Weiqiang Wang
We give canonical resolutions of singularities of several cone varieties arising from invariant theory. We establish a connection between our resolutions and resolutions of singularities of closure of conjugacy classes in classical Lie algebras.
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A cone singularity is a normal affine variety $X$ with an effective one-dimensional torus action with a unique fixed point $x\in X$ which lies in the closure of any orbit of the $k^*$-action. In this article, we prove a boundedness theorem for cone singularities in terms of their dimension, singularities, and isotropies. Given $d$ and $N$ two positive integers and $ε$ a positive real number, we prove that the class of $d$-dimensional $ε$-log canonical cone singularities with isotropies bounded by $N$ forms a bounded family.
Resolution of singularities of the odd nilpotent cone of orthosymplectic Lie superalgebras
本文构造了正交辛李超代数osp(m|2n)奇nilpotent锥的Springer型奇点消解方法，解决奇点消解问题。
Bilipschitz geometry of real surface singularities whose tangent cone is a plane
本文研究实曲面奇点在双Lipschitz等价下Nash锥与例外射线的行为，针对切锥为平面的孤立奇点情形。
Double Veronese cones with singularities
研究具有终端Gorenstein奇点的双重Veronese锥，确定节点数上限，分析自同构群结构，并建立有理性和无理性的判据。
We study the null-cone of a semi-simple algebraic group acting on a number of copies of its Lie algebra via the diagonal adjoint action. We show that the null cone has rational singularities in the case of SL3. We observe by example that the null-cone is not normal in general and that the normalization of the null-cone does not have rational singularities in general. This is achieved by computing cohomology of certain vector bundles on flag varieties.
The decomposition of a two dimensional complex germ with non-isolated singularity into semi-algebraic sets is given. This decomposition consists of four classes: Riemannian cones defined over a Seifert fibered manifold, a topological cone over thickened tori endowed with Cheeger-Nagase metric, a topological cone over mapping torus endowed with Hsiang-Pati metric and a topological cone over the tubular neighbourhoods of the link's singularities. In this decomposition there exist semi-algebraic sets that are metrically conical over the manifolds constituting the link. The germ is reconstituted u…
We describe the generic singularity of a Schubert variety of type A on each irreducible component of its singular locus. This singularity is given either by a cone of rank one matrices, or a quadratic cone.
Spectrum of cones of projective hypersurfaces with singularities isolated
本文研究具有孤立奇点的射影超曲面锥的谱，利用奇点谱数与全局次数给出描述，并推广至非约化平面曲线及普通奇点情形。
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。
It is known that ambient bilipschitz equivalence preserves tangent cones. This paper explores the behavior of the Nash cone and, in particular, exceptional rays under ambient bilipschitz equivalence for real surfaces in $R^3$ with isolated singularity.
K-semistability and singularities of normal affine cones
本文在Collins-Székelyhidi意义下，通过K-稳定性给出了Q-Gorenstein正规仿射奇点的完整且最优刻画。
Let G be a semi-simple algebraic group over the complex numbers, B a Borel subgroup of G, T a maximal torus in B and P a parabolic in G containing B. This paper deals with singularities of T-stable subvarieties of G/P. It turns out that under the restriction that G doesn't contain any G_2-factors, the key geometric invariant determining the singular T-fixed points of X is the linear span of the reduced tangent cone to X at a T-fixed point x provided the singularity is isolated. The goal of this paper is to describe this invariant at the maximal singular T-fixed points when X is a Schubert vari…
Let $N$ be a point of an orbit closure $\bar{O_M}$ in a module variety such that its orbit $O_N$ has codimension two in $\bar{O_M}$. We show that under some additional conditions the pointed variety $(\bar{O_M},N)$ is smoothly equivalent to a cone over a rational normal curve.
We describe a geometric-combinatorial algorithm that allows one, using solely the system of weights and roots, to determine the Hesselink strata of the null-cone of a linear representation of a reductive algebraic group and calculate their dimensions. In particular, it provides a constructive approach to calculating the dimension of the null-cone and determining all its irreducible components of maximal dimension. In the case of the adjoint representation (and, more generally, a $θ$-representation), the algorithm turns into the classification algorithm for the conjugacy classes of nilpotent el…
K-semistability of log Fano cone singularities
本文给出了log Fano锥奇点K-半稳定性的非阿基米德刻画，证明其与Collins--Székelyhidi的原始定义一致，并揭示了测试K-半稳定性只需测试特殊测试配置及其与lc位置的联系。
We explicitly fix a mistake in a preliminary statement of our previous paper on the conductor at a multiplanar singularity. The correction is not immediate and, though the mistake does not affect correctness of the subsequent results, the wrong statement could easily be misleading.
In this note we study the geometry of principally polarized abelian varieties (ppavs) with a vanishing theta-null (i.e. with a singular point of order two and even multiplicity lying on the theta divisor). We describe the locus within the theta-null divisor where this singularity is not an ordinary double point. By using theta function methods we first show that this locus does not equal the entire theta-null divisor (this was shown previously by O. Debarre). We then show that this locus is contained in the intersection of the theta-null divisor with the other irreducible components of the And…
This is the first of a series of three papers which provide proofs of results announced recently in arXiv:1210.7494.
Boundedness of log Fano cone singularities and discreteness of local volumes
本文证明固定维度下体积有下界的K-半稳定log Fano锥奇点有界，进而得出klt奇点局部体积集合仅以零为聚点。
Let G be a complex reductive group and V a G-module. Let π: V \to V//G be the quotient morphism and set N(V) = π^{-1}(π(0)). We consider the following question. Is the null cone N(V) reduced, i.e., is the ideal of N(V) generated by G-invariant polynomials? We have complete results when G is SL_2, SL_3 or a simple group of adjoint type, and also when G is semisimple of adjoint type and the G-module V is irreducible.
We study the scalar curvature of Kähler metrics that have cone singularities along a divisor, with a particular focus on certain specific classes of such metrics that enjoy some curvature estimates. Our main result is that, on the projective completion of a pluricanonical bundle over a product of Kähler--Einstein Fano manifolds with the second Betti number 1, momentum-constructed constant scalar curvature Kähler metrics with cone singularities along the $\infty$-section exist if and only if the log Futaki invariant vanishes on the fibrewise $\mathbb{C}^*$-action, giving a supporting evidence t…
Categorical absorptions of cone singularities
研究Kuznetsov-Shinder关于锥奇点的范畴吸收，推广其对节点奇点的结果。给出特定法诺簇上锥的范畴吸收中倾斜对象自同态代数的描述，得到代数奇点范畴与锥奇点范畴的三角等价及负K理论的消失结果，附录给出加权射影空间倾斜对象描述。
We study the notion of algebraic tangent cones at singularities of reflexive sheaves. These correspond to extensions of reflexive sheaves across a negative divisor. We show the existence of optimal extensions in a constructive manner, and we prove the uniqueness in a suitable sense. The results here are an algebro-geometric counterpart of our previous study on singularities of Hermitian-Yang-Mills connections.
We study the null fiber of a moment map related to dual pairs. We construct an equivariant resolution of singularities of the null fiber, and get conormal bundles of closed $ K_C $-orbits in the Lagrangian Grassmannian as the categorical quotient. The conormal bundles thus obtained turn out to be a resolution of singularities of the closure of nilpotent $ K_C $-orbits, which is a "quotient" of the resolution of the null fiber.
We use twistor methods to promote Namikawa's universal Poisson deformations of conic affine symplectic singularities to families of hyperkähler structures deforming hyperkähler cone metrics. The metrics we produce are generally incomplete, but for specific classes of hyperkähler cones, even these incomplete metrics have some interest and applications: we study in detail the case of the nilpotent cone of a simple complex Lie algebra, with applications to hyperkähler metrics with symmetries and hyperkähler quotients, and the case of Kleinian singularities, with applications to codimension-4 sing…
We generalize the notion of Thom polynomials from singularities of maps between two complex manifolds to invariant cones in representations, and collections of vector bundles. We prove that the generalized Thom polynomials, expanded in the products of Schur functions of the bundles, have nonnegative coefficients. For classical Thom polynomials associated with maps of complex manifolds, this gives an extension of our former result for stable singularities to nonnecessary stable ones. We also discuss some related aspects of Thom polynomials, which makes the article expository to some extent.
The following multi-determinantal algebraic variety plays a central role in algebra, algebraic geometry and computational complexity theory: ${\rm SING}_{n,m}$, consisting of all $m$-tuples of $n\times n$ complex matrices which span only singular matrices. In particular, an efficient deterministic algorithm testing membership in ${\rm SING}_{n,m}$ will imply super-polynomial circuit lower bounds, a holy grail of the theory of computation. A sequence of recent works suggests such efficient algorithms for memberships in a general class of algebraic varieties, namely the null cones of linear grou…
We study the cones in the first Voronoi or perfect cone decomposition of quadratic forms with respect to the question which of these cones are basic or simplicial. As a consequence we deduce that the singular locus of the moduli stack ${\mathcal A_g^{\mathop{Perf}}}$, the toroidal compactification of the moduli space of principally polarized abelian varieties of dimension $g$ given by this decomposition, has codimension $10$ if $g \geq 4$. Moreover we describe the non-simplicial locus in codimension $10$. We also show that the second Voronoi compactification ${\mathcal A_{g}^{\mathop{Vor}}}$ h…
This is the second of a series of three papers which provide proofs of results announced in arXiv:1210.7494. In this paper we consider the Gromov-Hausdorff limits of metrics with cone singularities in the case when the limiting cone angle is less than 2π. We show that these are in a natrual way projective algebraic varieties. In the case when the limiting variety and the limiting divisor are smooth we show that the limiting metric also has standard cone singularities.
On boundedness of singularities and minimal log discrepancies of Kollár components, II
本文证明K-半稳定log Fano锥奇点的有界性由局部体积下界和Kollár分量极小对数差异上界决定，并据此确认了三维情形的有界性猜想及klt奇点局部体积的聚集性质。
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