Tom Bridgeland
We describe the spaces of stability conditions on certain triangulated categories associated to Dynkin diagrams. These categories can be defined either algebraically via module categories of preprojective algebras, or geometrically via coherent sheaves on resolutions of Kleinian singularities. The resulting spaces are related to regular subsets of the corresponding complexified Cartan algebras.
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
On certain Lagrangian subvarieties in minimal resolutions of Kleinian singularities
本文分类Kleinian奇点的反Poisson对合，并描述其不动点轨迹在极小消解中的概形论原像，证明这些原像是不可约分量为 $\mathbb{P}^1$ 和 $\mathbb{A}^1$ 的奇异Lagrangian子簇。
We describe the derived category of coherent sheaves on the minimal resolution of the Kleinian singularity associated to a finite subgroup G of SL(2). Then, we give an application to the Euler-characteristic version of the Hall algebra of the category of coherent sheaves on an algebraic surface.
The exceptional configuration of the minimal resolution $\hat{S}_G $ of a Kleinian quotient surface $S_G (:= \CZ^2/G)$ is depicted by a $A$-$D$-$E$ Coxeter-Dynkin diagram. In this article, we show that branching indices of the affine $A$-$D$-$E$ diagram is geometrically characterized by a certain special function $F$ of $S_G$ as the multiplicities of its divisor components in $\hat{S}_G$, a version parallel to the elliptic fibration near certain types of simple singular fibers in Kodaira's elliptic surface theory. We further obtain the uniqueness property of the function $F$ (modular local uni…
We establish faithfulness of braid group actions generated by twists along an ADE configuration of $2$-spherical objects in a derived category. Our major tool is the Garside structure on braid groups of type ADE. This faithfulness result provides the missing ingredient in Bridgeland's description of a space of stability conditions associated to a Kleinian singularity.
For a finite subgroup $Γ\subset \mathrm{SL}(2,\mathbb{C})$ and $n\geq 1$, we construct the (reduced scheme underlying the) Hilbert scheme of $n$ points on the Kleinian singularity $\mathbb{C}^2/Γ$ as a Nakajima quiver variety for the framed McKay quiver of $Γ$, taken at a specific non-generic stability parameter. We deduce that this Hilbert scheme is irreducible (a result previously due to Zheng), normal, and admits a unique symplectic resolution. More generally, we introduce a class of algebras obtained from the preprojective algebra of the framed McKay quiver by a process called cornering, a…
We give a conceptual proof that the Poincaré series of the coordinate algebra of a Kleinian singularity and of a Fuchsian singularity of genus 0 is the quotient of the characteristic polynomials of two Coxeter elements. These Coxeter elements are interpreted geometrically, using triangulated categories and spherical twist functors.
Grothendieck proved that any locally free sheaf on a projective line over a field (uniquely) decomposes into a direct sum of line bundles. Ishii and Uehara construct an analogue of Grothendieck's theorem for pure sheaves on the fundamental cycle of the Kleinian singularity $A_n$. We first study the analogue for the other Kleinian singularities except for $A_n$. We also study the classification of rigid pure sheaves on the reduced scheme of the fundamental cycles. The classification is related to the classification of spherical objects in a certain Calabi-Yau $2$-dimensional category.
Quiver Varieties and Framed Sheaves on Compactified Kleinian Singularities
该研究针对Kleinian奇点$\mathbb{C}^2/\Gamma$，引入带光滑栈式除子的自然紧化栈，证明其上满足特定条件的带标架层集合与一类Nakajima箭图簇相关，部分推广了此前点Hilbert概型的相关研究。
Quivers variety for affine orthogonal quiver
本文研究正交箭图的箭图簇，利用预投影代数中的图形演算，确定了由仿射Dynkin图产生的Kleinian曲面奇点类型。
Revisiting the Classical McKay Correspondence, Derived Equivalences and the Spectrum of Kleinian Surface Singularities: A Look Through the Mirror
本文通过同调镜对称重访经典McKay对应，将其表述为Kleinian曲面奇点消没范畴与商轨形完美复形范畴间的导出等价，并借此用二元多面体群的表示论数据解释奇点的谱。
Moduli spaces of framed sheaves on compactified Kleinian singularities
该研究针对Kleinian奇点$\mathbb{C}^2/\Gamma$（$\Gamma$为$SL_2(\mathbb{C})$有限子群），构造了紧化该奇点的射影Deligne-Mumford栈上的框架层模空间，证明其为拟射影概形。
In this paper we prove an analogue of a recent result of Gordon and Stafford that relates the representation theory of certain noncommutative deformations of the coordinate ring of the n-th symmetric power of C^2 with the geometry of the Hilbert scheme of n points in C^2 through the formalism of Z-algebras. Our work produces, for every regular noncommutative deformation O^λof a Kleinian singularity X=C^2/Γ, as defined by Crawley-Boevey and Holland, a filtered Z-algebra which is Morita equivalent to O^λ, such that the associated graded Z-algebra is Morita equivalent to the minimal resolution of…
Automorphism groups of deformations and quantizations of Kleinian singularities
本文证明 A 或 D 型 Kleinian 奇点的量子化同构与形变 Poisson 同构本质相同，并刻画了对应自同构群的抽象结构，首次计算了 D 型形变间的所有仿射同构。
We construct Bridgeland stability conditions on the derived category of smooth quasi-projective Deligne-Mumford surfaces whose coarse moduli spaces have ADE singularities. This unifies the construction for smooth surfaces and Bridgeland's work on Kleinian singularities. The construction hinges on an orbifold version of the Bogomolov-Gieseker inequality for slope semistable sheaves on the stack, and makes use of the Toën-Hirzebruch-Riemann-Roch theorem.
Kleinian singularities: some geometry, combinatorics and representation theory
本文综述了 Kleinian 奇点相关的几何、组合与表示论，重点介绍单位根替换演算，并讨论 A 型、D 型及 E 型情形的结果与问题。
C. Weibel and Thomason-Trobaugh proved (under some assumptions) that algebraic K-theory with coefficients is A1-homotopy invariant. In this article we generalize this result from schemes to the broad setting of dg categories. Along the way, we extend Bass-Quillen's fundamental theorem as well as Stienstra's foundational work on module structures over the big Witt ring to the setting of dg categories. Among other cases, the above A1-homotopy invariance result can now be applied to sheaves of (not necessarily commutative) dg algebras over stacks. As an application, we compute the algebraic K-the…
Stacky Resolutions of Kleinian Singularities and Nakajima Quiver Varieties
该研究构造克莱因奇点的非交换平展分解，证明其与典范轨形分解Morita等价，确定对应点希尔伯特概型为中岛箭图簇，并刻画其两类锥的结构。
A finite subgroup of ${\rm SL}_2(\CC)$ defines a (Kleinian) rational surface singularity. The McKay correspondence yields a relation between the Poincaré series of the algebra of invariants of such a group and the characteristic polynomials of certain Coxeter elements determined by the corresponding singularity. Here we consider some non-abelian finite subgroups $G$ of ${\rm SL}_3(\CC)$. They define non-isolated three-dimensional Gorenstein quotient singularities. We consider suitable hyperplane sections of such singularities which are Kleinian or Fuchsian surface singularities. We show that w…
Resolution of singularities via Tannaka duality
本文借助 Tannaka 对偶推广 Abdelgadir–Segal 方法，构造 Clebsch–Gordan 簇，为所有 Klein 奇点同时提供光滑与堆叠消解。
Kleinian orbifolds, Cohomological Hall Algebras, and Yangians
本文为亏格解析克莱因奇点的轨形建立支撑例外轨迹凝聚层的COHA存在性并计算为仿射杨代数正半部分完备化，证明布里奇兰德稳定性条件空间点源自克莱因轨形且仿射杨代数正半部分可从对应COHA恢复，提供首个稳定性条件空间上的点态COHA族例子。
Resolutions of Type $\mathbb{A}$ Quantum Surface Singularities
本文通过构造几何解析类别\mathcal{X}，展示了类型A量子克莱因奇点的非交换解析与导出等价性，并推广了经典克莱因奇点的几何 McKay对应。
We study the relationship between singularity categories and relative singularity categories and discuss constructions of differential graded algebras of relative singularity categories. As consequences, we obtain structural results, which are known or generalise known results, on singularity categories of algebras with radical square zero, of non-commutative deformations of Kleinian singularities, of $SL_3(\mathbb{C})$-quotient singularities and of Gorenstein toric threefolds.
Stability over cDV singularities and other complete local rings
本文研究 cDV 奇点非交换极小模型的半稳定模范畴，利用 silting 理论与 Dynkin 图组合刻画稳定性，并将二项 silting–稳定性对应推广至完备局部 Noether 环上的代数。
We introduce $\mathfrak{f}$-stability, a modification of fibration stability of Dervan-Sektnan [12], and show that $\mathfrak{f}$-semistable fibrations have only semi log canonical singularities. Moreover, $\mathfrak{f}$-stability puts restrictions on semi log canonical centers on Fano fibrations.
Stability of klt singularities
本文综述klt奇点稳定性理论的近期进展，重点围绕稳定退化猜想的求解展开相关研究梳理。
Stability and singularities of relative hypersurfaces
本文研究曲线上的相对超曲面，证明纤维的不稳定性条件，给出对数规范阈值上界，并与中山Zariski分解信息比较。
We study the spaces of locally-finite stability conditions on the derived categories of coherent sheaves on the minimal resolutions of $A_n$-singularities supported at the exceptional sets. Our main theorem is that they are connected and simply-connected. The proof is based on the study of spherical objects in math.AG/0409151 and the homological mirror symmetry for $A_n$-singularities.
We study the problem of $\mathfrak{m}$-adic stability of F-singularities, that is, whether the property that a quotient of a local ring $(R,\mathfrak{m})$ by a non-zero divisor $x \in \mathfrak{m}$ has good F-singularities is preserved in a sufficiently small $\mathfrak{m}$-adic neighborhood of $x$. We show that $\mathfrak{m}$-adic stability holds for F-rationality in full generality, and for F-injectivity, F-purity and strong F-regularity under certain assumptions. We show that strong F-regularity and F-purity are not stable in general. Moreover, we exhibit strong connections between stabilit…
K-polystability and reduced uniform K-stability of log Fano cone singularities
研究证明log Fano锥奇点在特定条件下K-多项稳定性与特殊测试配置的K-多项稳定性等价，并建立其减少均匀K稳定性等价于K-多项稳定性。
I use Bridgeland's definition of a stability condition on a triangulated category to investigate the stability of D-branes on Calabi-Yau cones given by the canonical line bundle over a del Pezzo surface. In this context, I prove the existence of the decay of a D3-brane into a set of fractional branes. This is an important aspect of the derivation of quiver gauge theories from branes at singularities via the technique of equivalences of categories. Some important technical aspects of this equivalence are discussed. I also prove that the representations corresponding to skyscraper sheaves suppor…
Let $R$ be an isolated Gorenstein singularity with a non-commutative resolution $A=End_R(R\oplus M)$. In this paper, we show that the relative singularity category $Δ_R(A)$ of $A$ has a number of pleasant properties, such as being Hom-finite. Moreover, it determines the classical singularity category $D_{sg}(R)$ of Buchweitz and Orlov as a certain canonical quotient category. If $R$ has finite CM type, which includes for example Kleinian singularities, then we show the much more surprising result that $D_{sg}(R)$ determines $Δ_R(Aus(R))$, where $Aus(R)$ is the corresponding Auslander algebra. …
The purpose of the present paper is to set up a formalism inspired from non-Archimedean geometry to study K-stability. We first provide a detailed analysis of Duistermaat-Heckman measures in the context of test configurations, characterizing in particular the trivial case. For any normal polarized variety (or, more generally, polarized pair in the sense of the Minimal Model Program), we introduce and study the non-Archimedean analogues of certain classical functionals in Kähler geometry. These functionals are defined on the space of test configurations, and the Donaldson-Futaki invariant is in…
In this paper we construct noncommutative resolutions of a certain class of Calabi-Yau threefolds studied by F. Cachazo, S. Katz and C. Vafa. The threefolds under consideration are fibered over a complex plane with the fibers being deformed Kleinian singularities. The construction is in terms of a noncommutative algebra introduced by V. Ginzburg, which we call the "N=1 ADE quiver algebra".
Stability conditions on surfaces and contractions of curves
研究源自双有理态射的光滑曲面上的Bridgeland稳定性条件，在ADE或循环商奇点下构造预稳定性条件并证明支持性质，同时证明此类态射不能收缩正亏格的光滑曲线。
Variations of GIT Quotients and Dimer Combinatorics for Toric Compound Du Val Singularities
本文研究有向复合Du Val奇点的二聚体模型，利用锯齿路径序列确定稳定性参数空间的墙-腔结构，实现跨墙时稳定表示变化的追踪，为三维Gorenstein有向奇点的射影平展解析提供新视角。
Simple, or Kleinian, singularities are classified by Dynkin diagrams of type ADE. Let g be the corresponding finite-dimensional Lie algebra, and W its Weyl group. The set of g-invariants in the basic representation of the affine Kac-Moody algebra g^ is known as a W-algebra and is a subalgebra of the Heisenberg vertex algebra F. Using period integrals, we construct an analytic continuation of the twisted representation of F. Our construction yields a global object, which may be called a W-twisted representation of F. Our main result is that the total descendant potential of the singularity, int…
Let $S$ be a complete intersection of a smooth quadric 3-fold $Q$ and a hypersurface of degree $d$ in ${\mathbb P}^4$. In this paper we analyze GIT stability of $S$ with respect to the natural $G=SO(5, {\mathbb C})$-action. We prove that if $d\ge 4$ and $S$ has at worst semi-log canonical singularities then $S$ is $G$-stable. Also, we prove that if $d\ge 3$ and $S$ has at worst semi-log canonical singularities then $S$ is $G$-semistable.
We introduce moduli spaces of quasi-admissible hyperelliptic covers with at worst A and D singularities. The stability conditions for these moduli spaces depend on two parameters describing allowable singularities. By varying these parameters, we interpolate between the stacks of stable limits of A and D singularities and the quotients of the versal deformation spaces of A and D singularities by a natural $\mathbb{G}_m$ action.
Categorical resolutions of curves and Bridgeland stability
本文研究了奇异曲线的范畴解析与Bridgeland稳定性条件，扩展了经典构造，并给出了特定奇点曲线的torsion-free sheaves模空间的显式描述。
Lech-Mumford constant and stability of local rings
本文研究局部环的半稳定性，证明温和条件下半稳定奇点具有对数典范性，并通过新例子与 Lech-Mumford 常数理论推进奇点稳定性分析。
On K-stability of singular hyperelliptic Fano 3-folds
本文研究了具有特定奇点和反canonical线性系性质的Fano 3折面的K稳定性问题。
Numerical Reduction and Sharp Thresholds for Adjoint Singularities of Foliated Surfaces
本文研究叶状表面伴随奇点的数值简化与阈值，通过简化负定例外配置，分类ε-伴随log canonical奇点，并确定ε=1/5和ε=1/4的尖锐阈值，应用于伴随极小模型程序的稳定性范围。
Given a smooth polarized Riemann surface (X, L) endowed with a hyperbolic metric $ω$ with cusp singularities along a divisor D, we show the L^2 projective embedding of (X, D) defined by L^k is asymptotically almost balanced in a weighted sense. The proof depends on sufficiently precise understanding of the behavior of the Bergman kernel in three regions, with the most crucial one being the neck region around D. This is the first step towards understanding the algebro-geometric stability of extremal Kähler metrics with singularities.
We introduce an inductive argument for proving birational superrigidity and K-stability of singular Fano complete intersections of index one, using the same types of information from lower dimensions. In particular, we prove that a hypersurface in $\mathbb{P}^{n+1}$ of degree $n+1$ with only ordinary singularities of multiplicity at most $n-5$ is birationally superrigid and K-stable if $n\gg0$. As part of the argument, we also establish an adjunction type result for local volumes of singularities.
We prove the K-moduli space of cubic threefolds is identical to their GIT moduli. More precisely, the K-(semi,poly)-stability of cubic threefolds coincide to the corresponding GIT stabilities, which could be explicitly calculated. In particular, this implies that all smooth cubic threefolds admit Kähler-Einstein metric as well as provides a precise list of singular KE ones. To achieve this, our main new contribution is an estimate in dimension three of the volumes of kawamata log terminal singularities introduced by Chi Li. This is obtained via a detailed study of the classification of three d…
Let $M$ be a hyperkaehler manifold, and $F$ a torsion-free and reflexive coherent sheaf on $M$. Assume that $F$ (outside of its singularities) admits a connection with a curvature which is invariant under the standard SU(2)-action on 2-forms. If the curvature is square-integrable, then $F$ is stable and its singularities are hyperkaehler subvarieties in $M$. Such sheaves (called hyperholomorphic sheaves) are well understood. In the present paper, we study sheaves admitting a connection with SU(2)-invariant curvature which is not necessarily square-integrable. This situation arises often, for i…
We extend the framework of K-stability (Tian, Donaldson) to more general algebro-geometric setting, such as partial desingularisations of (fixed) singularities, (not necessarily flat) families over higher dimensional base and the classical birational geometry of surfaces. We also observe that "concavity" of the volume function implies decrease of the (generalised) Donaldson-Futaki invariants along the Minimal Model Program, in our generalised settings. Several related results on the connection with the MMP theory, some of which are new even in the original setting of families over curves, are …
We prove stability of logarithmic tangent sheaves of singular hypersurfaces D of the projective space with constraints on the dimension and degree of the singularities of D. As main application, we prove that determinants and symmetric determinants have stable logarithmic tangent sheaves and we describe an open dense piece of the associated moduli space.
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。