Wolfgang Ebeling, Sabir M. Gusein-Zade
We introduce a certain index of a collection of germs of 1-forms on a germ of a singular variety which is a generalization of the local Euler obstruction corresponding to Chern numbers different from the top one.
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Following Bloch-Esnault-Kerz and Green-Griffiths' recent works on deformation of algebraic cycle classes, we use Chern character from K-theory to negative cyclic homology to show how to eliminate obstructions to deforming cycles.
In this note we introduce the concept of reflective projective varieties. These are stratified projective varieties with certain dimension constraints on their dual varieties. We prove that for such varieties, the Chern-Schwartz-MacPherson classes of the strata completely determine the local Euler obstructions and the polar degrees. We also propose an algorithm to compute the local Euler obstructions when such varieties are formed by group orbits. As examples we compute the local Euler obstructions of quadratic hypersurfaces and ordinary determinantal varieties to illustrate our method.
This work has two complementary parts, in the first part we compute the local Euler obstruction of generic determinantal varieties and apply this result to compute the Chern--Schwartz--MacPherson class of such varieties. In the second part we compute the Euler characteristic of the stabilization of an essentially isolated determinantal singularity (EIDS). The formula is given in terms of the local Euler obstruction and Gaffney's $m_{d}$ multiplicity.
We give explicit formulas for the dimensions and the degrees of $A$-discriminant varieties introduced by Gelfand-Kapranov-Zelevinsky. Our formulas can be applied also to the case where the $A$-discriminant varieties are higher-codimensional and their degrees are described by the geometry of the configurations $A$. Moreover combinatorial formulas for the Euler obstructions of general (not necessarily normal) toric varieties will be also given.
Rational points on varieties and the Brauer-Manin obstruction
本讲义介绍了有理点存在性的Brauer-Manin障碍，重点探讨理论与计算之间的相互作用。
We use Matsui and Takeuchi's formula for toric A-discriminants to give algorithms for computing local Euler obstructions and dual degrees of toric surfaces and 3-folds. In particular, we consider weighted projective spaces. As an application we give counterexamples to a conjecture by Matsui and Takeuchi. As another application we recover the well-known fact that the only defective normal toric surfaces are cones.
We show how to transport descent obstructions from the category of covers to the category of varieties. We deduce examples of curves having $\QQ$ as field of moduli, that admit models over every completion of $\QQ$, but have no model over $\QQ$.
We investigate the question of whether the existence of a family of local zero-cycles of degree $d$ orthogonal to the Brauer group implies the non-emptiness of the Brauer-Manin set for certain varieties. We provide various examples of Brauer-Manin obstruction to the existence of zero-cycles of appropriate degrees.
Using Chern character, we construct a natural transformation from the local Hilbert functor to a functor of Artin rings defined from Hochschild homology, which allows us to reconstruct the semi-regularity map and the infinitesimal Abel-Jacobi map. Combining that construction of the semi-regularity map with obstruction theory of functors of Artin rings, we give a different proof of a theorem of Bloch stating that the semi-regularity map annihilates certain obstructions to embedded deformations of a closed subvariety which is a locally complete intersection.
Nous montrons comment associer à une gerbe définie sur un corps de nombres une obstruction de Brauer-Manin mesurant, comme dans le cas des variétés, le défaut d'existence d'une section globale. Ceci nous conduit à une généralisation de la dualité de Tate-Poitou au cas non-abélien. We find a way to associate to a gerbe defined over a number field a Brauer-Manin obstruction, which measures, as for varieties, the defect to the existence of a global section. This leads us to a generalization of Tate-Poitou duality in the non-abelian case.
Recently Dasheng Wei proved that the Brauer-Manin obstruction is the only obstruction to the Hasse principle for 0-cycles of degree 1 on some fibrations over the projective line defined by bi-cyclic normic equations. In the present paper, we prove the exactness of the global-to-local sequence for Chow groups of 0-cycles of such varieties, which signifies that the Brauer-Manin obstruction is also the only obstruction to weak approximation for 0-cycles of arbitrary degree. Our main theorem also generalizes several existing results.
Chern-Schwartz-MacPherson classes in the point of view of Obstruction Theory and Lipschitz framework
本文针对奇异簇陈类的定义问题，通过在利普希茨框架下利用Marie-Hélène Schwartz基于障碍理论的原始思想，简化了复解析簇陈类的定义与构造，弥补了该原始定义被遗忘的不足。
The goal of this note is to explain a derivation of the formulas for the local Euler obstructions of determinantal varieties of general, symmetric and skew-symmetric matrices, by studying the invariant de Rham complex and using character formulas for simple equivariant $D$-modules. These calculations are then combined with standard arguments involving Kashiwara's local index formula and the description of characteristic cycles of simple equivariant $D$-modules. The formulas are implicit in the work of Boe and Fu, and in the case of general matrices they have also been obtained recently by Gaff…
We give a simplified proof (in characteristic zero) of the decomposition theorem for complex projective varieties with klt singularities and numerically trivial canonical bundle. The proof rests in an essential way on most of the partial results of the previous proof obtained by many authors, but avoids those in positive characteristic by S. Druel. The single to some extent new contribution is an algebraicity and bimeromorphic splitting result for generically locally trivial fibrations with fibres without holomorphic vector fields. We give first the proof in the easier smooth case, following t…
Let $X$ be a complete $\mathbb{Q}$-factorial toric variety. We explicitly describe the space $H^2(X,T_X)$ and the cup product map $H^1(X,T_X)\times H^1(X,T_X)\to H^2(X,T_X)$ in combinatorial terms. Using this, we give an example of a smooth projective toric threefold for which the cup product map does not vanish, showing that in general, smooth complete toric varieties may have obstructed deformations.
We study the Brauer-Manin obstruction to the existence of zero-cycles of degree $d$ on certain classes of varieties over number fields. We generalise existing results in the literature and prove some results about fibrations over the projective line, where the geometric Brauer group of the generic fibre is not assumed to be finite. The idea is to assume that the Brauer-Manin obstruction to the Hasse principle is the only one for certain fibres and then deduce analogous results for zero-cycles.
We give explicit MacPherson cycles for the Chern-MacPherson class of a closed affine algebraic variety $X$ and for any constructible function $α$ with respect to a complex algebraic Whitney stratification of $X$. We define generalized degrees of the global polar varieties and of the MacPherson cycles and we prove a global index formula for the Euler characteristic of $α$. Whenever $α$ is the Euler obstruction of $X$, this index formula specializes to the Seade-Tibar-Verjovsky global counterpart of the Le-Teissier formula for the local Euler obstruction.
We prove that for a large class of subvarieties of abelian varieties over global function fields, the Brauer-Manin condition on adelic points cuts out exactly the rational points. This result is obtained from more general results concerning the intersection of the adelic points of a subvariety with the adelic closure of the group of rational points of the abelian variety.
Several authors have proved Lefschetz type formulae for the local Euler obstruction. In particular, a result of this type is proved in [BLS].The formula proved in that paper turns out to be equivalent to saying that the local Euler obstruction, as a constructible function, satisfies the local Euler condition (in bivariant theory) with respect to general linear forms. The purpose of this work is to understand what prevents the local Euler obstruction of satisfying the local Euler condition with respect to functions which are singular at the considered point. This is measured by an invariant (or…
The local obstruction to semi-stable reduction for abelian varieties
本文综述了格罗滕迪克定义的阿贝尔簇半稳定约化局部障碍群的研究进展，并首次纳入等特征局部域情形，旨在给出仅依赖维数的群论刻画。
Intersection theory and Chern classes on normal varieties
本文研究正规簇上自反层的相交理论与陈类，将Mumford理论推广至更高维，定义第二陈类，并证明正特征下的Bogomolov型不等式及有界性结果。
For $m\geq n$, Let $K$ be an algebraic closed base field, and define $τ_{m,n,k}$ to be the set of $m\times n$ matrices over $K$ with kernel dimension $\geq k$. This is a projective subvariety of $\mathbb{P}^{mn-1}$, and is usually called determinantal variety. In most cases $τ_{m,n,k}$ is singular with singular locus $τ_{m,n,k+1}$. In this paper we compute the local Euler obstruction of $τ_{m,n,k}$, and we prove that the characteristic cycle of the intersection cohomology complex of $τ_{m,n,k}$ is irreducible. We also give an explicit formula for the Chern-Mather class of $τ_{m,n,k}$ as a clas…
It is known that projective minimal models satisfy the celebrated Miyaoka-Yau inequalities. In this article, we extend these inequalities to the set of all smooth, projective and non-uniruled varieties.
Reflective obstructions of unitary modular varieties
本文利用 Prasad 体积公式定量估计酉模簇的反射障碍，证明高维时障碍足够小，并将其 Kodaira 维数问题归结为小权尖点形式构造。
We give upper and lower bounds for the order of the top Chern class of the Hodge bundle on the moduli space of principally polarized abelian varieties.
We give upper and lower bounds for the order of the top Chern class of the Hodge bundle on the moduli space of principally polarized abelian varieties. We also give a generalization to higher genera of the famous formula $12 λ_1=δ$ for genus 1.
Product of Brauer--Manin obstruction for 0-cycles over number fields and function fields
本文证明光滑簇乘积上1次0-循环的Brauer–Manin障碍等价于各因子同时存在该障碍，并将类似结果推广到$\mathbb{C}((t))$-曲线的函数域上的光滑簇。
In part I we reduced the arithmetic (characteristic zero) version of the P \not \subseteq NP conjecture to the problem of showing that a variety associated with the complexity class NP cannot be embedded in the variety associated the complexity class P. We call these class varieties. In this paper, this approach is developed further, reducing the nonexistence problems, such as the P vs. NP and related lower bound problems, to existence problems: specifically to proving existence of obstructions to such embeddings among class varieties. It gives two results towards explicit construction of such…
A complex analytic approach to orbifold Chern classes on singular varieties and its applications
本文提出了一种复分析方法，用于研究奇异变种上的环面 Chern 类，并应用于稳定有理 sheaf 的 Bogomolov-Gieseker 不等式及复杂 torus 商的特征化。
The purpose of this short note is to prove a formula for the Chern-Mather classes of a toric variety in terms of its orbits and the local Euler obstructions at general points of each orbit (Theorem 2). We use the general definition of the Chern-Schwartz-MacPherson classes (see [7]) and their special expression in case of a toric variety (see [2]). As a corollary, we obtain a formula by Matsui-Takeuchi [8, Corollary 1.6]. Alternatively, one could deduce the formula of Theorem 2 from the Matsui-Takeuchi formula, by using our general result [11, Theoreme 3] for the degree of the polar varieties i…
Let $A^{[[n]]}$ denote the $2(n - 1)$-dimensional generalised Kummer variety constructed from the abelian surface $A$. Further, let $X$ be an arbitrary smooth projective surface with $\int_X c_1(X)^2 \neq 0$, and $X^{[k]}$ the Hilbert scheme of zero-dimensional subschemes of $X$ of length $k$. We give a formula which expresses the value of any complex genus on $A^{[[n]]}$ in terms of Chern numbers of the varieties $X^{[k]}$. It is shown by Ellingsrud and Stroemme how to use Bott's residue formula to effectively calculate the Chern numbers of the Hilbert schemes $(\IP^2)^{[k]}$ of points on the…
Euler obstructions and Verdier specializations
该研究为完全交簇的Milnor纤维与单值化的层论研究奠定基础，阐明Verdier特殊化层结构，结合Thom条件推导非孤立奇点完全交簇的欧拉障碍公式。
Birational invariance of motivic zeta functions of $K$-trivial varieties, and obstructions to smooth fillings
本研究证明$K$-平凡簇的动机zeta函数及相关单值性猜想具有双有理不变性，扩展了已知适用范围，并揭示该函数可作为簇在$t=0$处光滑填充存在性的障碍。
Obstructions to embedding singular curves in toric varieties
研究存在能嵌入\(\mathbb{P}^d\)却不能嵌入维度小于\(d\)的射影正规环面簇及加权射影空间的不可约约化曲线，通过证明得出此结论。
We give a generalization to higher genera of the famous formula $12 λ=δ$ for genus 1. We also compute the classes of certain strata in the Satake compactification as elements of the push down of the tautological ring.
We extend the descent theory of Colliot-Thélène and Sansuc to arbitrary smooth algebraic varieties by removing the condition that every invertible regular function is constant. This links the Brauer--Manin obstruction for integral points on arithmetic schemes to the obstructions defined by torsors under groups of multiplicative type.
Reciprocity obstruction to strong approximation over p-adic function fields
本文在p进曲线函数域上构造SL_n齐性空间反例，证明强逼近失效，并提出且刻画了作为唯一障碍的互反障碍。
In 1954 Hirzebruch asked which linear combinations of Chern numbers are topological invariants of smooth complex projective varieties. We give a complete answer to this question in small dimensions, and also prove partial results without restrictions on the dimension.
We give explicit formulas for the Chern-Schwartz-MacPherson classes of all Schubert varieties in the Grassmannian of $d$-planes in a vector space, and conjecture that these classes are effective. We prove this is the case for (very) small values of $d$.
We show that the Chern-Schwartz-MacPherson class of a Schubert cell in a Grassmannian is represented by a reduced and irreducible subvariety in each degree. This gives an affirmative answer to a positivity conjecture of Aluffi and Mihalcea.
A theorem of Batyrev's asserts that if two nonsingular varieties V,W are birational, and their canonical bundles agree after pull-back to a resolution of indeterminacies of a birational map between them, then the Betti numbers of V and W coincide. We prove that, in the same hypotheses, the total homology Chern classes of V and W are push-forwards of the same class in the Chow group of the resolution. For example, it follows that the push-forward of the total Chern class of a crepant resolution of a singular variety is independent of the resolution.
Let $k$ be a number field. In the spirit of a result by Yongqi Liang, we relate the arithmetic of rational points over finite extensions of $k$ to that of zero-cycles over $k$ for Kummer varieties over $k$. For example, for any Kummer variety $X$ over $k$, we show that if the Brauer-Manin obstruction is the only obstruction to the Hasse principle for rational points on $X$ over all finite extensions of $k$, then the ($2$-primary) Brauer-Manin obstruction is the only obstruction to the Hasse principle for zero-cycles of any given odd degree on $X$ over $k$. We also obtain similar results for pr…
Suppose $X$ is a smooth complex algebraic variety. A necessary condition for a complex topological vector bundle on $X$ (viewed as a complex manifold) to be algebraic is that all Chern classes must be algebraic cohomology classes, i.e., lie in the image of the cycle class map. We analyze the question of whether algebraicity of Chern classes is sufficient to guarantee algebraizability of complex topological vector bundles. For affine varieties of dimension $\leq 3$, it is known that algebraicity of Chern classes of a vector bundle guarantees algebraizability of the vector bundle. In contrast, w…
This is my PhD Thesis, part of it has published in Acta Mathematica Sinica. In this paper, a class of morphisms which have a kind of singularity weaker than normal crossing is considered. We construct the obstruction such that the so-called semi-stable log structures exists if and only if the obstruction vanishes. In the case of no power, if the obstruction vanishes, then the semi-stable log structure is unique up to a unique isomorphism. So we obtain a kind of canonical structures on this family of morphisms.
We generalize the Chern class relation for the transversal intersection of two nonsingular varieties to a relation for possibly singular varieties, under a 'splayedness' assumption. The relation is shown to hold for both the Chern-Schwartz-MacPherson class and the Chern-Fulton class. The main tool is a formula for Segre classes of splayed subschemes. We also discuss the Chern class relation under the assumption that one of the varieties is a general very ample divisor.
Obstructions to semiorthogonal decompositions for singular projective varieties II: Representation theory
本文通过比较射影簇与有限维 Gorenstein 代数的奇点范畴，证明具有倾斜对象且仅含 ADE 超曲面奇点的奇数维射影簇必为节点型，从而给出半正交分解存在性的新障碍。
In this paper, we study obstructions to the Dirichlet property by two approaches: density of non-positive points and functionals on adelic R-divisors. Applied to the algebraic dynamical systems, these results provide examples of nef adelic arithmetic R-Cartier divisor which does not have the Dirichlet property. We hope the obstructions obtained in the article will give ways toward criteria of the Dirichlet property.
Let $k$ be a number field, let $X$ be a Kummer variety over $k$, and let $δ$ be an odd integer. In the spirit of a result by Yongqi Liang, we relate the arithmetic of rational points over finite extensions of $k$ to that of zero-cycles over $k$ for $X$. For example, we show that if the Brauer-Manin obstruction is the only obstruction to the existence of rational points on $X$ over all finite extensions of $k$, then the $2$-primary Brauer-Manin obstruction is the only obstruction to the existence of a zero-cycle of degree $δ$ on $X$ over $k$.
Homotopy continuation provides a numerical tool for computing the equivalence of a smooth variety in an intersection product. Intersection theory provides a theoretical tool for relating the equivalence of a smooth variety in an intersection product to the degrees of the Chern classes of the variety. A combination of these tools leads to a numerical method for computing the degrees of Chern classes of smooth projective varieties in P^n. We illustrate the approach through several worked examples.
We study the obstruction degrees of translates of sub-tori of multiplicative tori and we show how they are connected to lower bounds for the essential minimum of these varieties. In particular, we combine our computations with results of A. Schinzel and of F. Amoroso - R. Dvornicich, that we thus extend to translates of sub-tori defined over CM fields. This shows that the essential minimum of these varieties have a geometric behavior similar to that of varieties which are not translates of sub-tori. We also refine and extend to translates of sub-tori a lower bound obtained by D. Bertrand in co…
Kawamata proposed a conjecture predicting that every nef and big line bundle on a smooth projective variety with trivial first Chern class has nontrivial global sections. We verify this conjecture for several cases, including (i) all hyperkähler varieties of dimension $\leq 6$; (ii) all known hyperkähler varieties except for O'Grady's 10-dimensional example; (iii) general complete intersection Calabi-Yau varieties in certain Fano manifolds (e.g. toric ones). Moreover, we investigate the effectivity of Todd classes of hyperkähler varieties and Calabi-Yau varieties. We prove that the fourth Todd…
Motivic integration and MacPherson's transformation are combined in this paper to construct a theory of "stringy" Chern classes for singular varieties. These classes enjoy strong birational invariance properties, and their definition encodes data coming from resolution of singularities. The singularities allowed in the theory are those typical of the minimal model program; examples are given by quotients of manifolds by finite groups. For the latter an explicit formula is proven, assuming that the canonical line bundle of the manifold descends to the quotient. This gives an expression of the s…
It was noticed in a very recent preprint of T. Eguchi, K. Hori, and Ch.-Sh. Xiong (hep-th/9703086) that a curious identity between Betti numbers and Chern classes holds for many examples of Fano varieties. The goal of this paper is to prove that for varieties with trivial odd cohomology groups this identity is equivalent to having zero Hodge numbers $h^{p,q}$ for $p\neq q$.
The Euler characteristic of Milnor fibers over 2-generic symmetric determinantal varieties
本文推导了非退化函数f:X→C的米尔诺纤维欧拉特征公式，通过描述概形的托尔结构和计算牛顿多面体体积，应用Matsui-Takeuchi公式得出结果，并探讨了局部欧拉阻挠与多项式米尔诺数的关系。
We prove that a rational linear combination of Chern numbers is an oriented diffeomorphism invariant of smooth complex projective varieties if and only if it is a linear combination of the Euler and Pontryagin numbers. In dimension at least three we prove that only multiples of the top Chern number, which is the Euler characteristic, are invariant under diffeomorphisms that are not necessarily orientation-preserving. These results solve a long-standing problem of Hirzebruch's. We also determine the linear combinations of Chern numbers that can be bounded in terms of Betti numbers.
Let X be a normal projective Q-Gorenstein variety with at worst log-terminal singularities. We prove a formula expressing the total stringy Chern class of a generic complete intersection in X via the total stringy Chern class of X. This formula is motivated by its applications to mirror symmetry for Calabi-Yau complete intersections in toric varieties. We compute stringy Chern classes and give a combinatorial interpretation of the stringy Libgober-Wood identity for arbitrary projective Q-Gorenstein toric varieties. As an application we derive a new combinatorial identity relating d-dimensional…
Weak approximation of symmetric products and norm varieties
本文研究数域上簇的对称积及其光滑射影模型是否保持带Brauer-Manin障碍的弱逼近性质，并将同一方法推广到范数簇。
In this paper, I construct Chern classes in the rigid cohomology of P. Berthelot. We start by constructing Chern classes for proper varieties. To prove all the properties we have to reinterpret the construction in a crystalline way. Then we can deal with open varieties using a compactification theorem.
We discuss a method for calculating the Chern-Schwartz-MacPherson (CSM) class of a Schubert variety in the Grassmannian using small resolutions introduced by Zelevinsky. As a consequence, we show how to compute the Chern-Mather class and local Euler obstructions using small resolutions instead of the Nash blowup. The algorithm obtained for CSM classes also allows us to prove new cases of a positivity conjecture of Aluffi and Mihalcea.
Consider the natural torus action on a partial flag manifold $Fl$. Let $Ω_I\subset Fl$ be an open Schubert variety, and let $c^{sm}(Ω_I)\in H_T^*(Fl)$ be its torus equivariant Chern-Schwartz-MacPherson class. We show a set of interpolation properties that uniquely determine $c^{sm}(Ω_I)$, as well as a formula, of `localization type', for $c^{sm}(Ω_I)$. In fact, we proved similar results for a class $κ_I\in H_T^*(Fl)$ --- in the context of quantum group actions on the equivariant cohomology groups of partial flag varieties. In this note we show that $c^{SM}(Ω_I)=κ_I$.
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