Mitra Koley, A. J. Parameswaran
In this article we study asymptotic slopes of strongly semistable vector bundles on a smooth projective surface. A connection between asymptotic slopes and strong restriction theorem of a strongly semistable vector bundle is shown. We also give an equivalent criterion of strong semistability of a vector bundle in terms of its asymptotic slopes under some assumptions on the surface and on the bundle.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
Slope semistability of Veronese normal bundles
本文将有理正规曲线法丛均衡的经典结论推广到所有 Veronese 法丛，证明其斜率半稳定性，并给出次数为 2 的 Veronese 法丛在直线和有理正规曲线上限制的线丛分解。
We give an algebraic proof valid in arbitrary characteristic for the known equivalence between (strongly) slope semistable vector bundles with vanishing discriminant and vanishing determinant and numerically flat bundles. We also address a related question of Misra.
Previously, many people have studied a stability of vector bundles of given rank and Chern classes on algebraic varieties. Recently, we are interested in the slope stability of the rank 2 Lazarsfeld-Mukai bundle $E_{C,Z}$ on a K3 surface $X$ associated to a very ample smooth curve $C$ on $X$ and a base point free pencil $Z$ on $C$ with respect to $\mathcal{O}_X(C)$. In this paper, we will give a sufficient condition for such a Lazarsfeld-Mukai bundle $E_{C,Z}$ to be $\mathcal{O}_X(C)$-slope semistable by ACM line bundles with respect to $\mathcal{O}_X(C)$.
Slope-semistability and moduli of coherent sheaves: a survey
本文综述了代数几何与复几何中半稳定相干层模空间存在性的新旧结果，涵盖斜率半稳定性及相关构造。
CM-line bundles and slope $\K$-semistability for big and nef line bundles along subschemes
本文利用Chow-Mumford线丛理论，给出了沿子概形及big和nef线丛的Donaldson-Futaki $\K$-稳定性不变量的显式相交理论描述，推广了Ross-Thomas斜率稳定性理论，并利用了线丛关于丰富锥的连续性。
Seshadri slope $\K$-semistability for the blow-up of projective space along a linear subspace
本文应用 Chow-Mumford 线丛理论，证明了沿线性子空间爆破的射影空间在例外除子上的形变到法锥测试构型具有 $\K$-半稳定性。
Let $E$ be a vector bundle of rank $r$ on a smooth complex projective variety $X$. In this article, we compute the nef and pseudoeffective cones of divisors in the Grassmann bundle $Gr_X(k,E)$ parametrizing $k$-dimensional subspaces of the fibers of $E$, where $1\leq k \leq rank(E)$, under assumptions on $X$ as well as on the vector bundle $E$. In particular, we show that nef cone and the pseudoeffective cone of $Gr_X(k,E)$ coincide if and only if $E$ is a slope semistable bundle on $X$ with $c_2(End(E))=0$. We also discuss about the nefness and ampleness of the universal quotient bundle $Q_k$…
We discuss a minimization problem of the degree of the CM line bundle among all possible fillings of a polarized family with fixed general fibers. We show that such minimization implies the slope semistability of the fiber if the central fiber is smooth.
An obstruction to asymptotic semistability and approximate critical metrics
本文考虑了极化Kaehler代数流形的渐近Chow-半稳定性的一个障碍。即使当一个线性代数群的正维行为nontrivially和holomorphically极化Kaehler代数流形常数数量曲率曲率，消失的障碍，使我们能够推广唐纳森的建设近似解方程的平衡度量。
Semistability of Syzygy Bundles Associated to Ulrich Bundles on Projective Varieties of Arbitrary Dimension
证明在光滑射影簇上，Ulrich丛的全局合冲丛在充分大的扭曲后关于极丰沛线丛是斜率半稳定的，验证了Miró-Roig猜想3.11。
This paper is devoted to generalizing the Mehta-Ramanathan restriction theorem to the case of ε-semistable and k-semistable decorated sheaves. After recalling the definition of decorated sheaves and their usual semistability we define the ε and k-(semi)stablility. We first prove the existence of a (unique) ε-maximal destabilizing subsheaf for decorated sheaves (Section 3.1). After some others preliminar results (such as the opennes condition for families of ε-semistable decorated sheaves) we finally prove, in Section 3.7, a restriction theorem for slope ε-semistable decorated sheaves. In Secti…
A slope-unstable bundle on a surface with 1-homogeneous projectivization
回答了Fulger和Langer关于判别式为零的射影丛斜率半稳定性与正性相互作用的问题，核心方法和主要结果由Chatgpt 5.5 pro及基于Rethlas系统的Danus系统得出。
Slopes of trigonal fibred surfaces and of higher dimensional fibrations
本文推广Cornalba-Harris方法，在GIT半稳定条件下给出高维纤维化斜率下界，并证明偶数亏格三角纤维化斜率的锐利下界，从而证实Harris-Stankova猜想。
Given a polarized manifold there are obstructions for asymptotic Chow semistability described as integral invariants. One of them is an obstruction to the existence for the first Chern class of the polarization to admit a constant scalar curvature Kähler (cscK) metric. A natural question is whether or not the other obstructions are linearly dependent on the obstruction to the existence of a cscK metric. The purpose of this paper is to see that this is not the case by exhibiting toric Fano threefolds in which these obstructions span at least two dimension. To see this we show that on toric Fano…
Semistability and asymptotics of geometric flows
本文证明在斜率半稳定全纯向量丛上，Hermitian-Yang-Mills流的渐近行为由Haiden-Katzarkov-Kontsevich-Pandit迭代滤过代数几何地决定，并推广到Calabi流和矩映射流，将无限维流简化为有限维流。
Negative Effective Divisors and Bridgeland Stability of Line Bundles on Surfaces
针对连通光滑复射影曲面，证明Arcara–Miles猜想的有效除子版本及其严格类比，揭示线丛半稳定失效由负有效除子检测，相关数值条件与整数标度稳定性等价。
We study the basic properties of Higgs sheaves over compact Kähler manifolds and we establish some results concerning the notion of semistability; in particular, we show that any extension of semistable Higgs sheaves with equal slopes is semistable. Then, we use the flattening theorem to construct a regularization of any torsion-free Higgs sheaf and we show that it is in fact a Higgs bundle. Using this, we prove that any Hermitian metric on a regularization of a torsion-free Higgs sheaf induces an admissible structure on the Higgs sheaf. Finally, using admissible structures we proved some prop…
We show that for any stable sheaf $E$ of slope $> 2g-1$ on a smooth, projective curve of genus $g$, the associated Picard sheaf $\hat{E}$ on the Picard variety of the curve is stable. We introduce a homological tool for testing semistability of Picard sheaves.
We provide a complete description of the asymptotics of the gradient flow on the space of metrics on any semistable quiver representation. This involves a recursive construction of approximate solutions and the appearance of iterated logarithms and a limiting filtration of the representation. The filtration turns out to have an algebraic definition which makes sense in any finite length modular lattice. This is part of a larger project by the authors to study iterated logarithms in the asymptotics of gradient flows, both in finite and infinite dimensional settings.
We resolve pathological wall-crossing phenomena for moduli spaces of sheaves on higher-dimensional base manifolds. This is achieved by considering slope-semistability with respect to movable curves rather than divisors. Moreover, given a projective n-fold and a curve C that arises as the complete intersection of n-1 very ample divisors, we construct a modular compactification of the moduli space of vector bundles that are slope-stable with respect to C. Our construction generalises the algebro-geometric construction of the Donaldson-Uhlenbeck compactification by Joseph Le Potier and Jun Li. Fu…
Let Δ\subset \mathbb{R}^n be an n-dimensional Delzant polytope. It is well-known that there exist the n-dimensional compact toric manifold X_Δand the very ample (\mathbb{C}^\times)^n-equivariant line bundle L_Δon X_Δassociated with Δ. In the present paper, we show that if (X_Δ,L_Δ^i) is Chow semistable then the sum of integer points in iΔis the constant multiple of the barycenter of Δ. Using this result we get a necessary condition for the polarized toric manifold (X_Δ,L_Δ) being asymptotically Chow semistable. Moreover we can generalize the result of Futaki, Sano and the author to the case wh…
Studying degenerations of moduli spaces of semistable principal bundles on smooth curves leads to the problem of constructing and studying moduli spaces on singular curves. In this note, we will see that the moduli spaces of $δ$-semistable pseudo bundles on a nodal curve constructed by the first author become, for large values of $δ$, the moduli spaces for semistable singular principal bundles. The latter are reasonable candidates for degenerations and a potential basis of further developments as on irreducible nodal curves. In particular, we find a notion of semistability for principal bundle…
Moduli of rank 2 Higgs sheaves on elliptic surfaces
本文研究椭圆曲面上秩为2希格斯层的模空间，证明斜率半稳定性可遗传到一般纤维，并给出奇数纤维次数下模空间的典范乘积分解及希格斯场取值的结构性结论。
Asymptotic Chow semistability implies Ding polystability for Gorenstein toric Fano varieties
本文证明了 Gorenstein 环面 Fano 簇的渐近 Chow 半稳定性蕴含关于环面测试构型的 Ding 多稳定性，推广了已有结果，并给出了 Mabuchi 常数的可加性公式及相对 K-稳定性与 Ding 稳定性差异的系统实例。
Numerical semistability of projective toric varieties
本文给出光滑环面簇数值半稳定的充要条件，并证明极化环面簇的渐近数值半稳定性等价于对环面退化的K-半稳定性。
Let $X$ be a del Pezzo surface. When the degree of $X$ is at least 4, we compute the cohomology of a general sheaf in the moduli space of Gieseker semistable sheaves. We also classify the Chern characters for which the general sheaf in the moduli space is non-special, i.e. has at most one nonzero cohomology group. Our results hold for arbitrary polarizations, slope semistability, and semi-exceptional moduli spaces. When the degree of $X$ is at least 3, we further show our construction of certain vector bundles implies the existence of stable and semistable sheaves with respect to the anti-cano…
On the spectrum of asymptotic slopes
秩为$s$的极大子丛的斜率除以映射在各种拉回下的度，形成一个有界的数集合，称为丛的$s$ -谱。我们研究的上确界的$s$ -频谱，并确定它的更难Narasimhan过滤的束。
Ulrich Bundles on certain Grassmann Bundles Over Curves
本文证明曲线上相对Grassmann簇在任意非常丰沛极化下存在Ulrich丛，通过全局化纤维上的齐次Ulrich丛并利用上同调消没与Falting半稳定性准则完成构造。
Koszul property of Ulrich bundles and rationality of moduli spaces of stable bundles on Del Pezzo surfaces
本文研究 Del Pezzo 曲面上 Ulrich 丛的 Koszul 性质与迭代 syzygy 丛的斜率半稳定性，并证明无穷多个稳定丛模空间具有有理性，为 Costa 和 Miró-Roig 猜想提供新证据。
Quantized Volume Comparison for Fano Manifolds, II
本文解决了第二作者的量子体积比较猜想，证明光滑复K-半稳定法诺簇满足指定上界，且该结论在切丛斜率半稳定时仍成立，核心用jet维数计数技巧结合半稳定条件得到维数界。
The holomorphic invariants introduced by Futaki as obstruction to the asymptotic Chow semistability are studied by an algebraic-geometric point of view and are shown to be the Mumford weights of suitable line bundles on the Hilbert scheme. These invariants are calculated in two special cases. The first is a projective bundle over a curve of genus at least two, and it is shown that it is asymptotically Chow polystable (with every polarization) if and only the underlying vector bundle is slope polystable. This proves a conjecture of Morrison with the extra assumption that the involved polarizati…
We derive an explicit formula for the asymptotic slope of the Aubin-Yau functional along a Bergman geodesic on a surface of complex dimension 2, extending the work of Phong-Sturm on Riemann surfaces. This is equivalent to an explicit calculation of the Donaldson-Futaki invariant of a test configuration. The slope is given as a rational linear combination of period integrals of rational functions that sum to a rational number. The result gives a way to check directly whether a two dimensional projective variety is K-stable.
Bundle-extension inverse problems over elliptic curves
本文研究复椭圆曲线上的丛扩张反问题，证明在数值与行列式约束下，多数丛态射的（上）核可预先指定或满足刚性约束，给出多类开稠集结果及推广、假设必要性例证。
Semistable Higgs bundles on elliptic surfaces
本文分析了椭圆曲面上的一类 Higgs bundles，证明了在谱曲线既约等特定条件下 Higgs 场的垂直性或标量性，并将其应用于通过曲线拉回半稳定性刻画斜率半稳定 Higgs bundles 的问题。
Let $X$ be a normal and geometrically integral projective variety over a global field $K$ and let $\overline{D}$ be an adelic Cartier divisor on $X$. We prove a conjecture of Chen, showing that the essential minimum $ζ_{\mathrm{ess}}(\overline{D})$ of $\overline{D}$ equals its asymptotic maximal slope under mild positivity assumptions. As an application, we see that $ζ_{\mathrm{ess}}(\overline{D})$ can be read on the Okounkov body of the underlying divisor $D$ via the Boucksom--Chen concave transform. This gives a new interpretation of Zhang's inequalities on successive minima and a criterion …
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。
We introduce Seshadri constants for line bundles in a relative setting. They generalize the classical Seshadri constants of line bundles on projective varieties and their extension to vector bundles studied by Beltrametti-Schneider-Sommese and Hacon. There are similarities to the classical theory. In particular, we give a Seshadri-type ampleness criterion, and we relate Seshadri constants to jet separation and to asymptotic base loci. We give three applications of our new version of Seshadri constants. First, a celebrated result of Mori can be restated as saying that any Fano manifold whose ta…
The asymptotic Samuel function and invariants of singularities
本文将渐近Samuel函数推广到任意环，引入诺特局部环的Samuel斜率并研究其性质，证明簇奇点处局部环的该斜率与奇点消解算法的部分不变量相关。
Deformed Hermitian Yang-Mills equation on rational homogeneous varieties
本文证明了有理齐次簇上形变埃尔米特 Yang-Mills 方程在任意不变 Kähler 度量下解的存在性，利用 Lie 理论和 Cartan 矩阵刻画齐次解，并给出全纯向量丛斜率半稳定性的新判据及其在 Fano 三维簇上的应用。
We study stacks of slope-semistable twisted sheaves on orbisurfaces with projective coarse spaces and prove that in certain cases they have many of the asymptotic properties enjoyed by the moduli of slope-semistable sheaves on smooth projective surfaces.
Curve configurations in the projective plane and their characteristic numbers
本文研究射影平面中曲线构型的特征数与陈斜率，证明极值直线构型具有相同的渐近不变量，并给出某类光滑曲线构型特征数的上界$8/3$。
We study the growth of the ground state degeneracy in the Kronecker model of quiver quantum mechanics. This is the simplest quiver with two gauge groups and bifundamental matter fields, and appears universally in the context of BPS state counting in four-dimensional N=2 systems. For large ranks, the ground state degeneracy is exponential with slope a modular function that we are able to compute at integral values of its argument. We also observe that the exponential of the slope is an algebraic number and determine its associated algebraic equation explicitly in several examples. The speed of …
Quot-scheme limit of Fubini-Study metrics and Donaldson's functional for vector bundles
本文通过定义 Fubini-Study 度量的 Quot-概形极限，建立了向量丛斜率稳定性与 Donaldson 泛函渐近行为的直接联系，并给出强制性与新证明。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。