Fedor Bogomolov, Yuri Tschinkel
We prove that the function field of an algebraic variety of dimension greater than 1 over an algebraically closed field of characteristic zero is determined by its first and second Milnor K-groups.
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The aim of this note is to give a simplified proof of the surjectivity of the natural Milnor-Chow homomorphism $ρ: K^M_n(A) \to CH^n(A,n)$ between Milnor $K$-theory and higher Chow groups for essentially smooth (semi-)local $k$-algebras $A$ with $k$ infinite. It implies the exactness of the Gersten resolution for Milnor $K$-theory at the generic point. Our method uses the Bloch-Levine moving technique and some properties of the Milnor $K$-theory norm for fields.
We construct the homomorphism of presheaves ${\mathrm{K}}^\mathrm{MW}_* \to π^{*,*}$ over an arbitrary base scheme $S$, where $\mathrm{K}^\mathrm{MW}$ is the (naive) Milnor-Witt K-theory presheave. Also we discuss some partly alternative proof (or proofs) of the isomorphism of sheaves $\unKMW_n\simeq \underlineπ^{n,n}_s$, $n\in \mathbb Z$, over a filed $k$ originally proved in \cite{M02} and \cite{M-A1Top}.
We explicitly study Kato's residue homomorphisms in Milnor $K$-theory, which are closely related to Contou-Carrère symbols. As applications we establish several reciprocity laws for certain locally defined maps on $K$-groups that are associated to arithmetic surfaces.
We discuss combinatorial invariance of the betti numbers of the Milnor fiber for arrangements of lines with points of multiplicity at most three and describe a link between this problem and enumeration of solutions of the Catalan equation over function field in the case when its coefficients are products of linear forms and the equation defines an elliptic curve.
We study the cohomology theory and the canonical Milnor-Witt cycle module associated to a motivic spectrum. We prove that the heart of Morel-Voevodsky stable homotopy category over a perfect field (equipped with its homotopy t-structure) is equivalent to the category of Milnor-Witt cycle modules, thus generalising Déglise's thesis. As a corollary, we recover a theorem of Ananyevskiy and Neshitov and we prove that the Milnor-Witt K-theory groups are birational invariants.
We prove a restriction isomorphism for Chow groups of zero-cycles with coefficients in Milnor K-theory for smooth projective schemes over excellent henselian discrete valuation rings. Furthermore, we study torsion subgroups of these groups over local and finite fields.
Algebraic part of motivic cohomology with compact supports
本文将代数等价关系与正则同态推广至域上Voevodsky motives，证明了特定上同调群存在泛正则同态，恢复Murre定理等经典结论，并涵盖高阶Chow群与Milnor K-群。
Let $X$ be a smooth projective variety over a field of characteristic $0$ and $L$ an ample divisor. In this paper we study the Weak Lefschetz conjecture for Chow groups using the technique employed by Grothendieck in his study of the problem for Picard groups, and using Bloch's formula to interpret Chow groups in terms of Milnor $K$- theory we prove the infinitesimal part of the conjecture.
The definition of Milnor-Witt cycle modules in [Feld, N., Milnor-Witt cycle modules, Journal of Pure and Applied Algebra 224 (2020) 106298] can easily be adapted over general regular base schemes. However, there are simple examples to show that Gersten complex fails to be exact for cycle modules in general if the base is not a field. The goal of this article is to show that, for a restricted class of Milnor-Witt cycle modules over an excellent DVR satisfying an extra axiom, called here as R5, the expected properties of exactness of Gersten complex and $\mathbb{A}^1$-invariance hold. Moreover R…
We prove the version of Joyce-Song formula for the Behrend function identities in the motivic setting. The main method we use is the proof of Kontsevich-Soibelman conjecture about the motivic Milnor fibers by Q. T. Le, who uses the method of motivic integration for formal schemes and Cluckers-Loeser's motivic constructible functions. In the Appendix the motivic formula can be used to provide a different proof that there is an algebra homomorphism of Kontsevich-Soibelman from the motivic Hall algebra of the abelian category of coherent sheaves on a Calabi-Yau threefold $Y$ to the motivic quantu…
Given a real polynomial function and a point in its zero locus, we defined a set consisting of algebraic real Puiseux series naturally attached to these data. We prove that this set determines the topology and the geometry of the real Milnor fibre of the function at this point. To achieve this goal, we balance between the tameness properties of this set of Puiseux series, considered as a real algebraic object over the field of algebraic Puiseux series, and its behaviour as an infinite dimensional object over the real numbers.
We investigate properties of the contact exponent (in the sense of Hironaka [Hi]) of plane algebroid curve singularities over algebraically closed fields of arbitrary characteristic. We prove that the contact exponent is an equisingularity invariant and give a new proof of the stability of the maximal contact. Then we prove a bound for the Milnor number and determine the equisingularity class of algebroid curves for which this bound is attained. We do not use the method of Newton's diagrams. Our tool is the logarithmic distance developed in [GB-P1].
We show how formal and rigid geometry can be used in the theory of complex singularities, and in particular in the study of the Milnor fibration and the motivic zeta function. We introduce the so-called analytic Milnor fiber associated to the germ of a morphism f from a smooth complex algebraic variety X to the affine line. This analytic Milnor fiber is a smooth rigid variety over the field of Laurent series C((t)). Its etale cohomology coincides with the singular cohomology of the classical topological Milnor fiber of f; the monodromy transformation is given by the Galois action. Moreover, th…
We study homomorphisms of multiplicative groups of fields preserving algebraic dependence and show that such homomorphisms give rise to valuations.
This paper shows that algebraic (in)dependence is encoded in Milnor K-theory of fields. As an application, we show that the isomorphism type of a field is determined by its Milnor K-theory, up to purely inseparable extensions, in most situations.
Certain topologies on Milnor K-groups of higher local fields K are studied. These are related to the topology on the multiplicative group and important for explicit higher local class field theory. The structure of the quotient of Milnor K-groups modulo the intersection of all neighbourhoods of zero is described.
The Milnor Number of Plane Branches With Tame Semigroup of Values
本文研究任意特征代数闭域上平面分支的Milnor数，证明当值半群为驯顺半群（域特征不整除其极小生成元）时，Milnor数仍与半群导子相等，弥补了正特征域下原结论不成立的缺口。
Milnor-Witt K-theory and Witt K-theory of a field
本文回顾了域的Milnor-Witt K-理论的基本计算，并利用Kato对Milnor二次型猜想的解答，给出了特征为2的域的Witt K-理论同构于对称双线性形式Witt环基本理想的Rees代数的初等证明。
Tropical analogs of Milnor $K$-groups and tropicalizations of Zariski-Riemann spaces
本文引入有理 Milnor $K$-群的热带类似物并证明其 Zariski 层化具有 Gersten 分解，为后续建立平凡赋值域上光滑代数簇的热带 Hodge 猜想奠定基础。
An exponential homomorphism for a complete discrete valuation field of characteristic zero which relates differential forms and the Milnor K-groups of the field is studied. An application to explicit formulas is included.
This work contains a list of all known results on the quotient filtration on the Milnor K-groups of a complete discrete valuation field in terms of differential modules over the residue field . Author's recent study of the case of a tamely ramified field of characteristic 0 with residue field of characteristic p by using an exponential map and a syntomic complex is explained.
Geometric presentations of Milnor $K$-groups of certain Artin algebras and Bass-Tate-Kato norms
本文通过引入两个子复形及其等价关系，提供了Artin局部代数Milon K-群的几何表示，并推广了Bass-Tate和Kato范数的理论。
We show that operations in Milnor K-theory mod $p$ of a field are spanned by divided power operations. After giving an explicit formula for divided power operations and extending them to some new cases, we determine for all fields $k$ and all prime numbers $p$, all the operations $K^M_i/p \to K^M_j/p$ commuting with field extensions over the base field $k$. Moreover, the integral case is discussed and we determine the operations $K^M_i/p \to K^M_j/p$ for smooth schemes over a field.
For an arbitrary field p-torsion and cotorsion of the Milnor groups K_n(F) and K_n^{t}(F)=K_n(F)/\cap_{l\ge1} lK_n(F) are discussed. The work contains further discussions of an analogue of Satz 90 for K_n(F) and K_n^{t}(F) and computation of H^{n+1}_m(F) where F is either the rational function field in one variable F=k(t) or the formal power series F=k((t)).
In this paper, semilocal Milnor $K$-theory of fields is introduced and studied. A strongly convergent spectral sequence relating semilocal Milnor $K$-theory to semilocal motivic cohomology is constructed. In weight 2, the motivic cohomology groups $H^p_{Zar}(k,\mathbb Z(2))$, $p\leq 1$, are computed as semilocal Milnor $K$-theory groups $\widehat{K}^M_{2,3-p}(k)$. The following applications are given: (i) several criteria for the Beilinson-Soulé Vanishing Conjecture; (ii) computation of $K_4$ of a field; (iii) the Beilinson conjecture for rational $K$-theory of fields of prime characteristic i…
We construct a four-term exact sequence which provides information on the kernel and cokernel of the multiplication by a pure symbol in Milnor's K-theory mod 2 of fields of characteristic zero. As an application we establish, for fields of characteristics zero, the validity of three conjectures in the theory of quadratic forms - the Milnor conjecture on the structure of the Witt ring, the Khan-Rost-Sujatha conjecture and the J-filtration conjecture. The first version of this paper was written in the spring of 1996.
We present a formal scheme based cycle model for the motivic cohomology of the fat points defined by the truncated polynomial rings $k[t]/(t^m)$ with $m \geq 2$, in one variable over a field $k$. We compute their Milnor range cycle class groups when the field has sufficiently many elements. With some aids from rigid analytic geometry and the Gersten conjecture for the Milnor $K$-theory resolved by M. Kerz, we prove that the resulting cycle class groups are isomorphic to the Milnor $K$-groups of the truncated polynomial rings, generalizing a theorem of Nesterenko-Suslin and Totaro.
Let $F$ be a finitely generated regular field extension of transcendence degree $\geq 2$ over a perfect field $k$. We show that the multiplicative group $F^\times/k^\times$ endowed with the equivalence relation induced by algebraic dependence on $k$ determines the isomorphism class of $F$ in a functorial way. As a special case of this result, we obtain that the isomorphism class of the graded Milnor $K$-ring $K^M_*(F)$ determines the isomorphism class of $F$, when $k$ is algebraically closed or finite.
The Milnor conjecture has been a driving force in the theory of quadratic forms over fields, guiding the development of the theory of cohomological invariants, ushering in the theory of motivic cohomology, and touching on questions ranging from sums of squares to the structure of absolute Galois groups. Here, we survey some recent work on generalizations of the Milnor conjecture to the context of schemes (mostly smooth varieties over fields of characteristic not 2). Surprisingly, a version of the Milnor conjecture fails to hold for certain smooth complete p-adic curves with no rational theta c…
In this work, we present a generalization to varieties and sheaves of the fundamental ideal of the Witt ring of a field by defining a sheaf of fundamental ideals $\tilde{I}$ and a sheaf of Witt rings $\tilde{W}$ in the obvious way. The Milnor conjecture then relates the associated graded of $\tilde{W}$ to Milnor K-theory and so allows the classical invariants of a bilinear space over a field to be extended to our setting using étale cohomology. As an application of these results, we calculate the Witt ring of a smooth curve with good reduction over a non-dyadic local field.
Classically, regular homomorphisms have been defined as a replacement for Abel--Jacobi maps for smooth varieties over an algebraically closed field. In this work, we interpret regular homomorphisms as morphisms from the functor of families of algebraically trivial cycles to abelian varieties and thereby define regular homomorphisms in the relative setting, e.g., families of schemes parameterized by a smooth variety over a given field. In that general setting, we establish the existence of an initial regular homomorphism, going by the name of algebraic representative, for codimension-2 cycles o…
We construct nontrivial homomorphisms from the quasi group of some cubic surfaces over $\bbF_{\!p}$ into a group. We show experimentally that the homomorphisms constructed are the only possible ones and that there are no nontrivial homomorphisms in the other cases. Thereby, we follow the classification of cubic surfaces, due to A. Cayley.
The total spine of the Milnor fibration of a plane curve singularity
本文提出了一种方法，通过构造脊柱来实现平面曲线奇点的Milnor纤维的消失拓扑，并在商空间上建立了等价的光滑纤维结构。
Remarks on Milnor K-theory and Tate's conjecture for divisors
本文建立了有限域上除子 Tate 猜想与代数闭包上三变量函数域第三 Milnor K-群的一个明确代数问题之间的等价关系。
We introduce a new algebraic-cycle model for the motivic cohomology theory of truncated polynomials $k[t]/(t^m)$ in one variable. This approach uses ideas from the deformation theory and non-archimedean analysis, and is distinct from the approaches via cycles with modulus. We compute the groups in the Milnor range when the base field is of characteristic $0$, and prove that they give the Milnor $K$-groups of $k[t]/(t^m)$, whose relative part is the sum of the absolute Kähler differential forms.
A lower bound for the Milnor number of vector fields
本文研究了包含局部完全交光滑正维成分的全纯向量场，推导了其Milnor数的下界，并通过显式族展示最优性及奇点的重新分布机制。
On Goncharov's conjecture in next to Milnor degree
本文证明特征零域的 motivic 上同调在特定度与权下与多对数复形上同调有理同构，部分推广了 Suslin 定理。
Central simple algebras, Milnor $K$-theory and homogeneous spaces over complete discretely valued fields of dimension 2
本文研究维数为2的完备离散赋值域，证明中心单代数周期等于指数、Milnor K-理论模p类可由符号表示，并建立Serre猜想II。
Local to global principles for homomorphisms of abelian schemes
本文研究了定义在代数簇函数域上的阿贝尔簇同态存在性问题，利用限制映射到子簇建立了局部到全局的判定准则，并借助Hilbertianity方法和Tate猜想等工具证明了包括同源在内的各类同态的存在性。
Let $G$ be a commutative affine algebraic group over a field $F$, and let $H \colon \mathrm{Fields}_{F} \to \mathrm{AbGrps}$ be a functor. A (homomorphic) $H$-invariant of $G$ is a natural transformation $\mathrm{Tors}(-, G) \to H$, where $\mathrm{Tors}(-, G)$ is the functor $\mathrm{Fields}_{F} \to \mathrm{AbGrps}$ taking a field extension $L/F$ to the group of isomorphism classes of $G_{L}$-torsors over $\mathrm{Spec}(L)$. The goal of this paper is to compute the group $\mathrm{Inv}_{\mathrm{hom}}^{1}(G, H)$ of $H$-invariants of $G$ when $G$ is a group of multiplicative type, and $H$ is the …
We study analogues of Tate's conjecture on homomorphisms for abelian varieties when the ground field is finitely generated over an algebraic closure of a finite field. Our results cover the case of abelian varieties without nontrivial endomorphisms.
The kernel of the Gysin homomorphism for positive characteristic
本文在正特征域上扩展了Gysin核的定理，通过相同论证和étale基底论证，并利用比较定理和Tate猜想来证明其成立。
Hermitian $K$-theory and Milnor-Witt motivic cohomology over $\mathbb Z$
本文利用Milnor-Witt母题上同调与切片谱序列计算Hermitian $K$-群，证明Grothendieck-Riemann-Roch定理，并给出整数环的Hermitian $K$-群与高阶Witt群。
Given two algebraic groups $G$, $H$ over a field $k$, we investigate the representability of the functor of morphisms (of schemes) $\mathbf{Hom}(G,H)$ and the subfunctor of homomorphisms (of algebraic groups) $\mathbf{Hom}_{\rm gp}(G,H)$. We show that $\mathbf{Hom}(G,H)$ is represented by a group scheme, locally of finite type, if the $k$-vector space $\mathcal{O}(G)$ is finite-dimensional; the converse holds if $H$ is not étale. When $G$ is linearly reductive and $H$ is smooth, we show that $\mathbf{Hom}_{\rm gp}(G,H)$ is represented by a smooth scheme $M$; moreover, every orbit of $H$ acting…
On Kato and Kuzumaki's properties for the Milnor $K_2$ of function fields of $p$-adic curves
本文研究了 p 进域上曲线函数域的 Milnor K_2 群，证明了在超曲面次数满足 d^2 ≤ n 时该群可由具有有理点的有限扩张的范数像张成，并在曲线具有最大非分歧扩张点的情况下将结果推广到 d ≤ n。
Geometric properties of homomorphisms between the absolute Galois groups of mixed-characteristic complete discrete valuation fields with perfect residue fields
本文将 Mochizuki 与 Hoshi 关于 $p$-进局部域绝对伽罗瓦群几何同态的判别条件推广到具有完美剩余域的一般混合特征完备离散赋值域，并得到弱 Isom 远阿贝尔结果。
These are notes of my lectures at the summer school "Higher-dimensional geometry over finite fields" in Goettingen, June--July 2007. We present a proof of Tate's theorem on homomorphisms of abelian varieties over finite fields (including the $\ell=p$ case) that is based on a "quaternion trick". In fact, a a slightly stronger version of those theorems with "finite coefficients" is proven.
In this paper we study abstract group homomorphisms between the groups of rational points of linear algebraic groups which are not necessarily reductive. One of our main goal is to obtain results on homomorphisms from the groups of rational points of linear algebraic groups defined over certain specific fields to the groups of rational points of linear algebraic groups over number fields and non-archimedean local fields of characteristic zero; in this set-up we deal with the unexplored topic of abstract homomorphisms from the groups of rational points of anisotropic groups over non-archimedean…
Homological Milnor-Witt modules and Chow-Witt groups over general bases
本文提出了一种一般基上的同调Milnor-Witt模理论，扩展了Chow-Witt群，并证明了广义Bloch公式和有限性结果。
Notes on Milnor-Witt K-theory
本文为任意特征域建立 Milnor-Witt K-理论基础，利用 Grothendieck 迹映射处理不可分转移，完善其函子性公理。
The big de Rham-Witt forms over fields and motives of non-reduced schemes
本文通过代数循环证明了大德拉姆-维它形式群与Artin局部代数的相对改进米尔诺K群同构，并解释了其在非减方案动机上同调中的应用。
Let $k$ be a totally real field, and let $A/k$ be an absolutely irreducible, polarized Abelian variety of odd, prime dimension whose endomorphisms are all defined over $k$. Then the only strictly compatible families of abstract, absolutely irreducible representations of $\gal(\ov{k}/k)$ coming from $A$ are tensor products of Tate twists of symmetric powers of two-dimensional $λ$-adic representations plus field automorphisms. The main ingredients of the proofs are the work of Borel and Tits on the `abstract' homomorphisms of almost simple algebraic groups, plus the work of Shimura on the fields…
On Milnor $K$-theory in the imperfect residue case and applications to period-index problems
本文通过定义伪完美扩张，证明其可分裂混合特征完备离散赋值域的高阶上同调类，从而给出Brauer p-维数上界，并改进半全局域上的周期-指数界。
Let $E:y^2=x^3+Ax^2+Bx+C$ be a nonconstant elliptic curve over $\mathbb{Q}(t)$ with at least one nontrivial $\mathbb{Q}(t)$-rational $2$-torsion point. We describe a method for finding $t_0\in\mathbb Q$ for which the corresponding specialization homomorphism $t\mapsto t_0\in\mathbb{Q}$ is injective. The method can be directly extended to elliptic curves over $K(t)$ for a number field $K$ of class number $1$, and in principal for arbitrary number field $K$. One can use this method to calculate the rank of elliptic curves over $\mathbb Q(t)$ of the form as above, and to prove that given points a…
We present local classification results for isolated singularities of functions with respect to a Nambu structure (multi-vector field) of maximal degree, in a neighbourhood of a smooth point of its degeneracy hypersurface. The results depend on a logarithmic version of the Brieskorn-Sebastiani theorem, which guarantees the finiteness and freeness of the corresponding deformation module. This relates the functional moduli of the classification problem with the integrals of logarithmic forms along the vanishing cycles of the complement of the Milnor fibers of the restriction of the function on t…
Let G be a semisimple group over an algebraically closed field of very good characteristic for G. In the context of geometric invariant theory, G. Kempf has associated optimal cocharacters of G to an unstable vector in a linear G-representation. If the nilpotent element X in Lie(G) lies in the image of the differential of a homomorphism SL(2) --> G, we say that homomorphism is optimal for X, or simply optimal, provided that its restriction to a suitable torus of SL(2) is optimal for X in Kempf's sense. We show here that any two SL(2)-homomorphisms which are optimal for X are conjugate under th…
Let K be any field, and let G be a semisimple group over K. Suppose the characteristic of K is positive and is very good for G. We describe all group scheme homomorphisms phi:SL(2) --> G whose image is geometrically G-completely reducible -- or G-cr -- in the sense of Serre; the description resembles that of irreducible modules given by Steinberg's tensor product theorem. In case K is algebraically closed and G is simple, the result proved here was previously obtained by Liebeck and Seitz using different methods. A recent result shows the Lie algebra of the image of phi to be geometrically G-c…
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