Motivic equivalence of affine quadrics

Smooth models of motivic spheres

We study the representability of motivic spheres by smooth varieties. We show that certain explicit "split" quadric hypersurfaces have the $\mathbb A^1$-homotopy type of motivic spheres over the integers and that the $\mathbb A^1$-homotopy types of other motivic spheres do not contain smooth schemes as representatives. We then study some applications of these representability/non-representability results to the construction of new exotic $\mathbb A^1$-contractible smooth schemes. Then, we study vector bundles on even dimensional "split" quadric hypersurfaces by developing an algebro-geometric …

On the Invertibility of Motives of Affine Quadrics

We show that the reduced motive of a smooth affine quadric is invertible as an object of the triangulated category of motives DM(k, ZZ[1/e]) (where k is a perfect field of exponential characteristic e). We also establish a motivic version of the conjectures of Po Hu on products of certain affine Pfister quadrics. Both of these results are obtained by studying a novel conservative functor on (a subcategory of) DM(k, ZZ[1/e]), the construction of which constitutes the main part of this work.

双曲面的柔韧性

Flexibility of affine cones over a smooth complete intersection of two quadrics

本文研究了射影二次曲面和两个二次曲面交点的仿射锥的柔韧性，证明了其在特定条件下的几何性质。

On the Brauer group of affine diagonal quadrics

In a previous work, we introduced the notion of uniform generators of the Brauer group and proved that general diagonal cubic surfaces do not have such generators. In this paper, we prove that a similar non-existence result holds for affine diagonal quadrics.

Algebraic vector bundles on punctured affine spaces and smooth quadrics

We construct examples of non-isomorphic algebraic vector bundles on the punctured affine space with isomorphic pullbacks to the smooth quadric.

The tame and the wild automorphisms of an affine quadric threefold

We prove the existence of wild automorphisms on an affine quadric threefold. The method we use is an adaptation of the one used by Shestakov and Umirbaev to prove the existence of wild automorphisms on the affine three dimensional space.

On compactifications of affine homology 3-cells into quadric fibrations

In this paper we deal with compactifications of affine homology $3$-cells into quadric fibrations such that the boundary divisors contain fibers. We show that all such affine homology $3$-cells are isomorphic to the affine $3$-space $\mathbb{A}^{3}$. Moreover, we show that all such compactifications can be connected by explicit elementary links preserving $\mathbb{A}^{3}$ to the projective $3$-space $\mathbb{P}^{3}$.

动机同伦理论中仿射直线的保真性

Loyalty of the affine line in motivic homotopy theory

该研究在动机同伦理论中证明整数在导出概型S上可逆作用的等价条件，推导得出特定概型上动机球谱的A¹-不变性结论。

Integral points on affine quadric surfaces

It is well-known that the Hasse principle holds for quadric hypersurfaces. The Hasse principle fails for integral points on smooth quadric hypersurfaces of dimension 2 but the failure can be completely explained by the Brauer-Manin obstruction. We investigate how often the family of quadric hypersurfaces $ax^2 + by^2 +cz^2 = n$ has a Brauer-Manin obstruction. We improve previous bounds of Mitankin.

Embeddings of Affine Spaces into Quadrics

This article provides, over any field, infinitely many algebraic embeddings of the affine spaces $\mathbb{A}^1$ and $\mathbb{A}^2$ into smooth quadrics of dimension two and three respectively, which are pairwise non-equivalent under automorphisms of the smooth quadric. Our main tools are the study of the birational morphism $\mathrm{SL}_2 \to \mathbb{A}^3$ and the fibration $\mathrm{SL}_2 \to \mathbb{A}^3 \to \mathbb{A}^1$ obtained by projections, as well as degenerations of variables of polynomial rings, and families of $\mathbb{A}^1$-fibrations.

Arithmetic purity of the Hardy-Littlewood property and geometric sieve for affine quadrics

We establish the Hardy-Littlewood property (à la Borovoi-Rudnick) for Zariski open subsets in affine quadrics of the form $q(x_1,\cdots,x_n)=m$, where $q$ is a non-degenerate integral quadratic form in $n\geqslant 3$ variables and $m$ is a non-zero integer. This gives asymptotic formulas for the density of integral points taking coprime polynomial values, which is a quantitative version of the arithmetic purity of strong approximation property off infinity for affine quadrics.

The automorphism group of an affine quadric

We determine the automorphism group for a large class of affine quadric hypersurfaces over a field, viewed as affine algebraic varieties. In particular, we find that the group of real polynomial automorphisms of the n-sphere is just the orthogonal group O(n+1) whenever n is a power of 2. It is not known whether the same is true for arbitrary n. The proof uses Karpenko's theorem that certain projective quadrics over a field are not ruled. That is, they are not birational over the given field to the product of any variety with the projective line. We also formulate a general result on automorphi…

Grothendieck ring of varieties, D- and L-equivalence, and families of quadrics

We discuss a conjecture saying that derived equivalence of simply connected smooth projective varieties implies that the difference of their classes in the Grothendieck ring of varieties is annihilated by a power of the affine line class. We support the conjecture with a number of known examples, and one new example. We consider a smooth complete intersection $X$ of three quadrics in ${\mathbf P}^5$ and the corresponding double cover $Y \to {\mathbf P}^2$ branched over a sextic curve. We show that as soon as the natural Brauer class on $Y$ vanishes, so that $X$ and $Y$ are derived equivalent, …

Projective homogeneous varieties birational to quadrics

We will consider an explicit birational map between a quadric and the projective variety X(J) of traceless rank one elements in a simple reduced Jordan algebra J. X(J) is a homogeneous G-variety for the automorphism group G=Aut(J). We will show that the birational map is a blow up followed by a blow down. This will allow us to use the blow up formula for motives together with Vishik's work on the motives of quadrics to give a motivic decomposition of X(J).

A primer for unstable motivic homotopy theory

In this expository article, we give the foundations, basic facts, and first examples of unstable motivic homotopy theory with a view towards the approach of Asok-Fasel to the classification of vector bundles on smooth complex affine varieties. Our focus is on making these techniques more accessible to algebraic geometers.

The tame automorphism group of an affine quadric threefold acting on a square complex

We study the group Tame(SL$_2$) of tame automorphisms of a smooth affine 3-dimensional quadric, which we can view as the underlying variety of SL(2,$\mathbb{C}$). We construct a square complex on which the group admits a natural cocompact action, and we prove that the complex is CAT(0) and hyperbolic. We propose two applications of this construction: We show that any finite subgroup in Tame(SL$_2$) is linearizable, and that Tame(SL$_2$) satisfies the Tits alternative.

Critical varieties and motivic equivalence for algebras with involution

Motivic equivalence for algebraic groups was recently introduced in [9], where a characterization of motivic equivalent groups in terms of higher Tits indexes is given. As a consequence, if the quadrics associated to two quadratic forms have the same Chow motives with coefficients in F_2, this remains true for any two projective homogeneous varieties of the same type under the orthogonal groups of those two quadratic forms. Our main result extends this to all groups of classical type, and to some exceptional groups, introducing a notion of critical variety. On the way, we prove that motivic eq…

From exceptional collections to motivic decompositions via noncommutative motives

Making use of noncommutative motives we relate exceptional collections (and more generally semi-orthogonal decompositions) to motivic decompositions. On one hand we prove that the Chow motive M(X) of every smooth proper Deligne-Mumford stack X, whose bounded derived category D(X) of coherent schemes admits a full exceptional collection, decomposes into a direct sum of tensor powers of the Lefschetz motive. Examples include projective spaces, quadrics, toric varieties, homogeneous spaces, Fano threefolds, and moduli spaces. On the other hand we prove that if M(X) decomposes into a direct sum of…

Motivic homotopy of group scheme actions

To smooth schemes equipped with a smooth affine group scheme action, we associate an equivariant motivic homotopy category. Underlying our construction is the choice of an `equivariant Nisnevich topology' induced by a complete, regular, and bounded $cd$-structure. We show equivariant $K$-theory of smooth schemes is represented in the equivariant motivic homotopy category. This is used to characterize equivariantly contractible smooth affine curves and equivariant vector bundles on such curves. Generalizations of the purity and blow-up theorems in motivic homotopy theory are shown for actions o…

Framed motives of smooth affine pairs

The theory of framed motives by Garkusha and Panin gives computations in the stable motivic homotopy category $\mathbf{SH}(k)$ in terms of Voevodsky's framed correspondences. In particular the motivically fibrant $Ω$-resolution in positive degrees of the motivic suspension spectrum $Σ_{\mathbb P^1}^\infty X_+$, where $X_+=X\amalg *$, for a smooth scheme $X\in \mathrm{Sm}_k$ over an infinite perfect field $k$, is computed. The computation by Garkusha, Neshitov and Panin of the framed motives of relative motivic spheres $(\mathbb A^l\times X,(\mathbb A^l-0)\times X)$, $X\in \mathrm{Sm}_k$, is on…

Motivic decompositions of families with Tate fibers: smooth and singular cases

We apply Wildeshaus's theory of motivic intermediate extensions to the motivic decomposition conjecture, formulated by Deninger-Murre and Corti-Hanamura. We first obtain a general motivic decomposition for the Chow motive of an arbitrary smooth projective family $f:X \rightarrow S$ whose geometric fibers are Tate. Using Voevodsky's motives with rational coefficients, the formula is valid for an arbitrary regular base $S$, without assuming the existence of a base field or even of a prime integer $\ell$ invertible on $S$. This result, and some of Bondarko' ideas, lead us to a generalized formula…

二次曲面的$D$-affinity性质再探讨

$D$-affinity of Quadrics Revisited

本文重新探讨了二次曲面的$D$-affinity性质，证明在特定条件下二次超曲面和格拉斯曼流形不是$D$-affine，并补充了之前关于$D$-affine条件的研究。

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。

Algebraic cycles and motivic generic iterated integrals

Following the work of Gangl, Goncharov and Levin in [GGL], we will give a combinatorial framework for motivic study of iterated integrals on the affine line. We will show that under a certain genericity condition these combinatorial objects yield to elements in the motivic Hopf algebra constructed in Bloch-Kriz [BK]. It will be shown that the Hodge realization of these elements coincides with the Hodge structure induced from the fundamental torsor of path of punctured affine line.

仿射A型弓形簇的不动点计数与动机不变量

Fixed point counts and motivic invariants of bow varieties of affine type A

本文通过计算仿射A型Cherkis弓形簇环面不动点的等变K理论，给出欧拉数与动机层级生成级数，推广Nakajima–Yoshioka结果，并联系抛物型与经典Nekrasov配分函数。

Affine Grassmannians in A^1-algebraic topology

Let k be a field. Denote by Spc(k)_* the unstable, pointed motivic homotopy category and by Omega_Gm: Spc(k)_* \to Spc(k)_* the Gm-loops functor. For a k-group G, denote by Gr_G the affine Grassmannian of G. If G is isotropic reductive, we provide a canonical motivic equivalence Omega_Gm G = Gr_G. If k is perfect, we use this to compute the motive M(Omega_Gm G) in DM(k, Z).

Conservativity of realizations implies that numerical motives are Kimura-finite and motivic zeta functions are rational

We prove: if the (étale or de Rham) realization functor is conservative on the category $DM_{gm}$ of Voevodsky motives with rational coefficients then motivic zeta functions of arbitrary varieties are rational and numerical motives are Kimura-finite. The latter statement immediately implies that the category of numerical motives is (essentially) Tannakian. This observation becomes actual due to the recent announcement of J. Ayoub that the De Rham cohomology realization is conservative on $DM_{gm}(k)$ whenever $\operatorname{char} k=0$. We apply this statement to exterior powers of motives comi…

仿射二次三维簇的驯顺自同构的次数增长

Degree growth for tame automorphisms of an affine quadric threefold

本文利用赋值估计和方复形上的群作用，证明了保持仿射二次三维簇的驯顺自同构的动力次数避开 1 和 4/3 之间的值，并刻画了有限多个此类自同构随机乘积的次数增长指数为正的条件。

Symmetries of some motivic integrals

We give an explicit formula for the motivic integrals related to the Milnor number over spaces of parametrised arcs on the plane with fixed tangency orders with the axis. These integrals are rational functions of the parameters and the class of the affine line. Using a set of natural recurrence relations between them, we prove an unexpected invariance property with respect to the simultaneous inversion of the parameters and the class of the affine line. We also discuss a generalization of this system of recurrence relations whose solutions are also symmetric and satisfy additional differential…

光滑仿射二次三维簇的一些有趣双有理态射

Some interesting birational morphisms of smooth affine quadric $3$-folds

本文研究复光滑仿射二次三维簇上一族多项式诱导的双有理映射，揭示其上同调双曲性、非代数整数的第二动力学度及周期点增长与代数熵的差异，并结合 Bertini 实验验证。

On infinite effectivity of motivic spectra and the vanishing of their motives

This paper is dedicated to the study of the kernel of the "compact motivization" functor $M_{k}^c:SH^c(k)\to DM^c(k)$ (i.e., we try to describe those compact objects of $SH(k)$ whose associated motives vanish. Moreover, we study the question when the $m$-connectivity of $M^c_{k}(E)$ ensures the $m$-connectivity of $E$ itself (with respect to the corresponding homotopy t-structures). We prove that the kernel of $M_{k}^c$ vanishes and the corresponding "connectivity detection" statement is also valid if and only if $k$ is a non-orderable field; this is an easy consequence of the corresponding re…

$\mathbb{A}^1$-连通分量的仿射二次曲面

$\mathbb A^1$-connected components of affine quadrics

本文研究了光滑二次超曲面的$\mathbb{A}^1$-连通分量，通过迭代朴素$\mathbb{A}^1$-连通分量函子，证明了其在n=2时稳定，并给出了完整特征描述。

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。

Motivic degree zero Donaldson-Thomas invariants

Given a smooth complex threefold X, we define the virtual motive of the Hilbert scheme of n points on X. In the case when X is Calabi-Yau, this gives a motivic refinement of the n-point degree zero Donaldson-Thomas invariant of X. The key example is affine three-space, where the Hilbert scheme can be expressed as the critical locus of a regular function on a smooth variety, and its virtual motive is defined in terms of the Denef-Loeser motivic nearby fiber. A crucial technical result asserts that if a function is equivariant with respect to a suitable torus action, its motivic nearby fiber is …

Motivic Galois theory for motives of niveau $\leq$ 1

Let T be a Tannakian category over a field k of characteristic 0 and π(T) its fundamental group. In this paper we prove that there is a bijection between the otimes-equivalence classes of Tannakian subcategories of T and the normal affine group sub-T-schemes of π(T). We apply this result to the Tannakian category T_1(k) generated by motives of niveau \leq 1 defined over k, whose fundamental group is called the motivic Galois group G_mot(T_1(k)) of motives of niveau \leq 1. We find four short exact sequences of affine group sub-T_1(k)-schemes of G_mot(T_1(k)), correlated one to each other by in…

Tate motives on Witt vector affine flag varieties

Relying on recent advances in the theory of motives we develope a general formalism for derived categories of motives with Q-coefficients on perfect (ind-)schemes. As an application we give a motivic refinement of Zhu's geometric Satake equivalence for Witt vector affine Grassmannians in this set-up.

两个二次曲面交集的强逼近

Strong approximation for the intersection of two quadrics

本文研究两个仿射二次曲面交集的强逼近问题，并将其应用于证明秩四数域上非分裂纤维由二次扩张分裂情形下的弱逼近纤维化方法。

Motivic Poincaré series, toric singularities and logarithmic jacobian ideals

The geometric motivic Poincaré series of a variety, which was introduced by Denef and Loeser, takes into account the classes in the Grothendieck ring of the sequence of jets of arcs in the variety. Denef and Loeser proved that this series has a rational form. We describe it in the case of an affine toric variety of arbitrary dimension. The result, which provides an explicit set of candidate poles, is expressed in terms of the sequence of Newton polyhedra of certain monomial ideals, which we call logarithmic jacobian ideals, associated to the modules of differential forms with logarithmic poles…

On motivic obstructions to Witt cancellation for quadratic forms over schemes

The paper provides computations of the first non-vanishing $\mathbb{A}^1$-homotopy sheaves of the orthogonal Stiefel varieties which are relevant for the unstable isometry classification of quadratic forms over smooth affine schemes over perfect fields of characteristic $\neq 2$. Together with the $\mathbb{A}^1$-representability for quadratic forms, this provides the first obstructions for rationally trivial quadratic forms to split off a hyperbolic plane. For even-rank quadratic forms, this first obstruction is a refinement of the Euler class of Edidin and Graham. A couple of consequences are…

相对 $\mathbb{A}^1$-收缩性与 exotic 动机球体

Relative $\mathbb{A}^1$-Contractibility of Smooth Schemes and Exotic Motivic Spheres

本文研究了光滑动机球体的构造，证明了在无限完美域上，$\mathcal{X}_n \backslash \{\bullet\}$ 是exotic动机球体的模型。

作为无-$(d+2)$-在-$Q$-二次曲面上点集的有理正规曲线

Rational normal curves as no-$(d+2)$-on-$Q$-quadric sets

本文构造了大小为 n-o(n) 的 d 维点集，使任意仿射超平面至多交 d 点、任意超球面至多交 d+1 点，并将结论推广到 Q-二次曲面及有限域情形，核心工具是射影空间中的有理正规曲线。

Schön仿射簇中超曲面的稳定有理性

Stable rationality of hypersurfaces in schön affine varieties

本文通过动机方法，将[Nicaise--Ottem'22]的结果从代数环面扩展到schön仿射簇中的超曲面，还应用该方法证明了复Grassmann簇Gr(2,n)中某些超曲面的无理性，与先前研究结果一致。

Rost motives, affine varieties, and classifying spaces

In the present article we investigate ordinary and equivariant Rost motives. We provide an equivariant motivic decomposition of the variety X of full flags of a split semisimple algebraic group over a smooth base scheme, study torsion subgroup of the Chow group of twisted forms of X, define some equivariant Rost motives over a field and ordinary Rost motives over a general base scheme, and relate equivariant Rost motives with classifying spaces of some algebraic groups.

Cohomological invariants in positive characteristic

We determine the mod p cohomological invariants for several affine group schemes G in chararacteristic p. These are invariants of G-torsors with values in etale motivic cohomology, or equivalently in Kato's version of Galois cohomology based on differential forms. In particular, we find the mod 2 cohomological invariants for the symmetric groups and the orthogonal groups in characteristic 2, which Serre computed in characteristic not 2. We also determine all operations on the mod p etale motivic cohomology of fields, extending Vial's computations of the operations on the mod p Milnor K-theory …

Donaldson-Thomas invariants for 3-Calabi-Yau varieties of dihedral quotient type

We compute motivic Donaldson-Thomas invariants for crepant resolutions of quotients of affine three-space by even dihedral groups in terms of an affine type D root system, using double dimensional reduction and the representation theory of affine type D quivers.

Equivalence motivique des groupes algebriques semisimples

Two semisimple algebraic groups are said to be motivic equivalent if the motives of the associated twisted flag varieties are isomorphic modulo any prime p. The purpose of this note is to construct the combinatorial invariants which characterize motivic equivalence and which are the motivic analogues of the Tits indices which appear in the classification of semisimple algebraic groups. The expression of these invariants -the Tits p-indexes- in terms of the classical invariants associated to the natural underlying structures of semisimple algebraic groups allow to produce algebraic criteria of …

旗簇的T-等变母题

$T$-equivariant motives of flag varieties

本文利用Khan-Ravi稳定同伦范畴构造，将旗簇的T-等变K-理论环谱分解为以Weyl群为指标的BT的K-理论谱直和，并由此推导出T-等变相交理论与K-理论的经典结果。

On the Motive of a Fibre Bundle and its Applications

In this article we compute the motive associated to a cellular fibration $Γ$ over a smooth scheme $X$ inside Veovodsky's motivic categories. We implement this result to study the motive associated to a $G$-bundle, and additionally to study motives of varieties admitting a resolution of singularities by a tower of cellular fibrations (e.g. affine Schubert varieties in a twisted affine flag variety).

指数动机在仿射格拉斯曼流形上的研究

Exponential motives on the affine Grassmannian

本文提出指数动机的概念，并通过Gaitsgory的Kirillov模型与Whittaker动机比较，建立仿射旗形猜想中的Tate动机和t-结构基础结果，证明了指数Tate动机与分类堆的诱导凝聚sheaves等价。

Motives and periods in Bianchi IX gravity models

We show that, when considering the anisotropic scaling factors and their derivatives as affine variables, the coefficients of the heat kernel expansion of the Dirac-Laplacian on $SU(2)$ Bianchi IX metrics are algebro-geometric periods of motives of complements in affine spaces of unions of quadrics and hyperplanes. We show that the motives are mixed Tate and we provide an explicit computation of their Grothendieck classes.

环面作用、Morse 同调与仿射空间上点的 Hilbert 概形

Torus actions, Morse homology, and the Hilbert scheme of points on affine space

本文在 motivic 同伦论中提出乘法群作用的猜想，并利用奇异簇的 Morse 理论证明部分结果，进而得出仿射空间上点的 Hilbert 概形与原点处支持的子空间同伦等价。

Periods and motives in the spectral action of Robertson-Walker spacetimes

We show that, when considering the scaling factor as an affine variable, the coefficients of the asymptotic expansion of the spectral action on a (Euclidean) Robertson-Walker spacetime are periods of mixed Tate motives, involving relative motives of complements of unions of hyperplanes and quadric hypersurfaces and divisors given by unions of coordinate hyperplanes.

拟线性二次型的扩展Karpenko定理与Karpenko-Merkurjev定理

Extended Karpenko and Karpenko-Merkurjev theorems for quasilinear quadratic forms

本文针对特征2域上的拟线性二次型，引入新不变量Δ(p)，结合Izhboldin维数对函数域扩张后的迷向指数建立强约束，从而统一并推广了Karpenko与Karpenko-Merkurjev定理的拟线性版本，证明了作者此前猜想的一个强形式。