Motivic Chern classes of cones

奇异（复）algebraic varieties的Hirzebruch类和motivic Chern类

Hirzebruch classes and motivic Chern classes for singular (complex) algebraic varieties

本文研究了奇异复代数簇的特征同调类的一些新理论。首先我们引入了一个自然变换T_{y}：K_{0}（var/X）-> H_{*}（X，Q）[y]，它推广了相应的Hirzebruch特征.这里K_{0}（var/X）是X上复代数簇的相对Grothendieck群，由Looijenga和Bittner在motivic积分方面引入和研究。T_{y}是动机测度的同调类版本，对应于公知的Hodge多项式的适当专门化。这个变换统一了MacPherson和Schwartz的Chern类变换（y=-1）和Baum-Fulton-MacPherson的奇异Riemann-Roch定理中的Todd类变换（y=0）。事实上，T_{y}是Yokura的Todd类变换的一个推广版本和一个新的motivic Chern类变换mC_{*}的组合：K_{0}（var/X）-> G_{0}（X）[y]，它将余切丛的全类推广到奇异空间。这里G_{0}（X）是相干层的Grothendieck群，mC_{*}的构造是基于M. Saito的代数混合Hodge模理论的一些结果.在本文的最后部分，我们利用Levine和Morel的代数配边理论将mC_{*}进一步提升到从K_{0}（var/X）到一个合适的普适Borel-Moore弱同调论的自然变换mC '_{*}。此外，我们的结果还可以推广到基域k上的簇，这些簇可以嵌入到复数中…

交错退化区域的动机类和对称轨道闭包

Motivic Classes of isotropic degeneracy loci and symmetric orbit closures

本文提供计算交错退化区域的动机柴恩和希泽布鲁赫类的显式公式，特别涉及辛和奇正交格拉斯曼流形。结果扩展了Anderson-Chen-Tarasca在普通格拉斯曼退化区域的研究。

格拉斯曼流形中Schubert胞腔的motivic Chern类的一个Pieri型公式

A Pieri type formula for motivic Chern classes of Schubert cells in Grassmannians

本文证明了格拉斯曼流形中Schubert胞腔的motivic Chern类的Pieri公式，通过仿射Hecke代数转化计算，并应用于建立motivic Chern类与Segre motivic类的关系及寻找对偶层类的对称幂级数表示。

Motivic classes of degeneracy loci and pointed Brill-Noether varieties

Motivic Chern and Hirzebruch classes are polynomials with K-theory and homology classes as coefficients, which specialize to Chern-Schwartz-MacPherson classes, K-theory classes, and Cappell-Shaneson L-classes. We provide formulas to compute the motivic Chern and Hirzebruch classes of Grassmannian and vexillary degeneracy loci. We apply our results to obtain the Hirzebruch $χ_y$-genus of classical and one-pointed Brill-Noether varieties, and therefore their topological Euler characteristic, holomorphic Euler characteristic, and signature.

从Schubert胞的motivic Chern类到它们的Hirzebruch类和CSM类

From motivic Chern classes of Schubert cells to their Hirzebruch and CSM classes

本文证明了旗流形中Schubert胞的motivic Chern类的多个性质与猜想，利用motivic Chern变换定义了等变Hirzebruch变换，并借助Demazure-Lusztig递归给出了Hirzebruch类及相关对偶的递归公式。

The intrinsic stable normal cone

We construct an analog of the intrinsic normal cone of Behrend-Fantechi in the equivariant motivic stable homotopy category over a base-scheme B and construct a fundament class in E-cohomology for any cohomology theory E in SH(B). For affine B, a perfect obstruction theory gives rise to a virtual fundamental class in a twisted Borel-Moore E-homology for arbitrary E. This includes motivic cohomology (homotopy invariant) K-theory algebraic cobordism and the oriented Chow groups of Barge-Morel and Fasel. In the case of motivic cohomology, we recover the constructions of Behrend-Fantechi, with val…

Chern classes with modulus

In this paper, we construct Chern classes from the relative $K$-theory of modulus pairs to the relative motivic cohomology defined by Binda-Saito. An application to relative motivic cohomology of henselian dvr is given.

开放投影里奇变种的motivic Chern类及仿射Schubert单元

Motivic Chern Classes of Open Projected Richardson Varieties and of Affine Schubert Cells

本文研究了开放投影里奇变种和仿射Schubert单元的motivic Chern类，通过递推关系和格拉斯曼流形中的正体种公式，建立了与扭曲Kazhdan–Lusztig R多项式的关系。

Motivic local density

We develop a theory of local densities and tangent cones in a motivic framework, extending work by Cluckers-Comte-Loeser about $p$-adic local density. We prove some results about geometry of definable sets in Henselian valued fields of characteristic zero, both in semi-algebraic and subanalytic languages, and study Lipschitz continuous maps between such sets. We prove existence of regular stratifications satisfying analogous of Verdier condition $(w_f)$. Using Cluckers-Loeser theory of motivic integration, we define a notion of motivic local density with values in the Grothendieck ring of the …

双曲三维流形的陈-西蒙斯不变量、混合Tate motive与增广特征簇的motivic路径挠子

Chern-Simons invariants of hyperbolic three-manifolds, mixed Tate motives, and motivic path torsor of augmented character varieties

针对有限体积完备双曲三维流形，构造了定义在不变迹场上的混合Tate motive并关联其陈-西蒙斯不变量与复体积，分析单环面边界情形下的霍奇实现，解释motivic意义并验证假设，理论与多重对数motivic理论平行。

Hecke 代数在扭曲 motivic Chern 类与 K 理论稳定包络上的作用

Hecke algebra action on twisted motivic Chern classes and K-theoretic stable envelopes

本文通过引入作用于等变 K 理论的左右 Demazure-Lusztig 算子，推导了 Schubert 胞扭曲 motivic Chern 类的递归公式，揭示了其与 K 理论稳定包络的联系，并证明了算子生成的扭曲双重 Hecke 代数在 A 型中的作用。

Whittaker functions from motivic Chern classes

We prove a `motivic' analogue of the Weyl character formula, computing the Euler characteristic of a line bundle on a generalized flag manifold $G/B$ multiplied either by a motivic Chern class of a Schubert cell, or a Segre analogue of it. The result, given in terms of Demazure-Lusztig (D-L) operators, recovers formulas found by Brubaker, Bump and Licata for the Iwahori-Whittaker functions of the principal series representation of a $p$-adic group. In particular, we obtain a new proof of the classical Casselman-Shalika formula for the spherical Whittaker function. The proofs are based on local…

奇异空间的Hirzebruch类和motivic Chern类

Hirzebruch classes and motivic Chern classes for singular spaces

本文研究了奇异复代数（或紧解析）空间的特征同调类的一些新理论。本文引入了一个Motivic Chern类变换mC_{*}：K_{0}（var/X）-> G_{0}（X）[y]，它将余切丛的全λ-类推广到奇异空间.这里K_{0}（var/X）是X上的复代数簇的相对Grothendieck群，由Looijenga和Bittner在motivic积分方面引入和研究，而G_{0}（X）是O_{X} -模的相干层的Grothendieck群。我们定义了一个自然变换T_{y*}：K_{0}（var/X）-> H_{*}（X，Q）[y]，它通过适当的下推交换，推广了相应的Hirzebruch特征. T_{y*}是动机测量的同调类版本，对应于公知的Hodge多项式的适当专门化。这个变换统一了MacPherson和Schwartz的Chern类变换（y=-1）、Baum-Fulton-MacPherson的奇异Riemann-Roch定理的Todd类变换（y=0）和Cappell-Shaneson的L类变换（y=1）。在最简单的情况下，正常Gorenstein品种与"典型的奇异性“，我们也解释了之间的关系”stringy版本“我们的特征类，椭圆类的鲍里索夫-Libgober和stringy陈类的Bassffi和德Fernex-Lupercio-Nevins-Uribe。此外，我们的所有结果都可以推广…

Motivic Milnor classes

The Milnor class is a generalization of the Milnor number, defined as the difference (up to sign) of Chern--Schwartz--MacPherson's class and Fulton--Johnson's canonical Chern class of a local complete intersection variety in a smooth variety. In this paper we introduce a "motivic" Grothendieck group $K^{\mathcal Prop}_{\ell.c.i}(\mathcal V/X \to S)$ and natural transformations from this Grothendieck group to the homology theory. We capture the Milnor class, more generally Hirzebruch--Milnor class, as a special value of a distinguished element under these natural transformations. We also show a…

Characteristic classes of orbit stratifications, the axiomatic approach

Consider a complex algebraic group $G$ acting on a smooth variety $M$ with finitely many orbits, and let $Ω$ be an orbit. The following three invariants of $Ω\subset M$ can be characterized axiomatically: (1) the equivariant fundamental class $[\overlineΩ, M]\in H^*_G(M)$, (2) the equivariant Chern-Schwartz-MacPherson class $c(Ω, M)\in H^*_G(M)$, and (3) the equivariant motivic Chern class $mC(Ω, M) \in K_G(M)[y]$. The axioms for Chern-Schwartz-MacPherson and motivic Chern classes are motivated by the axioms for cohomological and K-theoretic stable envelopes of Okounkov and his coauthors. For …

关于紧化雅可比簇的陈类的反常性

On the Perversity of Chern Classes for Compactified Jacobians

本文证明了紧化雅可比簇纤维化陈类的反常性界，即第k个陈类的反常性≤k，并动机性地证明了arXiv:2402:08861中一个猜想的滤过版本。

An algebraic geometric realization of the Chern character

Using symmetrized Grassmannians we give an algebraic geometric presentation, in the level of classifying spaces, of the Chern character and its relation to Chern classes. This allows one to define, for any projective variety $X$, a Chern character map from the "holomorphic K-theory of X to its morphic cohomology (introduced by Friedlander and Lawson). The holomorphic K-theory of $X$, introduced by Lawson, Lima-Filho and Michelsohn and also by Friedlander and Walker, is defined in terms a group-completion of the space of algebraic morphisms from X into BU. It has been further studied by the aut…

Motivic Bivariant Characteristic Classes

Let K_0(V/X) be the relative Grothendieck group of varieties over X in obj(V), with V the category of (quasi-projective) algebraic (resp. compact complex analytic) varieties over a base field k. Then we constructed the motivic Hirzebruch class transformation in the algebraic context for k of characteristic zero and in the compact complex analytic context. It unifies the well-known three characteristic class transformations of singular varieties: MacPherson's Chern class, Baum-Fulton-MacPherson's Todd class and the L-class of Goresky-MacPherson and Cappell-Shaneson. In this paper we construct a…

非正则基格式上的Chern类、K-理论和Landweber正合性

Chern classes, K-theory and Landweber exactness over nonregular base schemes

本文件有两个目的。首先，利用动力Landweber正合函子定理，证明了Bott逆无限射影空间是同伦代数$K$ -理论。该论点是相当短比任何其他已知的证据，以及作为一个例证的有效性兰德韦伯精确性。其次，我们免除了经常隐含地需要在定向motivic环谱的概念的基础计划的正则性假设。后者使我们能够验证动机Landweber精确函子定理和有限克鲁尔维数的任意Noether基格式的代数配边谱的泛性质。

Schubert 胞的 Motivic Chern 类、Hecke 代数及其在 Casselman 问题中的应用

Motivic Chern classes of Schubert cells, Hecke algebras, and applications to Casselman's problem

本文计算了旗流形中 Schubert 胞的 motivic Chern 类，证明其由 Hecke 代数中的 Demazure-Lusztig 算子递归决定，并利用此递归与 Langlands 对偶性证明了关于标准基与 Casselman 基过渡矩阵系数分解及全纯性质的两个猜想。

天体积分、弦不变量和Chern-Schwartz-MacPherson类

Celestial integration, stringy invariants, and Chern-Schwartz-MacPherson classes

我们引入了一个正式的积分系统的品种映射正确和双理性到一个给定的，在一个相关的周群的价值。应用包括比较陈数的双有理品种，新的双有理不变量，“弦"陈类，和一个”天体“zeta函数专门的拓扑zeta函数。在其最简单的表现形式中，积分给出了可能是奇异变种的陈-施瓦茨-麦克弗森类的新表达式，将它们置于“变量变量变换”公式成立的上下文中。形式主义与动机整合有着密切的联系。

Chevalley公式用于Schubert胞的motivic Chern类以及稳定包中的稳定包

Chevalley formulae for the motivic Chern classes of Schubert cells and for the stable envelopes

本文基于Lenart和Postnikov的λ链，推导了广义旗 manifold 中 Schubert 胞 motivic Chern 类与线丛类相乘的 Chevalley 公式，并探讨了稳定包在 T*(G/B) 中的墙穿越公式及多项式系数的正性性质。

博雷尔 - 摩尔动机同调中的极限与边界特征类

The limit and boundary characteristic classes in Borel-Moore motivic homology

研究博雷尔 - 摩尔动机同调中极限与边界特征类，通过极限博雷尔 - 摩尔动机同调中的前特征类，证明零维部分可提升至零阶苏斯林同调，给出正特征下不成立的衡量方式及边界特征类前推公式并猜想与斯旺类一致。

The ample cone of moduli spaces of sheaves on the plane

Fix a Chern character of a stable sheaf on the plane. Assume either the rank is at most 6 or the rank and first Chern class are coprime and the discriminant is sufficiently large. We use recent results of Bayer and Macri on Bridgeland stability to compute the ample cone of the moduli space of Gieseker semistable sheaves with the given Chern character. We recover earlier results, such as those by Stromme and Yoshioka, as special cases.

Motivic characteristic classes in cohomological Hall algebras

The equivariant Chern-Schwartz-MacPherson (CSM) class and the equivariant Motivic Chern (MC) class are important characteristic classes of singular varieties in cohomology and K theory---and their theory overlaps with the theory of Okounkov's stable envelopes. We study CSM and MC classes for the orbits of Dynkin quiver representations. We show that the problem of computing the CSM and MC classes of all these orbits can be reduced to some basic classes $c^o_β$, $C^o_β$ parameterized by positive roots $β$. We prove an identity in a deformed version of Kontsevich-Soibelman's Cohomological (and K-…

第二动机陈类的簇构造

Cluster construction of the second motivic Chern class

本文通过簇坐标构造了任意分裂单连通单代数群 G 的第二泛动机陈类的显式余循环，并给出其分量构造、不同簇间的上边缘关系及在量子形变等方面的应用。

原代数簇的特征类与动机测度

Characteristic classes of proalgebraic varieties and motivic measures

Michael Gromov最近提出了他所谓的“符号代数几何”，其中对象是前代数簇：前代数簇根据定义是代数簇的投影系统的投射极限。本文以原代数簇的MacPherson Chern类变换的构造为模型，利用Fulton-MacPherson双变理论中的Grothendieck变换，引入原代数簇的特征类.我们表明，proalgebrical版本的欧拉-庞加莱特征的Grothendieck环中的值是一个推广的所谓motivic措施。

双 Whittaker 多项式与 Motivic Chern 类的格点模型

Lattice Models for Double Whittaker Polynomials and Motivic Chern Classes

本文通过构造依赖两组变量的可解格点模型，统一了 Motivic Chern 类与 Iwahori Whittaker 函数的格点表示，并揭示了 Kazhdan-Lusztig R-多项式形变的递推公式源于 Yang-Baxter 方程。

奇异簇的Stringy Chern类

Stringy Chern classes of singular varieties

本文将Motivic积分与MacPherson变换相结合，构造了奇异簇的“弦”Chern类理论。这些类具有很强的双有理不变性，它们的定义编码的数据来自奇点的分解。理论中允许的奇点是那些典型的极小模型程序;例子是由有限群的流形的约束给出的。对于后者，证明了一个明确的公式，假设流形的标准线丛下降到商。这给出了商的弦Chern类在不动点集数据的Chern-Schwartz-MacPherson类中的表达式。

由Bialynicki-Birula方法产生的动机分解

On motivic decompositions arising from the method of Bialynicki-Birula

最近，V. Chernousov，S. Gille和A. Merkurjev得到了一个类似于Bruhat分解的各向同性光滑射影齐次簇的动机的分解。使用的方法Bialynicki比鲁拉，我推广这种分解的情况下（可能各向异性）光滑的投影品种均匀的各向同性还原组的作用下。这就回答了一个N的问题。卡宾科

Characteristic classes of Hilbert schemes of points via symmetric products

We obtain a formula for the generating series of (the push-forward under the Hilbert-Chow morphism of) the Hirzebruch homology characteristic classes of the Hilbert schemes of points for a smooth quasi-projective variety of arbitrary pure dimension. This result is based on a geometric construction of a motivic exponentiation generalizing the notion of motivic power structure, as well as on a formula for the generating series of the Hirzebruch homology characteristic classes of symmetric products. We apply the same methods for the calculation of generating series formulae for the Hirzebruch cla…

Motivic twisted K-theory

This paper sets out basic properties of motivic twisted K-theory with respect to degree three motivic cohomology classes of weight one. Motivic twisted K-theory is defined in terms of such motivic cohomology classes by taking pullbacks along the universal principal BG_m-bundle for the classifying space of the multiplicative group scheme. We show a Kuenneth isomorphism for homological motivic twisted K-groups computing the latter as a tensor product of K-groups over the K-theory of BG_m. The proof employs an Adams Hopf algebroid and a tri-graded Tor-spectral sequence for motivic twisted K-theor…

Chow动机的高阶Tate迹

Higher Tate traces of Chow motives

本文通过定义动机的Tate迹这一新不变量，并利用Chernousov-Gille-Merkurjev分解与Bialynicki-Birula-Hesselink-Iversen滤过，完成了p-内半单代数群射影齐性簇Chow动机的完整分类，同时推广了内半单群动机等价性的刻画。

The Effective Cone of Moduli Spaces of Sheaves on a Smooth Quadric Surface

Let $ξ$ be a stable Chern character on $\mathbb{P}^1 \times \mathbb{P}^1$, and let $M(ξ)$ be the moduli space of Gieseker semistable sheaves on $\mathbb{P}^1 \times \mathbb{P}^1$ with Chern character $ξ$. In this paper, we provide an approach to computing the effective cone of $M(ξ)$ after showing that it is a Mori dream space for all $ξ$. We find Brill-Noether divisors spanning extremal rays of the effective cone using resolutions of the general elements of $M(ξ)$ which are found using the machinery of exceptional bundles. We use this approach to provide many examples of extremal rays in thes…

$\hbar$-deformed Schubert calculus in equivariant cohomology, K-theory, and elliptic cohomology

In this survey paper we review recent advances in the calculus of Chern-Schwartz-MacPherson, motivic Chern, and elliptic classes of classical Schubert varieties. These three theories are one-parameter ($\hbar$) deformations of the notion of fundamental class in their respective extraordinary cohomology theories. Examining these three classes in conjunction is justified by their relation to Okounkov's stable envelope notion. We review formulas for the $\hbar$-deformed classes originating from Tarasov-Varchenko weight functions, as well as their orthogonality relations. As a consequence, explici…

Motivic Pontryagin 类与双曲定向

Motivic Pontryagin classes and hyperbolic orientations

本文提出了motivic环谱的双曲定向概念，证明其对应Pontryagin类理论，并构造了万有双曲定向eta-周期交换motivic环谱，为研究向量丛示性类提供了新框架。

On higher Chern classes of vector bundles of conformal blocks

Here we consider higher Chern classes of vector bundles of conformal blocks on $\overline{\operatorname{M}}_{0,n}$, giving explicit formulas for them, and extending various results that hold for first Chern classes to them. We use these classes to form a full dimensional subcone of the Pliant cone on $\overline{\operatorname{M}}_{0,n}$.

On syntomic regulators I: constructions

We show that classical Chern classes from higher ($p$-adic) $K$-theory to syntomic cohomology extend to logarithmic syntomic cohomology. These Chern classes are compatible -- in a suitable sense -- with addition, products, and $λ$-operations. They are also compatible with the canonical Gysin sequences and, via period maps, with logarithmic étale Chern classes. Moreover, they induce logarithmic crystalline Chern classes. This uses as a critical new ingredient the recent comparison of syntomic cohomology with $p$-adic nearby cycles and $p$-adic motivic cohomology.

Framed transfers and motivic fundamental classes

We relate the recognition principle for infinite $\mathbf P^1$-loop spaces to the theory of motivic fundamental classes of Déglise, Jin, and Khan. We first compare two kinds of transfers that are naturally defined on cohomology theories represented by motivic spectra: the framed transfers given by the recognition principle, which arise from Voevodsky's computation of the Nisnevish sheaf associated with $\mathbf A^n/(\mathbf A^n-0)$, and the Gysin transfers defined via Verdier's deformation to the normal cone. We then introduce the category of finite E-correspondences for E a motivic ring spect…

拟阵的tautological类在nef锥中的极值性

Extremality of tautological classes of matroids in nef cones

本文研究了置换簇的多面体nef锥中由拟阵的tautological Chern类生成的射线的极值性，给出了拟阵的tautological Chern类生成极值射线的充要条件，并构造了介于秩函数和Bergman类之间的新极值射线族，进一步讨论了该定理通过增广tautological Chern类向星形簇的推广。

在排列胞菌簇上的Grothendieck权重与Matroids

Grothendieck Weights on Permutohedral Varieties and Matroids

本文研究排列胞菌簇上的Grothendieck权重，证明了K-平衡条件和环结构乘法规则，并应用于Matroids，计算了超平面排列补集的motivic Chern类，证明其仅依赖于Matroid而非具体实现。

Projective duality and a Chern-Mather involution

We observe that linear relations among Chern-Mather classes of projective varieties are preserved by projective duality. We deduce the existence of an explicit involution on a part of the Chow group of projective space, encoding the effect of duality on Chern-Mather classes. Applications include Plücker formulae, constraints on self-dual varieties, generalizations to singular varieties of classical formulas for the degree of the dual and the dual defect, formulas for the Euclidean distance degree, and computations of Chern-Mather classes and local Euler obstructions for cones.

有理曲线的动机分布与环面簇的扭曲乘积

Motivic distribution of rational curves and twisted products of toric varieties

本文研究光滑射影曲线上一族代数截面模空间的渐近稳定化，提出有理曲线情形的 Batyrev-Manin-Peyre 原理类比，并证明等分布性质对分裂环面簇成立，应用于扭曲乘积。

Explicit Formulas for h-Deformed Structure Constants of Grassmannians

The Chern-Schwartz-MacPherson (CSM) and motivic Chern (mC) classes of Schubert cells in a Grassmannian are one parameter deformations of the fundamental classes of the Schubert varieties in cohomology and K-theory respectively. Like the fundamental classes, the deformed classes form a basis for the cohomology and K-theory ring of the Grassmannian. The purpose of this paper is to initiate the study of the structure constants associated to the basis CSM and mC classes in terms of the combinatorics of polynomials. First, we prove formulas for the structure constants of projective spaces that invo…

Virtual fundamental classes of derived stacks I

We construct the étale motivic Borel-Moore homology of derived Artin stacks. Using a derived version of the intrinsic normal cone, we construct fundamental classes of quasi-smooth derived Artin stacks and demonstrate functoriality, base change, excess intersection, and Grothendieck-Riemann-Roch formulas. These classes also satisfy a general cohomological Bézout theorem which holds without any transversity hypotheses. The construction is new even for classical stacks and as one application we extend Gabber's proof of the absolute purity conjecture to Artin stacks.

Motivic bivariant characteristic classes

The relative Grothendieck group $K_0(\m V/X)$ is the free abelian group generated by the isomorphism classes of complex algebraic varieties over $X$ modulo the "scissor relation". The motivic Hirzebruch class ${T_y}_*: K_0(\m V /X) \to H_*^{BM}(X) \otimes \bQ[y]$ is a unique natural transformation satisfying that for a nonsingular variety $X$ the value ${T_y}_*([X \xrightarrow {\op {id}_X} X])$ of the isomorphism class of the identity $X \xrightarrow {id_X} X$ is the Poincaré dual of the Hirzebruch cohomology class of the tangent bundle $TX$. It "unifies" the well-known three characteristic cl…

Characteristic classes of singular toric varieties

In this paper we compute the motivic Chern classes and homology Hirzebruch characteristic classes of (possibly singular) toric varieties, which in the complete case fit nicely with a generalized Hirzebruch-Riemann-Roch theorem. As special cases, we obtain new (or recover well-known) formulae for the Baum-Fulton-MacPherson Todd (or MacPherson-Chern) classes of toric varieties, as well as for the Thom-Milnor L-classes of simplicial projective toric varieties. We present two different perspectives for the computation of these characteristic classes of toric varieties. First, we take advantage of …

The Duistermaat-Heckman formula and Chern-Schwartz-MacPherson classes

Let M be a smooth complex projective variety, bearing a Kähler symplectic form ωand a Hamiltonian action of a torus T, with finitely many fixed points M^T. One standard form of the Duistermaat-Heckman theorem gives a formula for M's Duistermaat-Heckman measure DH_T(M,ω) as an alternating sum of projections of cones, with overall direction determined by a Morse decomposition of M. Using Victor Ginzburg's construction of Chern-Schwartz-MacPherson classes, we show that these individual cone terms can themselves be interpreted as Duistermaat-Heckman measures of cycles in T^*M. (This has a similar …

虚基本类、全局normal cones和Fulton规范类

Virtual fundamental classes, global normal cones and Fulton's canonical classes

本文首次写于1997年，目的是简化和更基本的代数几何虚拟基本类的建设所定义的Li/Tian和Behrend/Fantechi。我们取代了使用阿丁堆栈在后者的方法由瑜伽锥束可能的独立利益。我们还观察到一个公式的虚拟基本类只涉及全陈类的指标丛和富尔顿的典型类的模空间。

Orbifold Chern classes inequalities and applications

In this paper we prove that given a pair $(X,D)$ of a threefold $X$ and a boundary divisor $D$ with mild singularities, if $(K_X+D)$ is movable, then the orbifold second Chern class $c_2$ of $(X,D)$ is pseudo-effective. This generalizes the classical result of Miyaoka on the pseudo-effectivity of $c_2$ for minimal models. As an application we give a simple solution to Kawamata's effective non-vanishing conjecture in dimension $3$, where we prove that $H^0(X, K_X+H)\neq 0$, whenever $K_X+H$ is nef and $H$ is an ample, effective, reduced Cartier divisor. Furthermore, we study Lang-Vojta's conjec…

Nearby cycles and characteristic classes of singular spaces

In this paper we give an introduction to our recent work on characteristic classes of complex hypersurfaces based on some talks given at conferences in Strasbourg, Oberwolfach and Kagoshima. We explain the relation between nearby cycles for constructible functions or sheaves as well as for (relative) Grothendieck groups of algebraic varieties and mixed Hodge modules, and the specialization of characteristic classes of singular spaces like the Chern-, Todd-, Hirzebruch- and motivic Chern-classes. As an application we get a description of the differences between the corresponding virtual and fun…

The structure of a local embedding and applications to Chern classes of weighted blow-ups

For a proper local embedding between two Deligne--Mumford stacks Y and X, we find, under certain mild conditions, a new (possibly non-separated) Deligne--Mumford stack X', with an etale, surjective and universally closed map to the target X, and whose fiber product with the image of the local embedding is a finite union of stacks with corresponding etale, surjective and universally closed maps to Y. Moreover, a natural set of weights on the substacks of X' allows the construction of a universally closed push-forward, and thus a comparison between the Chow groups of X' and X. We apply the const…

奇异空间的特征类综述

A survey of characteristic classes of singular spaces

向量丛和光滑流形的特征类理论在光滑流形理论中占有重要地位。奇异空间的特征类的合理概念的研究始于对奇异空间如奇异代数簇的系统研究。我们对奇异簇的特征类作了一个快速的综述，主要集中在一些重要的特征类的函子方面，如Chern类、Todd类和Thom-Hirzebruch的L-类的奇异形式.然后，我们解释了我们最近的“动机”特征类，在某种意义上统一了这三种不同的特征类理论。我们还讨论了它们的双变版本和特征类的proalgebraic簇，这是相关的motivic措施/集成。最后，我们解释了一些最近的工作，这些理论的“弦”的版本，连同一些参考“等变”的同行。

The nef cone of the moduli space of sheaves and strong Bogomolov inequalities

Let (X,H) be a polarized, smooth, complex projective surface, and let v be a Chern character on X with positive rank and sufficiently large discriminant. In this paper, we compute the Gieseker wall for v in a slice of the stability manifold of X. We construct explicit curves parameterizing non-isomorphic Gieseker stable sheaves that become S-equivalent along the wall. As a corollary, we conclude that if there are no strictly semistable sheaves of character v, the Bayer-Macri divisor associated to the wall is a boundary nef divisor on the moduli space of sheaves M_H(v). We recover previous resu…