Tom Bridgeland
Stability conditions are a mathematical way to understand $Π$-stability for D-branes in string theory. Spaces of stability conditions seem to be related to moduli spaces of conformal field theories. This is a survey article describing what is currently known about spaces of stability conditions, and giving some pointers for future research.
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Stability conditions on blowups
本文研究爆破下反常与几何稳定性条件的关系，证实Toda猜想于特例，并表明几何稳定性条件可由半正交分解诱导的反常稳定性条件导出。
Weak Stability Conditions as Limits of Bridgeland Stability Conditions
本文提出三角范畴上允许核中对象具有非极大相位的弱稳定性条件定义，并在 Weierstrass 椭圆曲面上构造了四类作为 Bridgeland 稳定性条件极限的弱稳定性条件。
A note on stability conditions on projective spaces
给出李定理关于射影空间上凝聚层有界导出范畴几何布里奇兰德稳定性条件存在性的新证明，此条件可限制到任意光滑射影簇诱导布里奇兰德稳定性条件。
Stability conditions on some families of Calabi-Yau threefolds via orbifolding
该研究通过轨形构造方法，证明了源自具有Bridgeland稳定性条件的Calabi-Yau三维流形族的衍生族也满足该条件，并验证了五次镜像簇的相关性质，推进了代数几何中稳定性条件的研究。
Stability conditions on the canonical line bundle of $\mathbb{P}^3$
本文研究三维射影空间上典范线丛全空间的稳定性条件空间，构造了几何稳定性条件族及其代数边界子集，并利用球面扭转构造其他稳定性条件。
Given a stability condition on a smooth projective variety $X$, we construct a family of stability conditions on $X\times C$, where $C$ is a smooth projective curve. In particular, this gives the existence of stability conditions on arbitrary products of curves. The proof uses, by following an idea of Toda, the positivity lemma established by Bayer and Macrì and weak stability conditions on the Abramovich-Polishchuk heart of a bounded t-structure in $D(X\times C)$.
We study stability conditions induced by functors between triangulated categories. Given a finite group acting on a smooth projective variety we prove that the subset of invariant stability conditions embeds as a closed submanifold into the stability manifold of the equivariant derived category. As an application we examine stability conditions on Kummer and Enriques surfaces and we improve the derived version of the Torelli Theorem for the latter surfaces already present in the litterature. We also study the relationship between stability conditions on projective spaces and those on their can…
We introduce the notion of a polynomial stability condition, generalizing Bridgeland stability conditions on triangulated categories. We construct and study a family of polynomial stability conditions for any normal projective variety. This family includes both Simpson-stability, and large volume limits of Bridgeland stability conditions. We show that the PT/DT-correspondence relating stable pairs to Donaldson-Thomas invariants (conjectured by Pandharipande and Thomas) can be understood as a wall-crossing in our family of polynomial stability conditions. Similarly, we show that the relation be…
Stability conditions on degenerated elliptic curves
本文研究退化椭圆曲线的稳定性条件，描述了稳定性条件空间的连通分量并计算了deck变换群。
Stability conditions for coherent systems on Integral Curves
针对整曲线上的凝聚系统范畴，本文利用倾斜方法构造三参数预稳定性条件族，判别其成为真稳定性条件的情形，并分析三类特定对象的半稳定性。
We shall study stability conditions and Fourier-Mukai transforms on an elliptic surface. In particular we shall explain duality of elliptic surfaces by Fourier-Mukai transforms.
We study some examples of Bridgeland-Douglas stability conditions on triangulated categories. From one side we give a complete description of the stability manifolds for smooth projective curves of positive genus. From the other side we study stability conditions on triangulated categories generated by an exceptional collection. In the case of the projective line this leads to the connectedness and simply-connectedness of the stability manifold.
We give some remarks on our papers with Minamide and Yanagida on Bridgeland stability conditions. We also give a remark on stability conditions on Enriques surfaces, and give another proof of the projectivity of the coarse moduli spaces of semi-stable objects, which were obtained by Nuer.
In this paper, we prove BG-type inequality conjecture for threefolds in the title. In particular, there exist Bridgeland stability conditions on these threefolds.
Bridgeland/Weak Stability Conditions under Spherical Twist Associated to A Torsion Sheaf
本文研究挠层诱导的球面扭转对K3曲面导出范畴上Bridgeland及弱稳定性条件的作用，构造了与非nef除子相关的稳定性条件，并讨论了线丛的稳定性。
Stability conditions on threefolds
研究满足Li条件的\(\PP^n\)上Bridgeland稳定性条件的子空间，证明当\(X\)为三维时受限稳定性条件与通过Bayer-Macrì-Toda双倾斜构造得到的一致，应用此证明了弱BMT猜想。
In this paper, we prove the Bogomolov-Gieseker type inequality conjecture for threefolds with nef tangent bundles. As a corollary, there exist Bridgeland stability conditions on these threefolds.
We show that, under mild conditions, the space of numerical Bridgeland stability conditions Stab(T) on a triangulated category T is complete. In particular the metric on a full component of Stab(T) for which the central charges factor through a finite rank quotient of the Grothendieck group K(T) is complete. As an example, we compute the metric on the space of numerical stability conditions on a smooth complex projective curve of genus greater than one, and show that in this case the quotient Stab(T)/C by the natural action of the complex numbers is isometric to the upper half plane equipped w…
Stability conditions on surface root stacks
本文在曲面根栈上构造了倾斜稳定性条件，并证明其关于有理Chen-Ruan上同调满足支撑性质。
The space of Bridgeland stability conditions on the bounded derived category of coherent sheaves on P2 has a principle connected component Stab^†(P2). We show that Stab^†(P2) is the union of geometric and algebraic stability conditions. As a consequence, we give a cell decomposition for Stab^†(P2) and show that Stab^†(P2) is contractible.
In this paper we describe a simply connected component of the complex manifold of stability conditions on the bounded derived category of a generic complex torus of any dimension. A torus is called generic if there are no nontrivial integral (p,p)-classes. We give an explicit description of all stability conditions in that connected component.
Weak stability conditions on coherent systems of genus four curves
本文研究四曲面相干系统的稳定性条件，证明其退化为节点三次三折面Kuznetsov组件上的稳定性条件。
This paper contains a description of one connected component of the space of stability conditions on the bounded derived category of coherent sheaves on a complex algebraic K3 surface.
Let $X$ be a smooth complex projective variety. In 2002, Bridgeland defined a notion of stability for the objects in $D^b(X)$, the bounded derived category of coherent sheaves on $X$, which generalized the notion of slope stability for vector bundles on curves. There are many nice connections between stability conditions on $X$ and the geometry of the variety. We construct new stability conditions for surfaces containing a curve $C$ whose self-intersection is negative. We show that these stability conditions lie on a wall of the geometric chamber of ${\rm Stab}(X)$, the stability manifold of $…
Bridgeland stability conditions on normal surfaces
本文在正规正常曲面上证明了新版本的Bogomolov不等式，从而构造了Bridgeland稳定性条件，并给出了非射影正常概形上稳定性条件的首批已知实例。
We prove the conjectural Bogomolov-Gieseker type inequality for tilt slope stable objects on each Fano threefold X of Picard number one. Based on the previous works on Bridgeland stability conditions, this induces an open subset of geometric stability conditions on D^b(X). We also get a new stronger bound for the Chern characters of slope semistable sheaves on X.
We investigate the behaviour of Bridgeland stability conditions under change of base field with particular focus on the case of finite Galois extensions. In particular, we prove that for a variety X over a field K and a finite Galois extension L/K the stability manifold of X embeds as a closed submanifold into the stability manifold of the base change variety.
Stability conditions on products of curves and Hilbert schemes of surfaces
研究曲线积导出范畴的稳定性条件，构造特定曲面点希尔伯特方案的稳定性条件。
We show that the stability conditions on the Kuznetsov component of a Gushel-Mukai threefold, constructed by Bayer, Lahoz, Macrì and Stellari, are preserved by the Serre functor, up to the action of the universal cover of $\text{GL}^+_2(\mathbb{R})$. As application, we construct stability conditions on the Kuznetsov component of special Gushel-Mukai fourfolds.
Wall in the stability space of the gluing stability conditions on Hirzebruch surfaces
本文研究Hirzebruch面上稳定性条件空间的墙结构，通过粘合构造分析几何室与墙的交集，并确定半稳定对象的模空间。
We give a conjectural construction of Bridgeland stability conditions on the derived category of fibred threefolds. The construction depends on a conjectural Bogomolov-Gieseker type inequality for certain stable complexes. It can be considered as a relative version of the construction of Bayer, Macrì and Toda. We prove the conjectural Bogomolov-Gieseker type inequality in the case of relative projective planes over curves. This gives the the existence of Bridgeland stability conditions on such threefolds.
Bridgeland stability condition is preserved under the Fourier-Mukai transform by its definition. We explain the relation with Gieseker stability. By studying the wall-crossing behavior, we reprove that the moduli spaces of stable sheaves on abelian surfaces are birationally equivalent, if the associated Mukai vectors are related by isometries of the Mukai lattice.
Noncommutative varieties and stability conditions: an overview
本文综述了非交换代数几何中稳定性条件和模空间理论的最新进展及其在超kähler几何和经典代数几何中的应用。
In this paper, we describe the spaces of stability conditions for the triangulated categories associated to three dimensional Calabi-Yau fibrations. We deal with two cases, the flat elliptic fibrations and smooth K3 (Abelian) fibrations. In the first case, we will see there exist chamber structures similar to those of the movable cone used in birational geometry. In the second case, we will compare the space with the space of stability conditions for the closed fiber of the fibration.
Stability conditions on crepant resolutions of quotients of product varieties
本文在乘积簇之商的crepant消解上构造稳定条件，得到任意维数严格Calabi-Yau簇的首批例子，并证明其与商stack导出等价，验证Bondal-Orlov猜想的一个实例。
We give a complete description of the Bridgeland stability manifold for the bounded derived category of holomorphic triples over a smooth projective curve of genus 1 as a connected, four dimensional complex manifold.
Polynomial Bridgeland Stability Conditions on the Category of Coherent Sheaves
本文提出了凝聚层范畴上一类推广Gieseker稳定性的适配型稳定性条件，证明其具备Harder–Narasimhan与Jordan–Hölder滤过，并探究了Bayer多项式Bridgeland稳定性条件与对应规范论概念的关联。
A Remark on Stability Conditions on Smooth Projective Varieties
本文证明了平滑射影簇的有界导出范畴总能接纳Bridgeland意义下的稳定性条件。
In this survey we borrow from Coskun and Huizenga an example of application of Bridgeland stability conditions to birational geometry and we rephrase it without assuming any previous knowledge about derived categories.
Stability conditions supported on Lefschetz classes
本文推广C. Li的工作，在莱夫谢茨型簇上构造满足Kontsevich-Soibelman支撑性质的全支撑稳定性条件，覆盖维数≤3的光滑复射影簇及光滑射影环面簇。
We derive constraints on the existence of walls for Bridgeland stability conditions for general projective surfaces. We show that in suitable planes of stability conditions the walls are bounded and derive conditions for when the number of walls is globally finite. In examples, we show how to use the explicit conditions to locate walls and sometimes to show that there are no walls at all.
We prove that a connected component of the space of stability conditions of a CY3 triangulated category generated by an A_2 collection of 3-spherical objects is isomorphic to the universal cover of the C^*-bundle of non-zero holomorphic differentials on the moduli space of elliptic curves.
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。
Ishii's conjecture and Bridgeland stability conditions for dihedral reflection groups
本文从Bridgeland稳定性条件的角度，给出了Ishii猜想在任意二面体反射群上的新证明，通过导出的McKay对应关系，将问题转化为根堆栈上的几何构造。
We show the existence of Gepner type Bridgeland stability conditions on the triangulated categories of graded matrix factorizations associated with homogeneous polynomials which define general cubic fourfolds containing a plane. The key ingredient is to describe the grade shift functor of matrix factorizations in terms of sheaves of Clifford algebras on the projective plane under Orlov/Kuznetsov equivalence.
We show that any extremal contraction from a smooth projective variety with dimension less than or equal to three appears as a moduli space of (semi)stable objects in the derived category of coherent sheaves.
Derived category of coherent systems on curves and stability conditions
针对亏格$g>0$的光滑射影曲线$C$，刻画其凝聚层有界导出范畴的Bridgeland稳定性条件的开轨迹，证明该稳定性流形可检测曲线的Brill--Noether理论。
We prove that ideal sheaves of lines in a Fano threefold $X$ of Picard rank one and index two are stable objects in the Kuznetsov component $\mathsf{Ku}(X)$, with respect to the stability conditions constructed by Bayer, Lahoz, Macrì and Stellari, giving a modular description to the Hilbert scheme of lines in $X$. When $X$ is a cubic threefold, we show that the Serre functor of $\mathsf{Ku}(X)$ preserves these stability conditions. As an application, we obtain the smoothness of non-empty moduli spaces of stable objects in $\mathsf{Ku}(X)$. When $X$ is a quartic double solid, we describe a conn…
We show that the construction of Bayer, Bertram, Macri and Toda gives rise to a Bridgeland stability condition on a principally polarized abelian threefold with Picard rank one by establishing their conjectural generalized Bogomolov-Gieseker inequality for certain tilt stable objects. We do this by proving that a suitable Fourier-Mukai transform preserves the heart of a particular conjectural stability condition. We also show that the only reflexive sheaves with zero first and second Chern classes are the flat line bundles.
These are notes of a course given at the 'school on moduli spaces' at the Newton Institute in January 2011. The abstract theory of stability conditions (due to Bridgeland and Douglas) on abelian and triangulated categories is developed via tilting and t-structures. Special emphasis is put on the bounded derived category of coherent sheaves on smooth projective varieties (in particular for curves and K3 surfaces). The lectures were targeted at an audience with little prior knowledge of triangulated categories and stability conditions but with an interest in vector bundles on curves.
The key result in the theory of Bridgeland stability conditions is the property that they form a complex manifold. This comes from the fact that given any small deformation of the central charge, there is a unique way to correspondingly deform the stability condition. We give a short direct proof of a strong version of this deformation property.
An object in the bounded derived category D^b(Coh(X)) of coherent sheaves on a complex projective K3 surface X is spherical if it is rigid and simple. Although spherical objects form only a discrete set in the moduli stack of complexes, they determine much of the structure of X and D^b(Coh(X)). Here we show that a stability condition on D^b(Coh(X)) is determined by the stability of spherical objects.
We define and study a gluing procedure for Bridgeland stability conditions in the situation when a triangulated category has a semiorthogonal decomposition. As an application we construct stability conditions on the derived categories of ${\mathbb{Z}}_2$-equivariant sheaves associated with ramified double coverings of ${\mathbb{P}}^3$. Also, we study the stability space for the derived category of ${\mathbb{Z}}_2$-equivariant coherent sheaves on a smooth curve $X$, associated with a degree 2 map $X\to Y$, where $Y$ is another smooth curve. In the case when the genus of $Y$ is $\geq 1$ we give …
We prove the Bogomolov-Gieseker type inequality conjectured by Bayer, Macri and Toda for threefolds with semistable tangent bundles and vanishing Chern classes in any characteristic, which was originally proved by Bayer, Macri and Stellari in characteristic zero. This gives the existence of Bridgeland stability conditions on such threefolds. As applications, we obtain Reider type theorem and confirm Fujita's conjecture for such threefolds in any characteristic.
Quasi-convergence of stability conditions
本文提出了一种将三角范畴的半正交分解与稳定性条件空间中的路径联系起来的框架，并证明当范畴是光滑且良好的idempotent complete预三角dg-范畴时，所有因子具有Bridgeland稳定性条件的半正交分解均可由此框架获得。
We prove the Bogomolov-Gieseker type inequality conjectured by Bayer, Macri and Toda for some products of three curves. This gives the first examples of Bridgeland stability conditions on some threefolds of general type. The key ingredients are the spreading out technique, Frobenius morphism and Bogomolov's inequality for product type varieties in positive characteristic, proved by the author recently.
Bridgeland stability conditions on some higher-dimensional Calabi--Yau manifolds and generalized Kummer varieties
本文在四类高维Calabi--Yau或超Kähler流形上构造Bridgeland稳定性条件，覆盖广义Kummer簇、Enriques曲面Hilbert概形的万有覆盖、双椭圆曲面覆盖的Albanese纤维及Cynk--Hulek流形，提供各维度的例子。
Using Bridgeland stability conditions we give sufficient criteria for a stable vector bundle on a surface to remain stable when restricted to a curve. We give a stronger criterion when the vector bundle is a general vector bundle on the plane. As an application, we compute the cohomology of such bundles for curves that lie in the plane or on Hirzebruch surfaces.
We show that the minimal model program on any smooth projective surface is realized as a variation of the moduli spaces of Bridgeland stable objects in the derived category of coherent sheaves.
Stability conditions and positivity of invariants of fibrations
本文研究证明极化簇族数值不变量正性的三种方法，揭示二维情形下它们与线性稳定性的联系，并在高维中提出推测并证明新结果。
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