Akira Ishii, Kazushi Ueda, Hokuto Uehara
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.
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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。
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…
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 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.
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…
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。
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…
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。
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。
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.
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.
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。
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.
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。
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.
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。
Bekka's $(c)$-regularity condition and families of line singularities with constant Lê numbers
本文研究了具有恒定 Lê 数的线奇异性族，证明其满足 Bekka 的 $(c)$-regularity 条件，并得出其拓扑等奇异的结论。
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。
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…
Hodge adjacency conditions for singularities
本文研究孤立 $k$-Du Bois 奇点及其平坦射影族，证明纤维混合 Hodge 数与泛形变间的相容关系，并将极化关系推广到正规交叉边界之外，分析弱极化关系图的组合性质。
Greuel, Lossen and Shustin gave a general sufficient numerical condition for the T-smoothness (smoothness and expected dimension) of equisingular families of plane curves. This condition involves a new invariant γfor plane curve singularities, and it is conjectured to be asymptotically proper. In math.AG/0308247, similar sufficient numerical conditions are obtained for the T-smoothness of equisingular families on various classes surfaces. These conditions involve a series of invariants γ_a, 0 <= a <= 1, with γ_1=γ. In the present paper we compute (respectively give bounds for) these invariants…
We investigate the equisingularity question for $1$-parameter deformation families of mixed polynomial functions $f_t(\mathbf{z},\bar{\mathbf{z}})$ from the Newton polygon point of view. We show that if the members $f_t$ of the family satisfy a number of elementary conditions, which can be easily described in terms of the Newton polygon, then the corresponding family of mixed hypersurfaces $f_t^{-1}(0)$ is Whitney equisingular (and hence topologically equisingular) and satisfies the Thom condition.
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 give some necessary conditions for the existence of a symplectic resolution for quotient singularities. The McKay correspondence is also worked out for these resolutions.
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 …
The aim of this paper is to study the singularities of certain moduli spaces of sheaves on K3 surfaces by means of Nakajima quiver varieties. The singularities in question arise from the choice of a non--generic polarization, with respect to which we consider stability, and admit natural symplectic resolutions corresponding to choices of general polarizations. For sheaves that are pure of dimension one, we show that these moduli spaces are, locally around a singular point, isomorphic to a quiver variety and that, via this isomorphism, the natural symplectic resolutions correspond to variations…
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.
We study adjacency of equisingularity types of planar curve singularities in terms of their Enriques diagrams. For linear adjacency a complete answer is obtained, whereas for arbitrary (analytic) adjacency a necessary condition and a sufficient condition are proved. We also show an example of singular curve of type D' that can be deformed to a curve of type D without D' being adjacent to D.
We give sufficient conditions, on data including the monodromy representation, the Stokes matrices and the Poincare ranks at prescribed singularities, to solve the generalized Riemann-Hilbert problem with irregular singularities. We recover in particular the irreducibility condition on the monodromy given by Bolibrukh and Kostov in the classical case. We apply the above criteria to solve the inverse problem in differential Galois theory with a better control of the singularities.
Let f be a definable function, enough differentiable. Under the condition of having strongly isolated singularities at infinity at a regular value c we give a sufficient condition expressed in terms of the total absolute curvature function to ensure the local triviality of the function f over a neighbourhood of c and doing so providing the tame version of Parusinski's Theorem on complex polynomials with isolated singularities at infinity.
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。
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