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.
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 associate (under a minor assumption) to any analytic isolated singularity of dimension $n\geq 2$ the `analytic lattice cohomology' ${\mathbb H}^*_{an}=\oplus_{q\geq 0}{\mathbb H}^q_{an}$. Each ${\mathbb H}^q_{an}$ is a graded ${\mathbb Z}[U]$--module. It is the extension to higher dimension of the `analytic lattice cohomology' defined for a normal surface singularity with a rational homology sphere link. This latest one is the analytic analogue of the `topological lattice cohomology' of the link of the normal surface singularity, which conjecturally is isomorphic to the Heegaard Floer cohom…
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…
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).
On the cotangent cohomology of rational surface singularities with almost reduced fundamental cycle
通过计算余切空间$T^{1}$和$T^{2}$的一般维数公式中的修正项，证明了一类有理曲面奇点的余切空间的维数公式.我们得到了有理曲面奇点$X$的对偶图不包含特定类型的构形时，修正项为零，这推广了Theo de Jong关于有理行列式曲面奇点的修正项$c(X)$为零的一个结果.特别地，我们的结果意味着$c(X)$是零的Riemenschneiders拟行列式有理曲面奇性，这也推广了结果的qoutient奇性。
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.
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…
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 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 …
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.
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…
A homotopical treatment of intersection cohomology recently developed by Chataur-Saralegui-Tanré associates a "perverse algebraic model" to every topological pseudomanifold, extending Sullivan's presentation of rational homotopy to intersection cohomology. In this context, there is a notion of "intersection-formality", measuring the vanishing of Massey products in intersection cohomology. In the present paper, we study the perverse algebraic model of complex projective varieties with isolated singularities. We endow such invariant with natural mixed Hodge structures. This allows us to prove so…
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…
The subject of this paper is the big quantum cohomology rings of symplectic isotropic Grassmannians $\text{IG}(2, 2n)$. We show that these rings are regular. In particular, by "generic smoothness", we obtain a conceptual proof of generic semisimplicity of the big quantum cohomology for $\text{IG}(2, 2n)$. Further, by a general result of Claus Hertling, the regularity of these rings implies that they have a description in terms of isolated hypersurface singularities, which we show in this case to be of type $A_{n-1}$. By the homological mirror symmetry conjecture, these results suggest the exis…
We study the analytic and topological invariants associated with complex normal surface singularities. Our goal is to provide topological formulae for several discrete analytic invariants whenever the analytic structure is generic (with respect to a fixed topological type), under the condition that the link is a rational homology sphere. The list of analytic invariants include: the geometric genus, the cohomology of certain natural line bundles, the cohomology of their restrictions on effective cycles (supported on the exceptional curve of a resolution), the cohomological cycle of natural line…
Over a field of characteristic zero, every deformation problem with cohomology constraints is controlled by a pair consisting of a differential graded Lie algebra together with a module. Unfortunately, these pairs are usually infinite-dimensional. We show that every deformation problem with cohomology constraints is controlled by a typically finite-dimensional L-infinity pair. As a first application, we show that for complex algebraic varieties with no weight-zero 1-cohomology classes, the components of the cohomology jump loci of rank one local systems containing the constant sheaf are tori. …
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…
Drinfeld's relative compactification plays a basic role in the theory of automorphic sheaves, and its singularities encode representation-theoretic information in the form of intersection cohomology. We introduce a resolution of singularities consisting of stable maps from nodal deformations of the curve into twisted flag varieties. As an application, we prove that the twisted intersection cohomology sheaf on Drinfeld's compactification is universally locally acyclic over the moduli stack of $G$-bundles at points sufficiently antidominant relative to their defect.
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.
We prove the generalised McKay correspondence for isolated singularities using Floer theory. Given an isolated singularity \C^n/G for a finite subgroup G in SL(n,\C) and any crepant resolution Y, we prove that the rank of positive symplectic cohomology SH_+(Y) is the number of conjugacy classes of G, and that twice the age grading on conjugacy classes is the \Z-grading on SH_+(Y) by the Conley-Zehnder index. The generalised McKay correspondence follows as SH_+(Y) is naturally isomorphic to ordinary cohomology H(Y), due to a vanishing result for full symplectic cohomology. In the Appendix we co…
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.
Quantum cohomology of the Hilbert scheme of points on A_n-resolutions
本文确定了与A_n型奇点有关的表面分解点的Hilbert方案的等变量子上同调的两点不变量。编码这些不变量的算子表示为仿射李代数\hat{gl}（n+1）在其基本表示上的作用。假设一定的非简并性猜想，这些算子决定了量子上同调环的完整结构。证明了量子上同调与A_n x P^1的Gromov-Witten/Donaldson-Thomas理论之间的关系。最后，我们讨论了相关量子微分方程的单值性质，并推广到D型和E型奇点。
We prove homological mirror symmetry for Milnor fibers of simple singularities in dimensions greater than one, which are among the log Fano cases of Conjecture 1.5 in arXiv:1806.04345. The proof is based on a relation between matrix factorizations and Calabi--Yau completions. As an application, we give an explicit computation of the Hochschild cohomology group of the derived $n$-preprojective algebra of a Dynkin quiver for any $n \geq 1$, and the symplectic cohomology group of the Milnor fiber of any simple singularity in any dimension greater than one.
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.
