Michael Larsen, Valery A. Lunts
We study the motivic Grothendieck group of algebraic varieties from the point of view of stable birational geometry. In particular, we obtain a counter-example to a conjecture of M. Kapranov on the rationality of motivic zeta-function.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
We construct an upgrade of the motivic volume by keeping track of dimensions in the Grothendieck ring of varieties. This produces a uniform refinement of the motivic volume and its birational version introduced by Kontsevich and Tschinkel to prove the specialization of birational types. We also provide several explicit examples of obstructions to stable rationality arising from this technique.
Unboundedness for motivic invariants of birational automorphisms
研究双射自同构的motivic不变量在三维流形上消失，而在高维投影空间上无界，并证明其自同构群不生成于特定结构映射。
Birational zeta functions
该研究定义了既约多项式动机ζ函数的双有理类似物，其具有弧接触轨迹的内蕴含义，并证明局部平面曲线奇点下该双有理ζ函数与动机ζ函数的极点本质一致。
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。
The main goal of this paper is to break up motivic cohomology into smaller pieces as suggested by the conjectural Bloch-Beilinson filtrations for the Chow groups.
We introduce a tower of localizing subcategories in Voevodsky's big (closed under infinite coproducts) triangulated category of motives. We show that the tower induces an interesting finite filtration on the motivic cohomology groups of smooth schemes over a perfect field. With rational coefficients, this finite filtration satisfies several of the properties of the still conjectural Bloch-Beilinson-Murre filtration.
Motivic invariants of birational maps
该研究构造了取值于Kontsevich–Tschinkel群与截尾Grothendieck簇群的双有理映射群胚同态不变量，用于研究Grothendieck环结构与L-等价性，并基于已有L-等价构造得到Cremona群的新结论。
Schematic Functorialities of Birational Motivic Homotopy Categories
该研究将 n-双有理动机同伦范畴提升为预层，推导单分支概型双有理动机同伦范畴的分解性质，给出双有理等价的检测方法，并证明稳定双有理态射等诱导完全忠实嵌入。
This paper is part of an endeavor to define an analogue of the slice filtration in the unstable motivic homotopy category. Our approach was inspired by the fact that the triangulated structures do not play a relevant role for the construction of birational homotopy categories as well as by the work of Kahn-Sujatha \cite{K-theory/0596} on birational motives, where the existence of a connection between the layers of the slice filtration and birational invariants is explicitly suggested. Our main result, shows that there is an equivalence of categories between the orthogonal components for the sl…
Bershadsky, Cecotti, Ooguri, and Vafa constructed a real-valued invariant for Calabi--Yau manifolds, which is now called the BCOV torsion. Based on it, a metric-independent invariant, called the BCOV invariant, was constructed by Fang--Lu--Yoshikawa and Eriksson--Freixas i Montplet--Mourougane. The BCOV invariant is conjecturally related to the Gromov--Witten theory via mirror symmetry. Based upon the previous work of the second author, we prove the conjecture that birational Calabi--Yau manifolds have the same BCOV invariant. We also extend the construction of the BCOV invariant to Calabi--Ya…
Complexes of stable birational invariants
本文引入了一个新的稳定双有理不变量，以链复形同伦类型的形式分类化动机体积，并由此构造了回缩有理性的动机障碍及一类普遍的不变量。
Inspired by p-adic (and real) principal value integrals, we introduce motivic principal value integrals associated to multi-valued rational differential forms on smooth algebraic varieties. We investigate the natural question whether (for complete varieties) this notion is a birational invariant. The answer seems to be related to the dichotomy of the Minimal Model Program.
Twisted jets, motivic measure and orbifold cohomology
本文通过引入Deligne-Mumford栈上的扭曲射流并研究其母题测度，证明了具有Gorenstein商奇点的双有理极小模型具有相同的带霍奇结构的轨形上同调。
Two semisimple algebraic groups of the same type are said to be motivic equivalent if the motives of the associated projective homogeneous varieties of the same type are isomorphic. We give general criteria of motivic equivalence in terms of the so-called higher Tits p-indices of algebraic groups. These results allow to give a complete classification of absolutely simple classical groups up to motivic equivalence in terms of the underlying algebraic structures. Among other applications of this classification, we deduce that there is a bijection between the stable birational equivalence classes…
We study the scheme of formal arcs on a singular algebraic variety and its images under truncations. We prove a rationality result for the Poincare series of these images which is an analogue of the rationality of the Poincare series associated to p-adic points on a p-adic variety. The main tools which are used are semi-algebraic geometry in spaces of power series and motivic integration (a notion introduced by M. Kontsevich). In particular we develop the theory of motivic integration for semi-algebraic sets of formal arcs on singular algebraic varieties, we prove a change of variable formula …
We investigate connections between Lipschitz geometry of real algebraic varieties and properties of their arc spaces. For this purpose we develop motivic integration in the real algebraic set-up. We construct a motivic measure on the space of real analytic arcs. We use this measure to define a real motivic integral which admits a change of variables formula not only for the birational but also for generically one-to-one Nash maps. As a consequence we obtain an inverse mapping theorem which holds for continuous rational maps and, more generally, for generically arc-analytic maps. These maps app…
Using the trivial fiber topology we describe motivic $\infty$-loop spaces and fibrant replacements in the motivic stable homotopy category $\mathbf{SH}_{\mathbb{A}^1,\mathrm{Nis}}(B)$ defined over one-dimensional base schemes $B$.
On the log motivic stable homotopy groups
在允许奇点解消的完美域上，通过比较对数 motivic 稳定同伦范畴与通常的 motivic 稳定同伦范畴，证明两者的稳定同伦群同构，揭示了对数 motivic 与通常 motivic 稳定同伦群的关系。
We propose a 'geometric Chevalley-Warning' conjecture, that is a motivic extension of the Chevalley-Warning theorem in number theory. It is equivalent to a particular case of a recent conjecture of F. Brown and O.Schnetz. In this paper, we show the conjecture is true for linear hyperplane arrangements, quadratic and singular cubic hypersurfaces of any dimension, and cubic surfaces in $\Pbb^3$. The last section is devoted to verifying the conjecture for certain special kinds of hypersurfaces of any dimension. As a by-product, we obtain information on the Grothendieck classes of the affine 'Pott…
Birational Algebraic Topology
本文在qcqs概形S上研究动机∞-范畴的双有理局部化，证明其与杠构造交换并保持连通性；在完美域k上，刻画了双有理群层为强A1-不变且Gm-收缩平凡；建立了双有理局部化与S^{2,1}-零化函子的等价，并证明了关于proper概形的若干双有理不变量性质。
We define a divisorial motivic zeta function for stable curves with marked points which agrees with Kapranov's motivic zeta function when the curve is smooth and unmarked. We show that this zeta function is rational, and give a formula in terms of the dual graph of the curve.
We study a K-theoretic characteristic class of singular varieties, namely the equivariant motivic Chern class. We prove that the motivic Chern class is characterized by an axiom system inspired by that of "K-theoretic stable envelopes," recently defined by Okounkov and studied in relation with quantum group actions on the K-theory algebra of moduli spaces. We also give explicit formulas for the equivariant motivic Chern classes of Schubert cells and matrix Schubert cells. Lastly, we calculate the equivariant motivic Chern class of the orbits of the A2 quiver representation, which yields formul…
We develop the formalism of universal torsors in equivariant birational geometry and apply it to produce new examples of nonbirational but stably birational actions of finite groups.
We give a survey of the background and recent development on the $K$-equivalence relation among birational manifolds. After a brief historical sketch of birational geometry, we define the $K$-partial ordering and $K$-equivalence in a birational class and discuss geometric situations that lead to these notions. One application to the filling-in problem for threefolds is given. We discuss the motivic aspect of $K$-equivalence relation. We believe that $K$-equivalent manifolds have the same Chow motive though we are unable to prove it at this moment. Instead we discuss various approaches toward t…
We introduce a formal integral on the system of varieties mapping properly and birationally to a given one, with value in an associated Chow group. Applications include comparisons of Chern numbers of birational varieties, new birational invariants, `stringy' Chern classes, and a `celestial' zeta function specializing to the topological zeta function. In its simplest manifestation, the integral gives a new expression for Chern-Schwartz-MacPherson classes of possibly singular varieties, placing them into a context in which a `change of variable' formula holds. The formalism has points of contac…
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).
On the Stable Birationality of Hilbert schemes of points on surfaces
本文研究不规则性为零的曲面上Hilbert概型之间的稳定双有理等价关系，证明几何有理曲面上仅有有限多个稳定双有理类，并由此推出motivic zeta函数的有理性。
We prove that stable rationality specializes in regular families whose fibers are integral and have at most ordinary double points as singularities. Our proof is based on motivic specialization techniques and the criterion of Larsen and Lunts for stable rationality in the Grothendieck ring of varieties.
We consider a complex smooth projective variety equipped with an action of an algebraic torus with a finite number of fixed points. We compare the motivic Chern classes of Białynicki-Birula cells with the $K$-theoretic stable envelopes of cotangent bundle. We prove that under certain geometric assumptions satisfied e.g. by homogenous spaces these two notions coincide up to normalization.
Birational maps of moduli spaces of stable sheaves on a K3 surface induced by Fourier-Mukai transforms
该研究探讨由Fourier-Mukai变换诱导的皮卡数为1的K3曲面上稳定层模空间的双有理映射问题。
We introduce and study the question how can stable birational types vary in a smooth proper family. Our starting point is the specialization for stable birational types of Nicaise and the author and our emphasis is on stable birational types of hypersurfaces. Building up on the work of Totaro and Schreieder on stable irrationality of hypersurfaces of high degree, we show that smooth Fano hypersurfaces of large degree over a field of characteristic zero are in general not stably birational to each other. In the appendix Claire Voisin proves a similar result in a different setting using the Chow…
This is an expository article, which contributes to the Proceedings of the conference "Groups of Automorphisms in Birational and Affine Geometry", held in Trento in 2012. We propose that (rational) fibrations on the projective space $\p^n$ by (birationally) Abelian hypersurfaces, for an arbitrary $n\ge 2$, provide an obstruction to stable rationality of algebraic varieties. We discuss the evidence for this proposition and derive some (almost straightforward) corollaries from it.
Already in the 1960s Grothendieck understood that one could obtain an almost entirely satisfactory theory of motives over a finite field when one assumes the full Tate conjecture. In this note we prove a similar result for motivic complexes. In particular Beilinson's Q-algebra of "correspondences at the generic point" is then defined for all connected varieties. We compute this for all smooth projective varieties (hence also for varieties birational to such a variety).
Stable degeneration and birational geometry
本文探讨了代数K稳定性理论中稳定退化的最新进展，聚焦于birational几何中的关键问题与方法。
Sheaf stable pairs, Quot-schemes, and birational geometry
本文建立层稳定对模空间理论与双有理几何的联系，重点研究光滑射影曲线上的稳定对模空间并给出低次数计算，同时提出若干后续问题。
We present a geometric construction of push-forward maps along projective morphisms for cohomology theories representable in the stable motivic homotopy category assuming that the element corresponding to the stable Hopf map is inverted in the coefficient ring of the theory. The construction is parallel to the one given by A. Nenashev for derived Witt groups. Along the way we introduce cohomology groups twisted by a formal difference of vector bundles as cohomology groups of a certain Thom space and compute twisted cohomology groups of projective spaces.
Using wall-crossing for K3 surfaces, we establish birational equivalence of moduli spaces of stable objects on generic Enriques surfaces for different stability conditions. As an application, we prove in the case of a Mukai vector of odd rank that they are birational to Hilbert schemes. The argument makes use of a new Chow-theoretic result, showing that moduli spaces on an Enriques surface give rise to constant cycle subvarieties of the moduli spaces of the covering K3.
Let X be a nonsingular algebraic variety in characteristic zero. To an effective divisor on X Kontsevich has associated a certain 'motivic integral', living in a completion of the Grothendieck ring of algebraic varieties. He used this invariant to show that birational Calabi-Yau varieties have the same Hodge numbers. Then Denef and Loeser introduced the motivic (Igusa) zeta function, associated to a regular function on X, which specializes to both the classical p-adic Igusa zeta function and the topological zeta function, and also to Kontsevich's invariant. This paper treats a generalization t…
The main goal of this paper is to construct an analogue of Voevodsky's slice filtration in the motivic unstable homotopy category. The construction is done via birational invariants, this is motivated by the existence of an equivalence of categories between the orthogonal components for Voevodsky's slice filtration and the birational motivic stable homotopy categories constructed in \cite{Pelaez:2011fk}. Another advantage of this approach is that the slices appear naturally as homotopy fibres (and not as in the stable setting, where they are defined as homotopy cofibres) which behave much bett…
In this paper, we study twisted algebraic $K$-theory from a motivic viewpoint. For a smooth variety $X$ over a field of characteristic zero and an Azumaya algebra $\mathcal{A}$ over $X$, we construct the $\mathcal{A}$-twisted motivic spectral sequence, by computing the slices of the motivic twisted algebraic $K$-theory spectrum as a twisted form of motivic cohomology. This generalizes previous results due to Kahn-Levine where $\mathcal{A}$ is assumed to be pulled back from a base field. Our methods use interaction between the slice filtration and birational geometry. Along the way, we prove a …
We determine systematic regions in which the bigraded homotopy sheaves of the motivic sphere spectrum vanish.
Periodic phenomena in stable motivic homotopy theory
研究稳定同伦理论工具对动机同伦理论的影响，探讨动机亚当斯谱序列、球的动机稳定同伦群周期性及合成谱，提出诸多未来研究问题。
Morel's stable connectivity theorems state that for any connective $S^1$-spectrum $F$ of motivic spaces (Nisnevich simplicial sheaves) over an arbitrary field, the spectrum $L_{\mathbb A^1}(F)$ is connective, and the same property for $\mathbb P^1$-spectra of motivic spaces. Here $L_{\mathbb A^1}$ denotes the $\mathbb A^1$-localisation in the category of motivic spectra over a field $k$. Originally the same property was conjectured for the case of motivic $S^1$-spectra over a base scheme $S$.In view of Ayoub's conterexamples the modified version of conjecture states that $L_{\mathbb A^1}(F)$ i…
An orbifold formula for algebraic stacks
本文推广orbifold公式至与光滑代数栈双有理的簇，提供计算弦论E-函数的新方法，并应用于klt奇点研究，涉及动机积分及Bruhat-Tits理论。
We study the moduli space of rank 0 semistable sheaves on some rational surfaces. We show the irreducibility and stable rationality of them under some conditions. We also compute several (virtual) Betti numbers of those moduli spaces by computing their motivic measures.
Motivic stable stems and Galois approximations of cellular motivic categories
本文证明小上同调维数域上的胞腔动机谱范畴可由绝对 Galois 群刻画，并据此确定几乎所有域在特征外的动机稳定茎。
The notion of d-critical loci was introduced by Joyce in order to give classical shadows of $(-1)$-shifted symplectic derived schemes. In this paper, we discuss birational geometry for d-critical loci, by introducing notions such as `d-critical flips', `d-critical flops', etc. They are not birational maps of the underlying spaces, but rather should be understood as virtual birational maps. We show that several wall-crossing phenomena of moduli spaces of stable objects on Calabi-Yau 3-folds are described in terms of d-critical birational geometry. Among them, we show that wall-crossing diagrams…
Motivic homotopy theory and stable homotopy groups
本文综述域上的 Morel-Voevodsky 动机同伦论及其同伦层计算，并说明动机 Adams 谱序列在确定稳定茎和催生综合同伦论中的作用。
Topological models for stable motivic invariants of regular number rings
本文为无穷多个数环建立稳定动机不变量的拓扑模型，并推广Morel定理至更深的基概形。
We construct well-behaved extensions of the motivic spectra representing generalized motivic cohomology and connective Balmer--Witt K-theory (among others) to mixed characteristic Dedekind schemes on which 2 is invertible. As a consequence we lift the fundamental fiber sequence of $η$-periodic motivic stable homotopy theory established in [arxiv:2005.06778] from fields to arbitrary base schemes, and use this to determine (among other things) the $η$-periodized algebraic symplectic and SL-cobordism groups of mixed characteristic Dedekind schemes containing 1/2.
Torsion birational motives of surfaces and unramified cohomology
该研究探讨了曲面的torsion birational motives与unramified cohomology的关系，并扩展了Kahn的结果，同时给出了违反积分Hodge猜想的例子。
We study the birational geometry of Hilbert schemes of points on non-minimal surfaces. In particular, we study the weak Lefschetz Principle in the context of birational geometry. We focus on the interaction of the stable base locus decomposition (SBLD) of the cones of effective divisors of $X^{[n]}$ and $Y^{[n]}$, when there is a birational morphism $f:X\rightarrow Y$ between surfaces. In this setting, $N^1(Y^{[n]})$ embeds in $N^1(X^{[n]})$, and we ask if the restriction of the stable base locus decomposition of $N^1(X^{[n]})$ yields the respective decomposition in $N^1(Y^{[n]})$ $i.e.$, if t…
We compute the 2-line of stable homotopy groups of motivic spheres over fields of characteristic not two in terms of motivic cohomology and hermitian K-groups.
Sheaf stable pairs on projective surfaces and birational geometry
本文研究了射影曲面上更高秩稳定对的模空间，并利用极小模型的birational几何分析其与基曲面的联系。
We study birational geometry of the moduli space of stable sheaves on a quadric surface with Hilbert polynomial $5m + 1$ and $c_{1} = (2, 3)$. We describe a birational map between the moduli space and a projective bundle over a Grassmannian as a composition of smooth blow-ups/downs.
I show that the cohomology of the generic points of algebraic complex varieties becomes {\sl stable} birational invariant, when considered `modulo the cohomology of the generic points of the affine spaces'.
In this paper we explore the isotropic stable motivic homotopy category constructed from the usual stable motivic homotopy category, following the work of Vishik on isotropic motives (see [29]), by killing anisotropic varieties. In particular, we focus on cohomology operations in the isotropic realm and study the structure of the isotropic Steenrod algebra. Then, we construct an isotropic version of the motivic Adams spectral sequence and apply it to find a complete description of the isotropic stable homotopy groups of the sphere spectrum, which happen to be isomorphic to the $Ext$-groups of …
We show that mapping spaces in the p-local motivic stable category over an Fp-scheme are strictly commutative monoids (whence HZ-modules) in a canonical way.
We describe a close relation between wall crossings in the birational geometry of moduli space of Gieseker stable sheaves $M_H(v)$ on $\bb{P}^2$ and mini-wall crossings in the stability manifold $Stab(D^b(\bb{P}^2))$.
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…
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。