Norm functors and effective zero cycles

The derived functors of unramified cohomology

We study the first "derived functors of unramified cohomology" in the sense of arXiv:1506.08385 [math.AG], applied to the sheaves $\mathbf{G}_m$ and $\mathcal{K}_2$. We find interesting connections with classical cycle-theoretic invariants of smooth projective varieties, involving notably a version of the Griffiths group, and the indecomposable $(2,1)$-cycles.

Duality and nearby cycles over general bases

This paper studies the sliced nearby cycle functor and its commutation with duality. Over a Henselian discrete valuation ring, we show that this commutation holds, confirming a prediction of Deligne. As an application we give a new proof of Beilinson's theorem that the vanishing cycle functor commutes with duality up to twist. Over an excellent base scheme, we show that the sliced nearby cycle functor commutes with duality up to modification of the base. We deduce that duality preserves universal local acyclicity over an excellent regular base. We also present Gabber's theorem that local acycl…

Weil限制与动机循环类映射

Weil restriction and the motivic cycle class map

本文研究了Weil限制映射与动机循环类映射的兼容性，并通过动机三角范畴提供了概念框架。

Nearby cycles on Drinfeld-Gaitsgory-Vinberg Interpolation Grassmannian and long intertwining functor

Let $G$ be a reductive group and $U,U^-$ be the unipotent radicals of a pair of opposite parabolic subgroups $P,P^-$. We prove that the DG-categories of $U(\!(t)\!)$-equivariant and $U^-(\!(t)\!)$-equivariant D-modules on the affine Grassmannian $Gr_G$ are canonically dual to each other. We show that the unit object witnessing this duality is given by nearby cycles on the Drinfeld-Gaitsgory-Vinberg interpolation Grassmannian defined in arXiv:1805.07721. We study various properties of the mentioned nearby cycles, in particular compare them with the nearby cycles studied in arXiv:1411.4206 and a…

Hamiltonian reduction and nearby cycles for Mirabolic D-modules

We study holonomic D-modules on SL_n(C)xC^n, called mirabolic modules, analogous to Lusztig's character sheaves. We describe the supports of simple mirabolic modules. We show that a mirabolic module is killed by the functor of Hamiltonian reduction from the category of mirabolic modules to the category of representations of the trigonometric Cherednik algebra if and only if the characteristic variety of the module is contained in the unstable locus. We introduce an analogue of the Verdier specialization functor for representations of Cherednik algebras which agrees, on category O, with the res…

Vanishing cycles and Hermitian duality

We show the compatibility between the moderate or nearby cycle functor for regular holonomic $\mathcal{D}$-modules, as defined by Beilinson, Kashiwara and Malgrange, and the Hermitian duality functor, as defined by Kashiwara.

Cycle groups for Artin stacks

We construct an algebraic homology functor for Artin stacks of finite type over a field, and we develop intersection-theoretic properties.

邻近周期层的 Hodge-Grothendieck 类与单值化不变量

Hodge-Grothendieck classes and monodromy invariants of nearby cycles sheaves

研究邻近周期层的 Hodge-Grothendieck 类与单值化不变量，评述邻近周期函子并证明广义局部不变周期结果。

Characteristic cycles of standard modules for the rational Cherednik algebra of type Z/lZ

We study the representation theory of the rational Cherednik algebra $H_κ= H_κ({\mathbb Z}_l)$ for the cyclic group ${\mathbb Z}_l = {\mathbb Z} / l {\mathbb Z}$ and its connection with the geometry of the quiver variety $M_θ(δ)$ of type $A_{l-1}^{(1)}$. We consider a functor between the categories of $H_κ$-modules with different parameters, called the shift functor, and give the condition when it is an equivalence of categories. We also consider a functor from the category of $H_κ$-modules with good filtration to the category of coherent sheaves on $M_θ(δ)$. We prove that the image of the reg…

An analogue of the field-of-norms functor and the Grothendieck Conjecture

The paper contains a construction of an analogue of the Fontaine-Wintenberger field-of-norms functor for higher dimensional local fields. This construction is done completely in terms of the ramification theory of such fields. It is applied to deduce the mixed characteristic case of a local analogue of the Grothendieck Conjecture for these fields from its characteristic p case, which was proved earlier by the author.

Compatible systems of $\ell$-adic sheaves

We introduce a notion of compatibility for families $(\mathcal{F}_{\ell})_{\ell}$ of bounded constructible $\ell$-adic complexes of étale sheaves on schemes. For schemes of finite type over a field, this notion is preserved by the usual six functors. We prove that the compatibility of a family is preserved by the nearby cycles functor and by the linearized $\varepsilon$-factors introduced recently by the author. We establish independence of $\ell$ for the characteristic cycles and characteristic $\varepsilon$-cycles of compatible families.

Norms in motivic homotopy theory

If $f:S' \to S$ is a finite locally free morphism of schemes, we construct a symmetric monoidal "norm" functor $f_\otimes: \mathcal H_*(S') \to\mathcal H_*(S)$, where $\mathcal H_*(S)$ is the pointed unstable motivic homotopy category over $S$. If $f$ is finite étale, we show that it stabilizes to a functor $f_\otimes: \mathcal{SH}(S') \to \mathcal{SH}(S)$, where $\mathcal{SH}(S)$ is the $\mathbb P^1$-stable motivic homotopy category over $S$. Using these norm functors, we define the notion of a normed motivic spectrum, which is an enhancement of a motivic $E_\infty$-ring spectrum. The main co…

概形上的范函子

The Norm Functor over Schemes

本研究将Ferrand的环上范函子全局推广到常秩有限局部自由概形态射场景，构造拟凝聚模/代数栈的范态射，还借助Segre嵌入给出有限平展覆盖下范态射的上同调描述，证明其导出代数栈等价关系$A_1^2 \equiv D_2$。

赋范纤维函子

Normed Fiber Functors

该研究借助Tannakian形式体系，将一般线性群Bruhat-Tits建筑的非阿基米德范数描述推广到非分歧约化群情形，并给出对应仿射群概形的模论刻画。

The Combinatorial Norm of a Morphism of Schemes

In this paper we will prove that there exists a covariant functor from the category of schemes to the category of graphs. This functor provides a combination between algebraic varieties and combinatorial graphs so that the invariants defined on graphs can be introduced to algebraic varieties in a natural manner. By the functor, we will define the combinatorial norm of a morphism of schemes. Then we will obtain some properties of morphisms of norm not great than one. The topics discussed here can be applied to study the discrete Morse theory on arithmetic schemes and Kontsevich's theory of grap…

混合特征下Gaitsgory中心函子与Arkhipov-Bezrukavnikov等价

Gaitsgory's central functor and the Arkhipov-Bezrukavnikov equivalence in mixed characteristic

本文通过发展Iwahori水平的Wakimoto滤过理论证明抛物水平p-adic Hecke栈邻近循环函子的反常t-正合性及ℰ₁-中心提升性，构造对应Arkhipov-Bezrukavnikov函子并证明其对经典群及E₆、E₇型例外群的等价性。

Weil限制、法丛与动机Thom空间

Weil restriction, normal bundles and motivic Thom spaces

本文通过Weil限制研究法丛与动机Thom空间，证明Weil限制保持向量丛并扩展法丛结果，进而建立与动机Thom空间的相容性，并在有理系数动机上同调中证明Weil限制映射与范数映射一致且保持Thom类。

Zero cycles on projective varieties and the norm principle

Using the Gille-Merkurjev norm principle we compute in a uniform way the image of the degree map for quadrics (Springer's theorem), for twisted forms of maximal orthogonal Grassmannians (theorem of Bayer-Fluckiger and Lenstra), for E6- (Rost's theorem), and E7-varieties.

Normal functions for algebraically trivial cycles are algebraic for arithmetic reasons

For families of smooth complex projective varieties we show that normal functions arising from algebraically trivial cycle classes are algebraic, and defined over the field of definition of the family. In particular, the zero loci of those functions are algebraic and defined over such a field of definition. This proves a conjecture of Charles.

Characteristic cycles and the microlocal geometry of the Gauss map, I

We propose two new approaches to the Tannakian Galois groups of holonomic D-modules on abelian varieties. The first is an interpretation in terms of principal bundles given by the Fourier-Mukai transform, which shows that they are almost connected. The second constructs a microlocalization functor relating characteristic cycles to Weyl group orbits of weights. This explains the ubiquity of minuscule representations, and we illustrate it with a Torelli theorem and with a bound for decompositions of a given subvariety as a sum of subvarieties. The appendix sketches a twistor variant that may be …

A short proof of a formula of Brasselet, Le and Seade for the Euler obstruction

Using the index theorem of Dubson, Le, Ginsburg and Sabbah for the vanishing cycle functor, we give a short proof of formula of Brasselet, Le and Seade for the Euler obstruction.

Drinfeld 结合子与 $A_\infty$ 函子

The Drinfeld associator and $A_\infty$ functors

本文比较了射影直线去掉三点上的幺幂混合 Hodge 结构经不同扩张与消失环得到的两个混合 Hodge 结构，计算了半完整同构的 Hodge 缺陷，其级数由逆 Drinfeld 结合子给出，并给出了 $A_\infty$ 解释。

A Note on the Two Approaches to Stringy Functors for Orbifolds

In this note, we reconcile two approaches that have been used to construct stringy multiplications. The pushing forward after pulling back that has been used to give a global stringy extension of the functors K_0,K^{top},A^*,H^* [CR, FG, AGV, JKK2], and the pulling back after having pushed forward, which we have previously used in our (re)-construction program for G-Frobenius algebras, notably in considerations of singularities with symmetries and for symmetric products. A similar approach was also used by [CH] in their considerations of the Chen-Ruan product in a deRham setting for Abelian or…

Compact subvarieties with ample normal bundles, algebraicity and cones of cycles

In this note we study two features of submanifolds (subvarieties) with ample normal bundles in a compact Kähler manifold X. First, we study how algebraic X can be, i.e. we investigate the algebraic dimension. Second, we study curves with ample normal bundles in case X is projective. We prove in various cases that the class of the curve is in the interior of the Mori cone.