Tom Bridgeland
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.
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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.
Contractibility of the geometric stability manifold of a surface
本研究基于几何稳定性流形的最新刻画，证明任意光滑复射影曲面的几何稳定性流形可缩，并借此发现无穷多族稳定性流形可缩的新曲面族。
We prove stability of rank two tautological bundles on the Hilbert square of a surface (under a mild positivity condition) and compute their Chern classes.
Let $L$ be a globally generated line bundle over a smooth irreducible complex projective surface $X$. The syzygy bundle $M_{L}$ is the kernel of the evaluation map $H^0(L)\otimes\mathcal O_X\to L$. We prove the $L$-stability of $M_L$ for Hirzebruch surfaces, del Pezzo surfaces and Enriques surfaces. The $(-K_X)$-stability of syzygy bundles $M_L$ over del Pezzo surfaces is also obtained.
We describe the GIT compactification of the moduli of (2,2)-type effective divisors of $\mathbb{P}^1\times\mathbb{P}^2$ (i.e., surfaces of the linear system $\vert π_1^*\mathcal{O}_{\mathbb{P}^1}(2)\otimes π_2^*\mathcal{O}_{\mathbb{P}^2}(2)\vert$ ) which are generically Del Pezzo surfaces of degree two. In order to get the compactification, we characterize stable and semi-stable (2,2)-type surfaces, and also determine the equivalence classes of strictly semi-stable (2,2)-type surfaces. Moreover, we describe the boundary of the moduli of (2,2)-type surfaces.
Stability and Fourier-Mukai transforms on an eliptic surface
本文研究椭圆面相干层的稳定性条件及其在相对傅里叶-穆卡伊变换下的行为。
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。
We verify that the Albanese fibration of the Cartwright-Steger surface is stable, answering a problem left open in [CKY].
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.
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。
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 shall study the chamber structure of positive cone of the albanese fiber of the moduli spaces of stable objects on an abelian surfaces via the chamber structure of stability conditions.
In this short note, we study the asymptotic Chow polystability of toric Del Pezzo surfaces appear in the moduli space of Kähler-Einstein Fano varieties constructed in [OSS16].
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。
Stability of Kernel Sheaves on Del Pezzo Surfaces
本文利用 Bridgeland 稳定性技术，首次证明了 Del Pezzo 曲面上由充分正的 Gieseker 稳定层诱导的核层具有斜率稳定性，填补了高秩层核层稳定性有效结果及 Gieseker 稳定层核层稳定性结果的空白。
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.
We study the Bridgeland stability of line bundles on surfaces using Bridgeland stability conditions determined by divisors. We show that given a smooth projective surface $S$, a line bundle $L$ is always Bridgeland stable for those stability conditions if there are no curves $C\subseteq S$ of negative self-intersection. When a curve $C$ of negative self-intersection is present, $L$ is destabilized by $L(-C)$ for some stability conditions. We conjecture that line bundles of the form $L(-C)$ are the only objects that can destabilize $L$, and that torsion sheaves of the form $L(C)|_C$ are the onl…
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.
Let $(X, H)$ be a polarized smooth projective algebraic surface and $E$ is globally generated, stable vector bundle on $X$. Then the Syzygy bundle $M_E$ associated to it is defined as the kernel bundle corresponding to the evaluation map. In this article we will study the stability property of $M_E$ with respect to $H$.
Let $F\subseteq\mathbb P ^{a+1}$ be a non-degenerate $K3$ surface of degree $2a$, where $a\ge2$. In this paper we deal with Ulrich bundles on $F$ of rank $2$. We deal with their stability and we construct $K3$ surfaces endowed with families of non-special Ulrich bundles of rank $2$ for each $a\ge2$.
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 $…
Let X be a smooth projective surface over C and let L be a line bundle on X generated by its global sections. Let f:X-->P^r be the morphism associated to L and let T be the tangent bundle of P^r; we investigate the μ-stability of f*T with respect to L when X is either a regular surface with p_g=0, a K3 surface or an abelian surface. In particular, we show that it is μ-stable when X is K3 and L is ample and when X is abelian and L^2>13.
In an algebro-geometric way, we completely determine whether smooth del Pezzo surfaces are K-(semi)stable or not.
In this article, we show that some semi-rigid $μ$-stable sheaves on a projective K3 surface $X$ with Picard number 1 are stable in the sense of Bridgeland's stability condition. As a consequence of our work, we show that the special set $U(X) \subset \Stab (X)$ reconstructs $X$ itself. This gives a sharp contrast to the case of an abelian surface.
Let $X$ be a smooth projective surface over an algebraically closed field $k$ of characteristic $p> 0$ with $Ω_{X}^{1}$ semistable and $μ(Ω_{X}^{1})>0$. For any semistable (resp. stable) bundle $W$ of rank $r$, we prove that $F_*W$ is semistable (resp. stable) when $p\geq r(r-1)^2+1$.
In this paper, we consider the preservation of stability by using the notion of Twisted stability. As applications, (1) we show that moduli spaces of vector bundles on K3 and abelian surfaces are irreducible, (2) we compute Hodge polynomials of some moduli spaces of vector bundles on Enriques surfaces and on elliptic surfaces with a section.
We show that some Gieseker stable sheaves on a projective K3 surface $X$ are stable with respect to a stability condition of Bridgeland on the derived category of $X$ if the stability condition is in explicit subsets of the space of stability conditions depending on the sheaves. Furthermore we shall give two applications of the result. As a part of these applications, we show that the fine moduli space of Gieseker stable torsion free sheaves on a K3 surface with Picard number one is the moduli space of $μ$-stable locally free sheaves if the rank of the sheaves is not a square number.
Stability of the Cotangent Bundle on Surfaces with Ample Canonical Class
本文研究K3曲面及具有丰富典范类的曲面上余切丛的Bridgeland稳定性，找到了阈值$t_0$使得余切丛在$t>t_0$时半稳定，并探讨了伪射影平面和伪二次曲面等特例。
Given a very ample line bundle L on a projective variety X, the syzygy bundle M_L associated to L is the kernel of the evaluation map on sections of L. Our main result is that if X is a smooth projective surface defined over an algebraically closed field, then M_L is slope-stable for any sufficiently positive L.
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.
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。
K-stability of del Pezzo surfaces with a single quotient singularity
本文研究带单个商奇点的del Pezzo曲面的K-稳定性，该曲面的极小解含两条满足E₁²=-n、E₂²=-m（n,m≥2）的例外曲线。
Over the last few years, wall-crossing for Bridgeland stability conditions has led to a large number of results in algebraic geometry, particular on birational geometry of moduli spaces. We illustrate some of the methods behind these result by reproving Lazarsfeld's Brill-Noether theorem for curves on K3 surfaces via wall-crossing. We conclude with a survey of recent applications of stability conditions on surfaces. The intended reader is an algebraic geometer with a limited working knowledge of derived categories. This article is based on the author's talk at the AMS Summer Institute on Algeb…
We study slope stability of smooth surfaces and its connection with exceptional divisors. We show that a surface containing an exceptional divisor with arithmetic genus at least two is slope unstable for some polarisation. In the converse direction we show that slope stability of surfaces can be tested with divisors, and prove that for surfaces with non-negative Kodaira dimension any destabilising divisor must have negative self-intersection and arithmetic genus at least two. We also prove that a destabilising divisor can never be nef, and as an application give an example of a surface that is…
Bridgeland Stability of Sheaves on del Pezzo Surface of Picard Rank Three
该研究针对皮卡秩3的德尔佩佐曲面（两点吹胀的$\u2119^{2}$）上的层，确定了线丛的失稳对象，证明了任意$(-1)$-曲线的限制层在所有除子型Bridgeland稳定性条件下均稳定，推进了代数几何中Bridgeland稳定性的具体计算研究。
In [Ma1] S. Ma established a bijection between Fourier--Mukai partners of a K3 surface and cusps of the Kähler moduli space. The Kähler moduli space can be described as a quotient of Bridgeland's stability manifold. We study the relation between stability conditions $σ$ near to a cusp and the associated Fourier--Mukai partner Y in the following ways. (1) We compare the heart of $σ$ to the heart of coherent sheaves on Y. (2) We construct Y as moduli space of $σ$-stable objects. An appendix is devoted to the group of auto-equivalences of the derived category which respect the component $Stab^\da…
Let f :S\to B be a non locally trivial fibred surface. We prove a lower bound for the slope of f depending increasingly from the relative irregularity of f and the Clifford index of the general fibres.
On a Weierstrass elliptic surface, we describe the action of the relative Fourier-Mukai transform on the geometric chamber of $\mathrm{Stab}(X)$, and in the K3 case we also study the action on one of its boundary components. Using new estimates for the Gieseker chamber we prove that Gieseker stability for polarizations on certain Friedman chamber is preserved by the derived dual of the relative Fourier-Mukai transform. As an application of our description of the action, we also prove projectivity for some moduli spaces of Bridgeland semistable objects.
Nets of quadric surfaces and plane cubics and their GIT stability
本文研究四次曲面网络与平面三次曲线的对应关系，通过Gale变换定义，并利用birational-geometric技术提供三重几何对象的GIT稳定性判据。
We give a combinatorial criterion for the tangent bundle on a smooth toric variety to be stable with respect to a given polarisation in terms of the corresponding lattice polytope. Furthermore, we show that for a smooth toric surface and a smooth toric variety of Picard rank 2, there exists an ample line bundle with respect to which the tangent bundle is stable if and only if it is an iterated blow-up of projective space.
We give a explicit description of gluing stability conditions on ruled surfaces by introducing gluing perversity. Moreover, we describe a destabilizing wall of skyscraper sheaves on ruled surfaces by deformation of stability conditions glued from $\widetilde{GL^{+}}(2,\mathbb{R})$-translates of the standard stability condition on the base curve.
Hurwitz-Brill-Noether theory via K3 surfaces and stability conditions
本文通过椭圆K3曲面上的Bridgeland稳定性条件，发展Hurwitz-Brill-Noether理论的新方法，证明椭圆K3曲面上的曲线是首个普遍的光滑k-gonal曲线例子，并给出主要存在性与不存在性结果的新证明，以及数域上普遍曲线的显式构造。
Let $X$ be a smooth compact complex surface with the canonical divisor $K_X$ ample and let $Θ_X$ be its holomorphic tangent bundle. Bridgeland stability conditions are used to study the space $H^1 (Θ_X)$ of infinitesimal deformations of complex structures of $X$ and its relation to the geometry/topology of $X$. The main observation is that for $X$ with $H^1 (Θ_X)$ nonzero and the Chern numbers $(c_2 (X), K^2_X)$ subject to $$ τ_X :=2ch_2 (Θ_X)=K^2_X -2c_2(X) >0 $$ the object $Θ_X [1]$ of the derived category of bounded complexes of coherent sheaves on $X$ is Bridgeland unstable in a certain pa…
Stability for Line Bundles and Deformed Hermitian-Yang-Mills Equation on Some Elliptic Surfaces
本文研究了曲面上的扭曲丰富性准则及其与变形Hermitian-Yang-Mills方程的等价关系，证明了Weierstrass椭圆K3曲面上特定线丛的扭曲丰富性蕴含Bridgeland稳定性，回答了Collins和Yau的问题。
We construct a topological embedding of the maximal connected component of Bridgeland stability conditions of a (twisted) Abelian surface into the distinguished connected component of the stability manifold of the associated (twisted) Kummer surface. We use methods developed for orbifold conformal field theories.
In this note, we prove that the syzygy bundle $M_L$ is cohomologically stable with respect to $L$ for any ample and globally generated line bundle $L$ on an Enriques (resp. bielliptic) surface over an algebraically closed field of characteristic $\neq 2$ (resp. $\neq 2,3$). In particular our result on complex Enriques surfaces improves a result of Torres-López and Zamora by removing a condition on Clifford index. Together with the results of Camere and Caucci--Lahoz, it implies that $M_L$ is stable with respect to $L$ for an ample and globally generated line bundle $L$ on any smooth minimal co…
Compactifications of moduli of del Pezzo surfaces via line arrangement and K-stability
本文利用K-稳定性和线构形研究光滑del Pezzo曲面模空间的紧化，构造了log del Pezzo对的K-模空间，并证明了其在$d=2,3,4$时同构于del Pezzo曲面的K-模空间，而在$d=1$时两者不同。
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 prove some general statements on stability conditions of Calabi-Yau surfaces and discuss the stability manifold of the cotangent bundle of P^1. Our primary interest is in spherical objects.
It is shown that there is a useful notion of a relative Bridgeland stability condition on the partially wrapped Fukaya category of a marked surface, relative to some part of the surface's boundary. This construction has nice functorial properties, obeying cutting and gluing relations. This reduces the calculation of stability conditions on the Fukaya category of any fully stopped surface into three types of base cases. Calculations of these cases shows that every Bridgeland stability condition on such categories can be described by flat surfaces. In other words, the map constructed by Haiden-K…
We identify spaces of half-translation surfaces, equivalently complex curves with quadratic differential, with spaces of stability structures on Fukaya-type categories of punctured surfaces. This is achieved by new methods involving the complete classification of objects in these categories, which are defined in an elementary way. We also introduce a number of tools to deal with surfaces of infinite area, where structures similar to those in cluster algebra appear.
Stability conditions on K3 surfaces via mass of spherical objects
该研究证明K3曲面上的稳定性条件由球状对象质量经自然$\mathbb{C}$-作用唯一确定，受相关紧化构造提议启发，还构造了与球状丛相关的松弛稳定性条件，推进了稳定性流形紧化研究。
In 1980, I. Morrison proved that slope stability of a vector bundle of rank 2 over a compact Riemann surface implies Chow stability of the projectivization of the bundle with respect to certain polarizations. We generalized Morrison's result to higher rank vector bundles over compact algebraic manifolds of arbitrary dimension that admit constant scalar curvature metric and have discrete automorphism group. In this article, we give a simple proof for polarizations $\mathcal{O}_{\mathbb{P}E^*}(d)\otimes π^* L^k$, where $d$ is a positive integer, $k \gg 0$ and the base manifold is a compact Riema…
The Thurston compactification of the stability manifold of a generic analytic K3 surface
该研究针对Picard群为0的解析K3曲面，借助半刚性对象的质量刻画了其Bridgeland稳定性流形的闭包，推进了相关几何结构的研究。
We establish some remarkable properties of the cohomology rings of the Hilbert schemes of n points on a projective surface X, from which one sees to what extent these cohomology rings are (in)dependent of X and n.
For every integer $a \geq 2$, we relate the K-stability of hypersurfaces in the weighted projective space $\mathbb{P}(1,1,a,a)$ of degree $2a$ with the GIT stability of binary forms of degree $2a$. Moreover, we prove that such a hypersurface is K-polystable and not K-stable if it is quasi-smooth.
We show that the K-moduli spaces of log Fano pairs $(\mathbb{P}^3, cS)$ where $S$ is a quartic surface interpolate between the GIT moduli space of quartic surfaces and the Baily-Borel compactification of moduli of quartic K3 surfaces as $c$ varies in the interval $(0,1)$. We completely describe the wall crossings of these K-moduli spaces. As the main application, we verify Laza-O'Grady's prediction on the Hassett-Keel-Looijenga program for quartic K3 surfaces. We also obtain the K-moduli compactification of quartic double solids, and classify all Gorenstein canonical Fano degenerations of $\ma…
Families of Bridgeland stability conditions induce families of stability data, wall-crossing structures and scattering diagrams on the motivic Hall algebra. These structures can be transferred to the quantum torus if the stability conditions of the family have global dimension at most 2. We show that geometric stability conditions on surfaces with nef anticanonical bundle have global dimension 2 and we study the resulting family of stability data. We formulate a conjecture relating this family to the family of stability data associated with a quiver with potential and we verify this conjecture…
We study the space of stability conditions on $K3$ surfaces from the perspective of mirror symmetry. It is done in the so called attractor backgrounds (moduli) which can be far from the conventional large complex limits and are selected by the attractor mechanism for certain black holes. We find certain highly non-generic behaviors of stability walls (a key notion in the study of wall crossings) in the space of stability conditions. They correspond via mirror symmetry to some non-generic behaviors of special Lagrangians in an attractor background. The main results can be understood as a mirror…
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.
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