Klaus Altmann, Jan Stevens
We show that the number of generators of the n-th cotangent cohomology group (n >=2) is the same for all rational surface singularities Y. For a large class of rational surface singularities, including quotient singularities, this number is also the dimension. For them we obtain an explicit formula for the corresponding Poincare series.
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We use representation theory and Bott's theorem to show vanishing of higher cotangent cohomology modules for the homogeneous coordinate ring of Grassmannians in the Plücker embedding. As a biproduct we answer a question of Wahl about the cohomology of the square of the ideal sheaf for the case of Plücker relations.
We give a new criterion for when a resolution of a surface of general type with canonical singularities has big cotangent bundle and a new lower bound for the values of $d$ for which there is a surface with big cotangent bundle that is deformation equivalent to a smooth hypersurface in $\mathbb{P}^3$ of degree $d$.
We prove dimension formulas for the cotangent spaces $T^{1}$ and $T^{2}$ for a class of rational surface singularities by calculating a correction term in the general dimension formulas. We get that it is zero if the dual graph of the rational surface singularity $X$ does not contain a particular type of configurations, and this generalizes a result of Theo de Jong stating that the correction term $c(X)$ is zero for rational determinantal surface singularities. In particular our result implies that $c(X)$ is zero for Riemenschneiders quasi-determinantal rational surface singularities, and this…
Simplicial complexes X provide commutative rings A(X) via the Stanley-Reisner construction. We calculated the cotangent cohomology, i.e., T1 and T2 of A(X) in terms of X. These modules provide information about the deformation theory of the algebro geometric objects assigned to X.
In this paper, we provide two different resolutions of structural sheaves of projectivized tangent bundles of smooth complete intersections. These resolutions allow in particular to obtain convenient (and completely explicit) descriptions of cohomology of twisted symmetric powers of cotangent bundles of complete intersections, which are easily implemented on computer. We then provide several applications. First, we recover the known vanishing theorems on the subject, and show that they are optimal via some non-vanishing theorems. Then, we study the symmetric algebra of global sections of symme…
In this paper we prove that for an almost complex orbifold, its virtual orbifold cohomology [math.AT/0606573] is isomorphic as algebras to the Chen-Ruan orbifold cohomology of its cotangent orbifold.
The cotangent cohomology groups T^1 and T^2 play an important role in deformation theory, the first as space of infinitesimal deformations, while the obstructions land in the second. Much work has been done to compute their dimension for rational surface singularities. For such singularities we give explicit dimension formulas for the groups T^i with i>2.
Surface quotient singularities and bigness of the cotangent bundle: Part II
本文提出一般型轨形曲面解消余切丛的大性与CMS判据，并针对A_n奇点给出未知不变量的求法、公式及对称微分形式的延拓结果。
The main problem studied is resolution of singularities of the cotangent sheaf of a complex- or real-analytic variety Y (or of an algebraic variety Y over a field of characteristic zero). Given Y, we ask whether there is a global resolution of singularities s: X -> Y such that the pulled-back cotangent sheaf of Y is generated by differential monomials in suitable coordinates at every point of X ("Hsiang-Pati coordinates''). Desingularization of the cotangent sheaf is equivalent to monomialization of Fitting ideals generated by minors of a given order of the logarithmic Jacobian matrix of s. We…
Let $G$ denote a complex semisimple linear algebraic group, $P$ a parabolic subgroup of $G$ and $\mathcal{P}=G/P$. We identify the quantum multiplication by divisors in $T^*\mathcal{P}$ in terms of stable basis, which is introduced by Maulik and Okounkov. Using this and the restriction formula for stable basis, we show that the $G\times\mathbb{C}^*$-equivariant quantum multiplication formula in $T^*\mathcal{P}$ is conjugate to the conjectured formula by Braverman.
A Frobenius splitting and cohomology vanishing for the cotangent bundles of the flag varieties of GL_n
本文证明了 GL_n 旗丛余切丛的 Frobenius 分裂具有最高次数，给出了分量函数公式并证明其在例外轨迹上消失，进而得出与支配权相关线丛的高阶上同调群为零。
We interpret the equivariant cohomology algebra H^*_{GL_n\times\C^*}(T^*F_λ;\C) of the cotangent bundle of a partial flag variety F_λparametrizing chains of subspaces 0=F_0\subset F_1\subset\dots\subset F_N =\C^n, \dim F_i/F_{i-1}=λ_i, as the Yangian Bethe algebra of the gl_N-weight subspace of a gl_N Yangian module. Under this identification the dynamical connection of [TV1] turns into the quantum connection of [BMO] and [MO]. As a result of this identification we describe the algebra of quantum multiplication on H^*_{GL_n\times\C^*}(T^*F_λ;\C) as the algebra of functions on fibers of a discr…
Surfaces of general type with positive second Segre number are known to have big cotangent bundle. We give a new criterion ensuring that a surface of general type with canonical singularities has a minimal resolution with big cotangent bundle. This provides many examples of surfaces with negative second Segre number and big cotangent bundle.
We prove the vanishing modulo torsion of the higher direct images of the sheaf of Witt vectors (and the Witt canonical sheaf) for a purely inseparable projective alteration between normal finite quotients over a perfect field. For this, we show that the relative Hodge-Witt cohomology admits an action of correspondences. As an application we define Witt-rational singularities which form a broader class than rational singularities. In particular, finite quotients have Witt-rational singularities. In addition, we prove that the torsion part of the Witt vector cohomology of a smooth, proper scheme…
In this paper we study the cohomology of tensor products of symmetric powers of the cotangent bundle of complete intersection varieties in projective space. We provide an explicit description of some of those cohomology groups in terms of the equations defining the complete intersection. We give several applications. First we prove a non-vanishing result, then we give a new example illustrating the fact that the dimension of the space of holomorphic symmetric differential forms is not deformation invariant. Our main application is the construction of varieties with ample cotangent bundle, prov…
Lattice cohomology and the embedded topological type of plane curve singularities
本文提出解析格上同调这一约化曲线奇点新不变量，证明其对不可约平面曲线奇点是完备嵌入拓扑不变量，并在多分支情形下与整Seifert形式比较。
Green and Griffiths have introduced several notions of singularities associated with normal functions, especially in connection with middle dimensional primitive Hodge classes. In this note, by using the more elementary aspects of the Decomposition Theorem, we define global and local singularities associated with primitive middle dimensional cohomology classes and by using the Relative Hard Lefschetz Theorem, we show that these singularities detect the global and local triviality of the primitive class. In a final section, we write-up a classical inductive argument relating the Hodge conjectur…
Cohomology on Cotangent Bundles of Partial Flag Varieties in Type A
针对A型特殊线性群的偏Flag簇余切丛上的向量丛，通过与Bott--Samelson簇上同调的比较得到高阶上同调消失等结论，进而给出相关函数环的生成元集并证明其仿射化具有终端奇点。
In this paper, we give partial answers to the following questions: Which contact manifolds are contactomorphic to links of isolated complex singularities? Which symplectic manifolds are symplectomorphic to smooth affine varieties? The invariant that we will use to distinguish such manifolds is called the growth rate of wrapped Floer cohomology. Using this invariant we show that if Q is a simply connected manifold whose unit cotangent bundle is contactomorphic to the link of an isolated singularity or whose cotangent bundle is symplectomorphic to a smooth affine variety then M must be rationall…
We construct the analytic lattice cohomology associated with the analytic type of any complex normal surface singularity. It is the categorification of the geometric genus of the germ, whenever the link is a rational homology sphere. It is the analytic analogue of the topological lattice cohomology, associated with the link of the germ whenever it is a rational homology sphere. This topological lattice cohomology is the categorification of the Seiberg--Witten invariant, and conjecturally it is isomorphic with the Heegaard Floer cohomology. We compare the two lattice cohomologies: in some simpl…
Based on the Brieskorn-Slodowy-Grothendieck diagram, we write the holomorphic structures (or filtrations) of the ADE Lie algebra bundles over the corresponding type ADE flag varieties, over the cotangent bundles of these flag varieties, and over the corresponding type $ADE$ singular surfaces. The main tool is the cohomology of line bundles over flag varieties and their cotangent bundles.
We construct the equivariant analytic lattice cohomology associated with the analytic type of a complex normal surface singularity whenever the link is a rational homology sphere. It is the categorification of the equivariant geometric genus of the germ. This is the analytic analogue of the topological lattice cohomology, associated with the link of the germ, and indexed by the spin$^c$--structures of the link (which is a categorification of the Seiberg--Witten invariant and conjecturally it is isomorphic with the Heegaard Floer cohomology).
We provide several results on splice-quotient singularities: a combinatorial expression of the dimension of the first cohomology of all `natural' line bundles, an equivariant Campillo-Delgado-Gusein-Zade type formula about the dimension of relative sections of line bundles (proving that the equivariant, divisorial multi-variable Hilbert series is topological), a combinatorial description of divisors of analytic function-germs, and an expression for the multiplicity of the singularity from its resolution graph. Additional, we establish a new formula for the Seiberg-Witten invariants of any rati…
Higher cotangent cohomology for Stanley-Reisner rings
受Altmann和Christophersen工作的启发，研究了与单纯复形K相关的Stanley-Reisner环S_K的余切上同调T^i_{S_K}（i≥3）的分次部分，给出了T^3和T^4的完全拓扑描述，并应用于球面等情形。
In this paper, we extract natural invariants of a singularity by using the Deligne weight filtration on the cohomology of an exceptional fibre of a resolution, and also on the intersection cohomology of the link. Our primary goal is to study and give natural bounds on the weights in terms of direct images of differential forms. These bounds can be made explicit for various standard classes such as rational, isolated normal Cohen-Macaulay and toroidal singularities, and lead to strong restrictions on the topology of these singularities. A secondary goal of this paper is to make the weight filtr…
Structural properties of the lattice cohomology of curve singularities
本文研究约化曲线奇点的格上同调，证明其约化上同调权分次非正、H^0模结构可判定Gorenstein性，且平面曲线奇点的H^0决定其重数。
It is proved that for projective varieties having Du Bois singularities is equivalent to the condition that the coherent cohomology groups of the structure sheaf coincide with the appropriate Hodge components of the singular cohomology groups.
Hodge theory of degenerations, (II): vanishing cohomology and geometric applications
本文研究孤立超曲面奇点的加权谱和消失上同调，揭示其对光滑化极限混合Hodge结构的作用，并应用于紧化与镜像对称中的多类奇点。
Cohomology of ideals in elliptic surface singularities
本文研究了二维正常奇点的正常还原数，证明了椭圆奇点的正常还原数为二，并探讨了非有理奇点的Gorenstein性质与$p_g$-理想的关系。
For a germ of a variety $\mathcal{V}, 0 \subset \mathbb C^N, 0$, a singularity $\mathcal{V}_0$ of type $\mathcal{V}$, is given by a germ $f_0 : \mathbb C^n, 0 \to \mathbb C^N, 0$ which is transverse to $\mathcal{V}$ in an appropriate sense so that $\mathcal{V}_0 = f_0^{-1}(\mathcal{V})$. For these singularities, we introduce "characteristic cohomology" to capture the contribution of the topology of $\mathcal{V}$ to that of $\mathcal{V}_0$, for the Milnor fiber (for $\mathcal{V}, 0$ a hypersurface), and complement and link of $\mathcal{V}_0$ (in the general case). The characteristic cohomology …
We use the G-invariant non-degenerate form on the Steinberg module to Frobenius split the cotangent bundle of a flag variety in good prime characteristics. This was previously only known for the general linear group. Applications are a vanishing theorem for pull back of line bundles to the cotangent bundle (proved for the classical groups and G_2 by Andersen and Jantzen and in characteristic zero by B. Broer (for all groups)), normality and rational singularities for the subregular nilpotent variety and good filtrations of the global sections of pull backs of line bundles to the cotangent bund…
We show that a natural isomorphism between the rational cohomology groups of the two zero-dimensional Hilbert schemes of $n$-points of two surfaces, the affine plane minus the axes and the cotangent bundle of an elliptic curve, exchanges the weight filtration on the first set of cohomology groups with the perverse Leray filtration associated with a natural fibration on the second set of cohomology groups. We discuss some associated hard Lefschetz phenomena.
We study the topology of some simple infinite dimensional singularities arising from spaces of \emph{algebraic formal loops}. We prove that in some simple cases the natural analogue of nearby cycles cohomology for a function on the loop space vanishes, and show further that a suitably renormalized version of the above cohomology produces a simple (non-zero) result.
We study Ruan's \textit{cohomological crepant resolution conjecture} for orbifolds with transversal ADE singularities. In the $A_n$-case we compute both the Chen-Ruan cohomology ring $H^*_{\rm CR}([Y])$ and the quantum corrected cohomology ring $H^*(Z)(q_1,...,q_n)$. The former is achieved in general, the later up to some additional, technical assumptions. We construct an explicit isomorphism between $H^*_{\rm CR}([Y])$ and $H^*(Z)(-1)$ in the $A_1$-case, verifying Ruan's conjecture. In the $A_n$-case, the family $H^*(Z)(q_1,...,q_n)$ is not defined for $q_1=...=q_n=-1$. This implies that the …
We generalize Macdonald's formula for the cohomology of Hilbert schemes of points on a curve from smooth curves to curves with planar singularities: we relate the cohomology of the Hilbert schemes to the cohomology of the compactified Jacobian of the curve. The new formula is a consequence of a stronger identity between certain perverse sheaves defined by a family of curves satisfying mild conditions. The proof makes an essential use of Ngo's support theorem for compactified Jacobians and generalizes this theorem to the relative Hilbert scheme of such families. As a consequence, we generalize …
We characterize quasihomogeneity of isolated singularities by the injectivity of the map induced by the first differential of the logarithmic differential complex in the top local cohomology supported in the singular point.
Let $\mathcal{V} \subset M$ denote any of the varieties of singular $m \times m$ complex matrices which may be general, symmetric, or skew-symmetric ($m$ even), or $m \times p$ matrices, in the corresponding space $M$ of such matrices. A "matrix singularity", $\mathcal{V}_0$ of "type $\mathcal{V}$", for any of the $\mathcal{V} \subset M$ is defined as $\mathcal{V}_0 = f_0^{-1}(\mathcal{V})$ by a germ $f_0 : \mathbb C^n, 0 \to M, 0$ (appropriately transverse to $\mathcal{V}$). In part I of this paper we introduced the notion of characteristic cohomology for a singularity $\mathcal{V}_0$ of type…
We construct a lattice cohomology ${\mathbb H}^*(C,o)=\oplus_{q\geq 0}{\mathbb H}^q(C,o)$ and a graded root ${\mathfrak R}(C,o)$ to any complex isolated curve singularity $(C,o)$. Each ${\mathbb H}^q(C,o)$ is a ${\mathbb Z}$-graded ${\mathbb Z}[U]$-module. The Euler characteristic of ${\mathbb H}^*(C,o)$ is the delta-invariant of $(C,o)$. The construction is based on the multivariable Hilbert series of the multifiltration provided by valuations of the normalization. Several examples are discussed, e.g. Gorenstein curves (where an additional symmetry is established), plane curves (in particular…
Let M be a moduli space of stable sheaves on a K3 or Abelian surface S. We express the class of the diagonal in the cartesian square of M in terms of the Chern classes of a universal sheaf. Consequently, we obtain generators of the cohomology ring of M. When S is a K3 and M is the Hilbert scheme of length n subschemes, this set of generators is sufficiently small in the sense that there aren't any relations among them in the stable cohomology ring. When S is the cotangent bundle of a Riemann surface, we recover the result of T. Hausel and M. Thaddeus: The cohomology ring of the moduli spaces o…
We generalize a vanishing theorem for the cohomology of symmetric powers of the cotangent bundle of subvarieties of projective space due to Schneider. From this we deduce new vanishing results for Green-Griffiths jet differential bundles, generalizing results of Diverio and Pacienza-Rousseau.
We study normal compact Kähler spaces whose rational cohomology ring is isomorphic to that of a complex torus. We call them rational cohomology tori. We classify, up to dimension three, those with rational singularities. We then give constraints on the degree of the Albanese morphism and the number of simple factors of the Albanese variety for rational cohomology tori of general type (hence projective) with rational singularities. Their properties are related to the birational geometry of smooth projective varieties of general type, maximal Albanese dimension, and with vanishing holomorphic Eu…
Iwahori-Coulomb branches, stable envelopes, and quantum cohomology of cotangent bundles of flag varieties
本文研究Iwahori-Coulomb分支在局部化equivariant量子上同调中的多项式性质，通过稳定包络和Demazure-Lusztig元素计算作用，并应用到旗流形切丛的量子上同调中。
Let $X$ be a complex, irreducible, quasi-projective variety, and $π:\widetilde X\to X$ a resolution of singularities of $X$. Assume that the singular locus ${\text{Sing}}(X)$ of $X$ is smooth, that the induced map $π^{-1}({\text{Sing}}(X))\to {\text{Sing}}(X)$ is a smooth fibration admitting a cohomology extension of the fiber, and that $π^{-1}({\text{Sing}}(X))$ has a negative normal bundle in $\widetilde X$. We present a very short and explicit proof of the Decomposition Theorem for $π$, providing a way to compute the intersection cohomology of $X$ by means of the cohomology of $\widetilde X…
We introduce a lifting property for local cohomology, which leads to a unified treatment of the dualizing complex for flat morphisms with semi-log-canonical, Du Bois or F-pure fibers. As a consequence we obtain that, in all 3 cases, the cohomology sheaves of the relative dualizing complex are flat and commute with base change. We also derive several consequences for deformations of semi-log-canonical, Du Bois and F-pure singularities.
Using the path lattice cohomology we provide a conceptual topological characterization of the geometric genus for certain complex normal surface singularities with rational homology sphere links, which is uniformly valid for all superisolated and Newton non--degenerate hypersurface singularities.
For an arithmetical scheme X, K. Kato introduced a certain complex of Gersten-Bloch-Ogus type whose component in degree a involves Galois cohomology groups of the residue fields of all the points of X of dimension a. He stated a conjecture on its homology generalizing the fundamental exact sequences for Brauer groups of global fields. We prove the conjecture over a finite field assuming resolution of singularities. Thanks to a recently established result on resolution of singularities for embedded surfaces, it implies the unconditional vanishing of the homology up to degree 4 for X projective …
Invariants of the singularities of secant varieties of curves
本文研究光滑射影曲线的 $k$-平面割线簇的奇点不变量，计算了任意曲线的交点上同调，并证明有理正规曲线的所有割线簇均为满足 Poincaré 对偶的有理同调流形，同时计算了最大非平凡割线簇的邻近圈与消没圈。
We study Ruan's "cohomological crepant resolution conjecture" (see math.AG/0108195) for orbifolds with transversal ADE singularities. Let [Y] be such an orbifold, Y its coarse moduli space and Z the crepant resolution of Y. Following Ruan [math.AG/0108195], we have a deformation of the cohomology ring H^*(Z) using some Gromov-Witten invariants of Z. The resulting family of rings will be denoted by H^*(Z)(q_1,...,q_n), where q_1,...,q_n are complex parameters. In the A_n case we compute both the orbifold cohomology ring H^*_{orb}([Y]) and the family H^*(Z)(q_1,...,q_n). The former is achieved i…
In this paper we study the local cohomology modules of Du Bois singularities. Let $(R,m)$ be a local ring, we prove that if $R_{red}$ is Du Bois, then $H_m^i(R)\to H_m^i(R_{red})$ is surjective for every $i$. We find many applications of this result. For example we answer a question of Kovács and the second author on the Cohen-Macaulay property of Du Bois singularities. We obtain results on the injectivity of $Ext$ that provide substantial partial answers of questions of Eisenbud-Mustata-Stillman in characteristic $0$, and these results can be viewed as generalizations of the Kodaira vanishing…
We compute symplectic cohomology for Milnor fibres of certain compound Du Val singularities that admit small resolution by using homological mirror symmetry. Our computations suggest a new conjecture that the existence of a small resolution has strong implications for the symplectic cohomology and conversely. We also use our computations to give a contact invariant of the link of the singularities and thereby distinguish many contact structures on connected sums of $S^2 \times S^3$.
We employ the perverse vanishing cycles to show that each reduced cohomology group of the Milnor fiber, except the top two, can be computed from the restriction of the vanishing cycle complex to only singular strata with a certain lower bound in dimension. Guided by geometric results, we alternately use the nearby and vanishing cycle functors to derive information about the Milnor fiber cohomology via iterated slicing by generic hyperplanes. These lead to the description of the reduced cohomology groups, except the top two, in terms of the vanishing cohomology of the nearby section. We use it …
For any negative definite plumbed 3-manifold M we construct from its plumbed graph a graded Z[U]-module. This, for rational homology spheres, conjecturally equals the Heegaard-Floer homology of Ozsvath and Szabo, but it has even more structure. If M is a complex singularity link then the normalized Euler-characteristic can be compared with the analytic invariants. The Seiberg--Witten Invariant Conjecture is discussed in the light of this new object.
For a large class of good moduli spaces $X$ of symmetric stacks $\mathcal{X}$, we define noncommutative motives $\mathbb{D}^{\text{nc}}(X)$ which can be regarded as categorifications of the intersection cohomology of $X$. These motives are summands of noncommutative resolutions of singularities $\mathbb{D}(X)\subset D^b(\mathcal{X})$ of $X$. The category $\mathbb{D}(X)$ is a global analogue of the noncommutative resolutions of singularities of $V//G$ for $V$ a symmetric representation of a reductive group $G$ constructed by Špenko-Van den Bergh.
Complexes of differential forms and singularities: The injectivity theorem
本文证明了对于具有(m-1)-Du Bois奇点的代数簇，从m阶Du Bois复形的Grothendieck对偶到其零阶上同调sheaf的Grothendieck对偶的自然映射在上同调上是单射的，验证了Popa, Shen和Vo的猜想G。
Let $\mathcal{T}$ be an arbitrary resolution graph and $(X, 0)$ a generic complex analytic normal surface singularity, and $\tilde{X}$ a generic resolution corresponding to it. Fix an effective integer cycle $Z$ supported on the exceptional curve and also an arbitrary Chern class $Z' \in L'$. In this article we aim to compute the cohomology numbers $h^1(\mathcal{O}_{Z}(Z'))$. Notice, that the case $Z'_v < 0, v \in |Z|$ was discussed in \cite{NNA2}, where the main theorem was, that in this special case these cohomology numbers equal to the cohomology numbers of the generic line bundle in $\text…
Differential forms on varieties with pre-$k$-rational singularities
本文针对具有预k-有理奇点的复代数簇，比较两类紧支上同调群的混合霍奇结构，研究其微分形式相关概念，重点考察Du Bois复形分次部分的深度。
In this manuscript we investigate the analouge of the Brill-Noether problem for smooth curves in the case of normal surface singularities. We determine the maximal possible value of $h^1$ of line bundles without fixed components in the Picard group $\pic^{l'}(\tX)$ in the following cases: for some special Chern classes $l'$ if $\tX$ is a resolution of a splice quotient singularity $(X, 0)$ and for arbitrary Chern classes in the case of weighted homogenous singularities. Motivated by this problem, we define the \emph{virtual cohomology numbers} $h^1_{virt}(l')$ for all Chern classes $l'$ such t…
We give a complete classification of Q_l-cohomology projective planes with isolated ADE-singularities and numerically trivial canonical bundle in odd characteristic. This leads to a beautiful relation with certain Enriques surfaces which parallels the situation in characteristic zero, yet displays intriguing subtleties.
In this paper we compute the cohomology of moduli space of cubic fourfolds with ADE type singularities relying on Kirwan's blowup and Laza's GIT construction. More precisely, we obtain the Betti numbers of Kirwan's resolution of the moduli space. Furthermore, by applying decomposition theorem we obtain the Betti numbers of the intersection cohomology of Baily-Borel compactification of the moduli space.
Let $V$ be a complex projective variety with isolated singularities. Let the smooth part be given the metric induced by a projective imbedding. Then we develop the $L_2$ harmonic theory and construct a pure Hodge structure on the $L_2$-cohomology of $V$. If the dimension of $V$ is two, we put a cohomological Hodge structure on its complex of sheaves of $L_2$ forms.
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