherent sheaves on a family of projective schemes, the Harder-Narasimhan type (in the sense of Gieseker semistability) of its restriction to each fiber is known to vary semicontinuously on the parameter scheme of the family. This defines a stratification of the parameter scheme by locally closed subsets, known as the Harder-Narasimhan stratification. In this note, we show how to endow each Harder-Narasimhan stratum with the structure of a locally closed subscheme of the parameter scheme, which enjoys the universal property that under any base change the…

Harder-Narasimhan 博弈

Harder-Narasimhan games

该研究从博弈论角度研究半稳定性与Harder-Narasimhan滤过，给出其存在唯一性的统一证明与半稳定性的概念阐释，还证明交换Noether环上有限型模余准素滤过的存在唯一性并将其归为Harder-Narasimhan滤过。

硬尔-纳拉辛汉理论的形式化

Formalization of Harder-Narasimhan theory

本文在Lean 4中形式化了硬尔-纳拉辛汉理论，基于序理论方法，避免传统代数几何依赖，并展示了有限生成模的共质滤子唯一性和半稳定滤子的存在性。

关于结构层直接像的Harder-Narasimhan滤波

On the Harder-Narasimhan filtration of the direct image of the structure sheaf

研究有限映射下结构层直接像向量束的Harder-Narasimhan滤波，提出计算方法及应用条件。

On the Gieseker Harder-Narasimhan filtration for principal bundles

We give an example of an orthogonal bundle where the Harder-Narasimhan filtration, with respect to Gieseker semistability, of its underlying vector bundle does not correspond to any parabolic reduction of the orthogonal bundle. A similar example is given for the symplectic case.

Hermitian-Einstein Inequalities and Harder-Narasimhan Filtration

Unstable holomorphic bundles can be described algebraically by Harder-Narasimhan filtrations. In this note we show how such filtrations correspond to the existence of special metrics defined by Hermitian-Einstein inequalities.

三角典范曲线的Harder-Narasimhan滤过

The Harder-Narasimhan Filtration of a Trigonal Canonical Curve

本研究利用三角典范曲线位于有理正规面scroll上的性质，计算了一般三角典范曲线及亏格6典范曲线法丛的Harder-Narasimhan滤过，为代数几何中相关丛的结构研究提供了具体结果。

Harder-Narasimhan Filtrations and K-Groups of an Elliptic Curve

Let $X$ be an elliptic curve over an algebraically closed field. We prove that some exact sub-categories of the category of vector bundles over $X$, defined using Harder-Narasimhan filtrations, have the same K-groups as the whole category.

Schematic Harder-Narasimhan stratification for families of principal bundles in higher dimensions

For any family of principal bundles with a reductive structure group G on a family X/S of smooth projective varieties in characteristic zero, it is known that the parameter scheme S has a set theoretic stratification by locally closed subsets which correspond to the Harder-Narasimhan types of the restriction of the principal bundle to the various fibers of X/S. We show that each of these subsets has in fact the structure of a locally closed subscheme of the parameter scheme S, with the following universal property: Under any base change, the pullback family admits a relative Harder-Narasimhan …

Harder-Narasimhan strata and $p$-adic period domains

We revisit the Harder-Narasimhan stratification on a minuscule $p$-adic flag variety, by the theory of modifications of $G$-bundles on the Fargues-Fontaine curve. We compare the Harder-Narasimhan strata with the Newton strata introduced by Caraiani-Scholze. As a consequence, we get further equivalent conditions in terms of $p$-adic Hodge-Tate period domains for fully Hodge-Newton decomposable pairs. Moreover, we generalize these results to arbitrary cocharacters case by considering the associated $B_{dR}^+$-affine Schubert varieties. Applying Hodge-Tate period maps, our constructions give appl…

关于 $B_{\mathrm{dR}}^+$-仿射 Grassmannian 上的弱 Harder-Narasimhan 分层

On the weak Harder-Narasimhan stratification on $B_{\mathrm{dR}}^+$-affine Grassmannian

本文在赋范等晶范畴上建立Harder-Narasimhan形式化，证明其滤过与张量积相容，推广Cornut结果，并由此在B_dR^+-仿射Grassmannian上对任意(G,b,μ)定义弱Harder-Narasimhan分层，研究其非空性、维数及与其他分层的关系。

Harder-Narasimhan stratification for the moduli stack of parabolic vector bundles

For every set of parabolic weights, we construct a Harder-Narasimhan stratification for the moduli stack of parabolic vector bundles on a curve. It is based on the notion of parabolic slope, introduced by Mehta and Seshadri. We also prove that the stratification is schematic, that each stratum is complete, and establish an analogue of Behrend's conjecture for parabolic vector bundles. A comparison with recent $Θ$-stratification approaches is discussed.

Schematic HN stratification for families of principal bundles and lambda modules

For a family of principal bundles with a reductive structure group on a family of curves in characteristic zero, it is known that the Harder Narasimhan type of its restriction to each fiber varies semicontinuously over the parameter scheme of the family. This defines a stratification of the parameter scheme by locally closed subsets, known as the Harder-Narasimhan stratification. In this note, we show how to endow each Harder-Narasimhan stratum with the structure of a locally closed subscheme of the parameter scheme, which enjoys the universal property that under any base change the pullback f…

On Harder-Narasimhan reductions for Higgs principal bundles

The existence and uniqueness of H-N reduction for the Higgs principal bundles over nonsingular projective variety is shown. We also extend the notion of H-N reduction for (Γ, G)-bundles and ramified G-bundles over a smooth curve.

Harder-Narasimhan滤过的导出模栈的显式构造

An explicit construction of derived moduli stacks of Harder-Narasimhan filtrations

本文借助Behrend等人的方法，显式构造了特征0代数闭域上连通射影概形的Harder-Narasimhan滤过导出模栈，还研究了带滤过层的导出形变理论并与Di Natale的构造做了对比。

Spherical Hall algebras of curves and Harder-Narasimhan stratas

Let X be any smooth projective curve defined over a finite field. We show that the characteristic functions of any Harder-Narasimhan strata S_a of Bun_{GL_n}X belongs to the spherical Hall algebra H_X^{sph} of X. We give a geometric analog of the above result: the intersection cohomology sheaf IC(S_a) belongs to the category of simple Eisenstein sheaves over Bun_{GL_n}X.

On the Harder-Narasimhan Filtration for Coherent Sheaves on P2: I

Let E be a torsion-free sheaf on P2. We give an effective method which uses the Hilbert function of E to construct a weak version of the Harder-Narasimhan filtration of a torsion-free sheaf on P2 subject only to the condition that E be sufficiently general among sheaves with that Hilbert function. This algorithm uses on a generalization of Davis' decomposition lemma to higher rank.

The Harder-Narasimhan system in quantum groups and cohomology of quiver moduli

Methods of Harder and Narasimhan from the theory of moduli of vector bundles are applied to moduli of quiver representations. Using the Hall algebra approach to quantum groups, an analog of the Harder-Narasimhan recursion is constructed inside the quantized enveloping algebra of a Kac-Moody algebra. This leads to a canonical orthogonal system, the HN system, in this algebra. Using a resolution of the recursion, an explicit formula for the HN system is given. As an application, explicit formulas for Betti numbers of the cohomology of quiver moduli are derived, generalizing several results on th…

A GIT interpretration of the Harder-Narasimhan filtration

An unstable torsion free sheaf on a smooth projective variety gives a GIT unstable point in certain Quot scheme. To a GIT unstable point, Kempf associates a "maximally destabilizing" 1-parameter subgroup, and this induces a filtration of the torsion free sheaf. We show that this filtration coincides with the Harder-Narasimhan filtration.

箭图表示的矩映射上的Harder-Narasimhan滤过

Harder-Narasimhan Filtration on Moment Map for Quiver Representations

本文针对无环多箭图推广已有学者的方法，计算矩映射零点集的Harder-Narasimhan层维数，发现其与该零点集等变上同调给出的Kac多项式项存在关联。

装饰向量丛的Harder-Narasimhan滤波

Harder-Narasimhan filtrations of decorated vector bundles

本文研究了装饰向量丛的Harder-Narasimhan滤波构造，并比较了不同方法的标准约简，同时建立了主丛的Harder-Narasimhan类型的障碍理论。

Harder-Narasimhan filtration for rank 2 tensors and stable coverings

We construct a Harder-Narasimhan filtration for rank $2$ tensors, where there does not exist any such notion a priori, as coming from a GIT notion of maximal unstability. The filtration associated to the 1-parameter subgroup of Kempf giving the maximal way to destabilize, in the GIT sense, a point in the parameter space of the construction of the moduli space of rank $2$ tensors over a smooth projective complex variety, does not depend on certain integer used in the construction of the moduli space, for large values of the integer. Hence, this filtration is unique and we define the Harder-Nara…

On the Harder-Narasimhan filtration for finite dimensional representations of quivers

We prove that the Harder-Narasimhan filtration for an unstable finite dimensional representation of a finite quiver coincides with the filtration associated to the 1-parameter subgroup of Kempf, which gives maximal unstability in the sense of Geometric Invariant Theory for the corresponding point in the parameter space where these objects are parametrized in the construction of the moduli space.

Eigenvalues of Curvature, Lyapunov exponents and Harder-Narasimhan filtrations

Inspired by Katz-Mazur theorem on crystalline cohomology and by Eskin-Kontsevich-Zorich's numerical experiments, we conjecture that the polygon of Lyapunov spectrum lies above (or on) the Harder-Narasimhan polygon of the Hodge bundle over any Teichmüller curve. We also discuss the connections between the two polygons and the integral of eigenvalues of the curvature of the Hodge bundle by using Atiyah-Bott, Forni and Möller's works. We obtain several applications to Teichmüller dynamics conditional to the conjecture.

Harder-Narasimhan 滤过的代数方法

An algebraic approach to Harder-Narasimhan filtrations

本文用挠类链给出 Harder-Narasimhan 滤过的代数构造，并刻画切片及推导完备 Hall 代数中的穿墙公式。

Quotients of unstable subvarieties and moduli spaces of sheaves of fixed Harder-Narasimhan type

When a reductive group $G$ acts linearly on a complex projective scheme $X$ there is a stratification of $X$ into $G$-invariant locally closed subschemes, with an open stratum $X^{ss}$ formed by the semistable points in the sense of Mumford's geometric invariant theory which has a categorical quotient $X^{ss} \to X//G$. In this article we describe a method for constructing quotients of the unstable strata. As an application, we construct moduli spaces of sheaves of fixed Harder-Narasimhan type with some extra data (an '$n$-rigidification') on a projective base.

沿$\mathrm{Bun}_G$的Harder-Narasimhan分层黏合层

Gluing sheaves along Harder-Narasimhan strata of $\mathrm{Bun}_G$

本文借助[FS21]构造的光滑图表局部闭分层上同调，给出沿Fargues-Scholze的$\mathrm{Bun}_G$的Harder-Narasimhan分层黏合素p挠层的方法，并在$G=\operatorname{GL}_2$时明确计算了几何常数项函子在特殊点的像。

Geometric Invariant Theory, holomorphic vector bundles and the Harder-Narasimhan filtration

This survey intends to present the basic notions of Geometric Invariant Theory (GIT) through its paradigmatic application in the construction of the moduli space of holomorphic vector bundles. Special attention is paid to the notion of stability from different points of view and to the concept of maximal unstability, represented by the Harder-Narasimhan filtration and, from which, correspondences with the GIT picture and results derived from stratifications on the moduli space are discussed.

GIT characterizations of Harder-Narasimhan filtrations

This Ph.D. thesis studies the relation between the Harder-Narasimhan filtration and a notion of GIT maximal unstability. When constructing a moduli space by using Geometric Invariant Theory (GIT), a notion of GIT stability appears, which is determined by 1-parameter subgroups. This thesis shows a correspondence between the 1-parameter subgroup giving maximal unstability from the GIT point of view and the Harder-Narasimhan filtration for different moduli problems: torsion free coherent sheaves, holomorphic pairs, Higgs sheaves, rank 2 tensors and quiver representations. The article [GSZ] contai…

关于直接像的Harder-Narasimhan斜率

On Harder-Narasimhan slopes of direct images

本文研究复射影簇平坦族中直接像的Harder-Narasimhan斜率渐近分布，建立Mehta-Ramanathan型定理，并用于估计线束重数。

无环箭图的表示的 Harder-Narasimhan 滤过的确定性算法

A deterministic algorithm for Harder-Narasimhan filtrations for representations of acyclic quivers

本文提出一个多项式时间确定性算法，计算无环箭图表示的 Harder-Narasimhan 滤过，并应用于代数闭域上不稳定表示，生成 Kempf 最大去稳定化单参数子群。

Harder-Narasimhan polygons and Laws of Large Numbers

We build on the recent techniques of Codogni and Patakfalvi, from \cite{Codogni:Patakfalvi:2021}, which were used to establish theorems about semi-positivity of the Chow Mumford line bundles for families of $\K$-semistable Fano varieties. Here we apply the Central Limit Theorem to ascertain the asymptotic probabilistic nature of the vertices of the \emph{Harder and Narasimhan polygons}. As an application of our main result, we use it to establish a filtered vector space analogue of the main technical result of \cite{Codogni:Patakfalvi:2021}. In doing so, we expand upon the slope stability theo…

Moduli Spaces of Unstable Objects: Sheaves of Harder-Narasimhan Length 2

Given a moduli problem posed using Geometric Invariant Theory, one can use Non-Reductive Geometric Invariant Theory to quotient unstable HKKN strata and construct 'moduli spaces of unstable objects', extending the usual moduli classifications. After giving a self-contained account of how to do this, we apply this method to construct moduli spaces for certain unstable coherent sheaves of HN length 2 on a projective scheme, which we call $τ$-stable sheaves. This extends a previous result of Brambila-Paz and Mata-Gutiérrez for rank two vector bundles on a curve.

李代数叠连接与哈德纳-纳拉辛汉约简

Lie algebroid connection and Harder-Narasimhan reduction

本文研究了在紧连通黎曼面上，解析李代数叠连接与哈德纳-纳拉辛汉约简的关系，证明了在特定条件下，约简存在解析连接。

持久模的Harder-Narasimhan滤过：度量稳定性

Harder-Narasimhan filtrations of persistence modules: metric stability

本文将skyscraper不变量从有限持久模推广到$\mathbb Z^n$和$\mathbb R^n$上的多参数持久模，并建立了该不变量的侵蚀型度量稳定性结果。

The Harder-Narasimhan stratification of the moduli stack of G-bundles via Drinfeld's compactifications

We use Drinfeld's relative compactifications and the Tannakian viewpoint on principal bundles to construct the Harder-Narasimhan stratification of the moduli stack Bun_G of G-bundles on an algebraic curve in arbitrary characteristic, generalizing the stratification for G=GL_n due to Harder and Narasimhan to the case of an arbitrary reductive group G. To establish the stratification on the set-theoretic level, we exploit a Tannakian interpretation of the Bruhat decomposition and give a new and purely geometric proof of the existence and uniqueness of the canonical reduction in arbitrary charact…

Harder-Narasimhan filtrations and optimal destabilizing vectors in complex geometry

We give a generalisation of the theory of optimal destabilizing 1-parameter subgroups to non-algebraic complex geometry. Consider a holomorphic action $G\times F\to F$ of a complex reductive Lie group $G$ on a finite dimensional (possibly non-compact) Kähler manifold $F$. Using a Hilbert type criterion for the (semi)stability of symplectic actions, we associate to any non semistable point $f\in F$ a unique optimal destabilizing vector in $\g$ and then a naturally defined point $f_0$ which is semistable for the action of a certain reductive subgroup of $G$ on a submanifold of $F$. We get a natu…

Stratifications of parameter spaces for complexes by cohomology types

We study a collection of stability conditions (in the sense of Schmitt) for complexes of sheaves over a smooth complex projective variety indexed by a positive rational parameter. We show that the Harder-Narasimhan filtration of a complex for small values of this parameter encodes the Harder-Narasimhan filtrations of the cohomology sheaves of this complex. Finally we relate a stratification into locally closed subschemes of a parameter space for complexes associated to these stability parameters with the stratification by Harder-Narasimhan types.

Harder-Narasimhan theory for linear codes

In this text we develop some aspects of Harder-Narasimhan theory, slopes, semistability and canonical filtration, in the setting of combinatorial lattices. Of noticeable importance is the Harder-Narasimhan structure associated to a Galois connection between two lattices. It applies, in particular, to matroids. We then specialize this to linear codes. This could be done from at least three different approaches: using the sphere-packing analogy, or the geometric view, or the Galois connection construction just introduced. A remarkable fact is that these all lead to the same notion of semistabili…

Théorie de la réduction pour les groupes p-divisibles

Starting from our work on Harder-Narasimhan filtrations of finite flat group schemes over a $p$-adic field, we developp a theory of Harder-Narasimhan filtrations for $p$-divisible groups. We apply this to the study of the geometry of period morphisms for Rapoport-Zink spaces and to the $p$-adic geometry of Shimura varieties. We define and study in particular some fundamental domains for the action of Hecke correspondences.

挠 $F$-规范丛的 Harder-Narasimhan 滤过

Harder-Narasimhan filtration of torsion F-gauges

本文在正特征完美域上建立了挠 $F$-规范丛的 Harder-Narasimhan 理论，完全分类了其稳定对象，并纠正了 Ekedahl 的分类，同时将其提升为 Bridgeland 稳定性条件。

持久模块的Harder-Narasimhan滤波器

Harder-Narasimhan Filtrations of Persistence Modules

本文研究了持久模块中Harder-Narasimhan类型的强度与局限性，引入sky scraper不变量并推广rank不变量，证明其比rank更精细，刻画了完整中心电荷集，并扩展至多维矩形可分解模块，最后证明有限个中心电荷可完全描述nestfree ladder模块。

关于Wess-Zumino-Witten方程和Harder-Narasimhan势

About Wess-Zumino-Witten equation and Harder-Narasimhan potentials

研究极化复射影流形中Wess-Zumino-Witten方程近似解的代数障碍，并推广为Monge-Ampère型方程，应用时证明了Andréotti-Grauert定理的反向结论。

主 $ρ$-层的模叠与 Gieseker-Harder-Narasimhan 滤过

The moduli stack of principal $ρ$-sheaves and Gieseker-Harder-Narasimhan filtrations

本文定义并紧化主 $ρ$-层模叠，构造 Gieseker 半稳定对象的模空间，并给出概形 Gieseker-Harder-Narasimhan 滤过及其分层，附录推广至任意特征。

模空间理论中的精细 Harder-Narasimhan 滤过

Refined Harder-Narasimhan filtrations in moduli theory

本文在具有良好模空间的诺特代数栈上定义 Harder-Narasimhan 滤过的典范精细化——迭代平衡滤过，统一并推广了 Haiden-Katzarkov-Kontsevich-Pandit 与 Kirwan 的工作，并猜想其描述 Kempf-Ness 流的渐近行为。

Harder-Narasimhan stacks for principal bundles in higher dimensions and arbitrary characteristics

Let $G$ be a split reductive group over a field $k$ of arbitrary characteristic, chosen suitably. Let $X\to S$ be a smooth projective morphism of locally noetherian $k$-schemes, with geometrically connected fibers. We show that for each Harder-Narasimhan type $τ$ for principal $G$-bundles, all pairs consisting of a principal $G$-bundle on a fiber of $X\to S$ together with a given canonical reduction of HN-type $τ$ form an algebraic stack $Bun_{X/S}^τ(G)$ over $S$. The forgetful $1$-morphism $Bun_{X/S}^τ(G) \to Bun_{X/S}(G)$ to the algebraic stack of all principal $G$-bundles on fibers of $X\to…

Harder-Narasimhan Filtrations which are not split by the Frobenius maps

Let $X$ be a smooth projective variety over a perfect field $k$ of characteristic $p>0$, and $V$ be a vector bundle over $X$. It is well known that if $X$ is a curve and $V$ is not strongly semistable, then some Frobenius pullback $(F^t)^*V$ is a direct sum of strongly semistable bundles. A natural question to ask is whether this still holds in higher dimension. Indranil Biswas, Yogish I. Holla, A.J. Parameswaran, and S. Subramanian showed that there is always a counterexample to this over any algebraically closed field of positive characteristic which is uncountable. However, we will produce …

Restricted tangent bundles for general free rational curves

Suppose that $X$ is a smooth projective variety and that $C$ is a general member of a family of free rational curves on $X$. We prove several statements showing that the Harder-Narasimhan filtration of $T_{X}|_{C}$ is approximately the same as the restriction of the Harder-Narasimhan filtration of $T_{X}$ with respect to the class of $C$. When $X$ is a Fano variety, we prove that the set of all restricted tangent bundles for general free rational curves is controlled by a finite set of data. We then apply our work to analyze Peyre's "freeness" formulation of Manin's Conjecture in the setting o…

Maximal rationally connected fibrations and movable curves

A well known result of Miyaoka asserts that a complex projective manifold is uniruled if its cotangent bundle restricted to a general complete intersection curve is not nef. Using the Harder-Narasimhan filtration of the tangent bundle, it can moreover be shown that the choice of such a curve gives rise to a rationally connected foliation of the manifold. In this note we show that, conversely, a movable curve can be found so that the maximal rationally connected fibration of the manifold may be recovered as a term of the associated Harder-Narasimhan filtration of the tangent bundle.

Optimal destabilizing vectors in some gauge theoretical moduli problems

We show that the ``classical'' Harder-Narasimhan filtration associated to a non semistable vector bundle $E$ can be viewed as a limit object for the action of the gauge group in the direction of an optimal destabilizing vector. This vector appears as an extremal value of the so called "maximal weight function". We give a complete description of these optimal destabilizing endomorphisms. Then we show that this principle holds for another important moduli problem: holomorphic pairs (i.e. holomorphic vector bundles coupled with morphisms with fixed source). We get a generalization of the Harder-N…

跳跃轨迹的余维数

Codimension of jumping loci

本文研究光滑射影簇上向量丛限制到曲线族时Harder-Narasimhan过滤的变化，确定了斜率跳跃轨迹余维数线性依赖跳跃量的几何条件，并应用于研究射影平面秩2丛及曲线模空间奇点轨迹。