Abstract:We study generically split octonion algebras over schemes using techniques of A^1-homotopy theory. By combining affine representability results with techniques of obstruction theory, we establish classification results over smooth affine schemes of small dimension. In particular, for smooth affine schemes over algebraically closed fields, we show that generically split octonion algebras may be classified by characteristic classes including the second Chern class and another ``mod 3" invariant. We review Zorn's ``vector matrix" construction of octonion algebras, generalized to rings by various authors, and show that generically split octonion algebras are always obtained from this construction over smooth affine schemes of low dimension. Finally, generalizing P. Gille's analysis of octonion algebras with trivial norm form, we observe that generically split octonion algebras with trivial associated spinor bundle are automatically split in low dimensions.
This paper is currently submitted for review; comments welcome!
Last Update: 2 Jun 2017
Abstract: We study cohomotopy sets in algebraic geometry using the Morel-Voevodsky A^1-homotopy category: these sets are defined in terms of maps from a smooth scheme to a motivic sphere. Following Borsuk, we show that in the presence of suitable dimension hypotheses on the source, our motivic cohomotopy sets can be equipped with abelian group structures. We then explore links between these motivic cohomotopy groups, Euler class groups a la Nori--Bhatwadekar--Sridharan and Chow-Witt groups.
Using these links, we show that, at least for k an infinite field having characteristic unequal to 2, the Euler class group of codimension d cycles on a smooth affine k-variety of dimension d coincides with the codimension d Chow-Witt group. As a byproduct, we describe the Chow group of zero cycles on a smooth affine k-scheme as the quotient of the free abelian group on zero cycles by the subgroup generated by reduced complete intersection ideals; this answers a question of S. Bhatwadekar and R. Sridharan.
This paper is currently submitted for review; comments welcome!
Last Update: 23 Aug 2016
Abstract:
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 <= 3, it is known that algebraicity of Chern classes of a vector bundle guarantees algebraizability of the vector bundle. In contrast, we show in dimension >=4, algebraicity of Chern classes is insufficient to guarantee algebraizability of vector bundles. To do this, we construct a new obstruction to algebraizability using Steenrod operations on Chow groups. By means of an explicit example, we observe that our obstruction is non-trivial in general.
This paper is currently submitted for review; comments welcome!
Last Update: 12 Oct 2015
Abstract:
We study the 0-th stable A^1-homotopy sheaf of a smooth proper variety over a field k assumed to be infinite, perfect and to have characteristic unequal to 2. We provide an explicit description of this sheaf in terms of the theory of (twisted) Chow-Witt groups as defined by Barge-Morel and developed by Fasel. We study the notion of rational point up to stable A^1-homotopy, defined in terms of the stable A^1-homotopy sheaf of groups mentioned above. We show that, for a smooth proper k-variety X, existence of a rational point up to stable A^1-homotopy is equivalent to existence of a 0-cycle of degree 1.
This paper is currently stalled; there is a technical ingredient for which we could not find good proofs in the literature (the comparison between Voevodsky's unstable duality construction and the usual construction of Atiyah duality), and we hoped to revise the presentation for clarity. Furthermore, we have received a number of comments on the posted draft that we would like to incorporate. As always, comments welcome!
Last update: a long time ago (sorry)
Abstract:
We establish a relative version of the abstract ``affine representability" theorem in A^1-homotopy theory from Part I of this paper. We then prove some A^1-invariance statements for generically trivial torsors under isotropic reductive groups over infinite fields analogous to the Bass-Quillen conjecture for vector bundles. Putting these ingredients together, we deduce representability theorems for generically trivial torsors under isotropic reductive groups and for associated homogeneous spaces in A^1-homotopy theory.
Abstract:
We prove that all rank 2 topological complex vector bundles on smooth affine quadrics of dimension 11 over the complex numbers are algebraizable.
Abstract:
We establish a general "affine representability" result in A^1-homotopy theory over a general base. We apply this result to obtain representability results for vector bundles in A^1-homotopy theory. Our results simplify and significantly generalize F. Morel's A^1-representability theorem for vector bundles.
Abstract:
We study the problem of classifying projectivizations of rank-two vector bundles over P^2 up to various notions of equivalence that arise naturally in A^1-homotopy theory, namely A^1-weak equivalence and A^1-h-cobordism.
First, we classify such varieties up to A^1-weak equivalence: over algebraically closed fields having characteristic unequal to two the classification can be given in terms of characteristic classes of the underlying vector bundle. When the base field is C, this classification result can be compared to a corresponding topological result and we find that the algebraic and topological homotopy classifications agree.
Second, we study the problem of classifying such varieties up to A^1-h-cobordism using techniques of deformation theory. To this end, we establish a deformation rigidity result for P^1-bundles over P^2 which links A^1-h-cobordisms to deformations of the underlying vector bundles. Using results from the deformation theory of vector bundles we show that if X is a P^1-bundle over P^2 and Y is the projectivization of a direct sum of line bundles on P^2, then if X is A^1-weakly equivalent to Y, X is also A^1-h-cobordant to Y.
Finally, we discuss some subtleties inherent in the definition of A^1-h-cobordism. We show, for instance, that direct A^1-h-cobordism fails to be an equivalence relation.
Abstract:
We study a version of the James model for the loop space of a suspension in unstable A^1-homotopy theory. We use this model to establish an analog of G.W. Whitehead's classical refinement of the Freudenthal suspension theorem in A^1-homotopy theory: our result refines F. Morel's A^1-simplicial suspension theorem. We then describe some E_1-differentials in the EHP sequence in A^1-homotopy theory. These results are analogous to classical results of G.W. Whitehead. Using these tools, we deduce some new results about unstable A^1-homotopy sheaves of motivic spheres, including the counterpart of a classical rational non-vanishing result.
Abstract:
We study the representability of motivic spheres by smooth varieties. We show that certain explicit “split” quadric hypersurfaces have the A^1-homotopy type of motivic spheres over the integers and that the A^1-homotopy types of other motivic spheres do not contain smooth schemes as representatives. We then study some applications of these representability/nonrepresentability results to the construction of new exotic A^1-contractible smooth schemes. Then, we study vector bundles on even dimensional “split” quadric hypersurfaces by developing an algebro-geometric variant of the classical construction of vector bundles on spheres via clutching functions.
Abstract:
We study the relationship between several notions of connectedness arising in A^1-homotopy theory of smooth schemes over a field k: A^1-connectedness, stable A^1-connectedness and motivic connectedness, and we discuss the relationship between these notations and rationality properties of algebraic varieties. Motivically connected smooth proper k-varieties are precisely those with universally trivial CH_0. We show that stable A^1-connectedness coincides with motivic connectedness, under suitable hypotheses on k. Then, we observe that there exist stably A^1-connected smooth proper varieties over the field of complex numbers that are not A^1-connected.
Abstract:
The goal of this note is to study the analog in unstable A^1-homotopy theory of the unit map from the motivic sphere spectrum to the Hermitian K-theory spectrum, i.e., the degree map in Hermitian K-theory. We show that "Suslin matrices", which are explicit maps from odd dimensional split smooth affine quadrics to geometric models of the spaces appearing in Bott periodicity in Hermitian K-theory, stabilize in a suitable sense to the unit map. As applications, we deduce that K^{MW}_i(F)=GW^i_i(F) for i <= 3, which can be thought of as an extension of Matsumoto's celebrated theorem describing K_2 of a field. These results provide the first step in a program aimed at computing the sheaf \pi_n^{A^1}(A^n \ 0) for n >= 4.
Abstract:
We establish the equality of two definitions of an Euler class in algebraic geometry: the first definition is as a "characteristic class" with values in Chow-Witt theory, while the second definition is as an "obstruction class." Along the way, we refine Morel's relative Hurewicz theorem in A^1-homotopy theory, and show how to define (twisted) Chow-Witt groups for geometric classifying spaces.
Abstract:
We discuss secondary (and higher) characteristic classes for algebraic vector bundles with trivial top Chern class. We then show that if X is a smooth affine scheme of dimension d over a field k of finite 2-cohomological dimension (with char(k) unequal to 2) and E is a rank d vector bundle over X, vanishing of the Chow-Witt theoretic Euler class of E is equivalent to vanishing of its top Chern class and these higher classes. We then derive some consequences of our main theorem when k is of small 2-cohomological dimension.
Abstract:
We discuss the relationship between the A^1-homotopy sheaves of A^n \ 0 and the problem of splitting off a trivial rank 1 summand from a rank n-vector bundle. We begin by computing \pi_1^{A^1}(A^3\0), and providing a host of related computations of "non-stable" A^1-homotopy sheaves. We then use our computation to deduce that a rank 3 vector bundle on a smooth affine 4-fold over an algebraically closed field having characteristic unequal to 2 splits off a trivial rank 1 summand if and only if its third Chern class (in Chow theory) is trivial. This result provides a positive answer to a case of a conjecture of M.P. Murthy.
Abstract:
We give a cohomological classification of vector bundles of rank 2 on a smooth affine threefold over an algebraically closed field having characteristic unequal to 2. As a consequence we deduce that cancellation holds for rank 2 vector bundles on such varieties. The proofs of these results involve three main ingredients. First, we give a description of the first non-stable A^1-homotopy sheaf of the symplectic group. Second, these computations can be used in concert with F. Morel's A^1-homotopy classification of vector bundles on smooth affine schemes and obstruction theoretic techniques (stemming from a version of the Postnikov tower in A^1-homotopy theory) to reduce the classification results to cohomology vanishing statements. Third, we prove the required vanishing statements.
Abstract:
We determine the first non-stable A^1-homotopy sheaf of SL_n. Using techniques of obstruction theory involving the A^1-Postnikov tower, supported by some ideas from the theory of unimodular rows, we classify vector bundles of rank >= d-1 on split smooth affine quadrics of dimension 2d-1. These computations allow us to answer a question posed by Nori, which gives a criterion for completability of certain unimodular rows. Furthermore, we study compatibility of our computations of A^1-homotopy sheaves with real and complex realization.
Abstract:
We study some aspects of the relationship between A^1-homotopy theory and birational geometry. We study the so-called A^1-singular chain complex and zeroth A^1-homology sheaf of smooth algebraic varieties over a field k. We exhibit some ways in which these objects are similar to their counterparts in classical topology and similar to their motivic counterparts (the (Voevodsky) motive and zeroth Suslin homology sheaf). We show that if k is infinite the zeroth A^1-homology sheaf is a birational invariant of smooth proper varieties, and we explain how these sheaves control various cohomological invariants, e.g., unramified \'etale cohomology. In particular, we deduce a number of vanishing results for cohomology of A^1-connected varieties. Finally, we give a partial converse to these vanishing statements by giving a characterization of A^1-connectedness by means of vanishing of unramified invariants.
Abstract:
We show how the techniques of Voevodsky's proof of the Milnor conjecture and the Voevodsky- Rost proof of its generalization the Bloch-Kato conjecture can be used to study counterexamples to the classical L\"uroth problem. By generalizing a method due to Peyre, we produce for any prime number l and any integer n >= 2, a rationally connected, non-rational variety for which non-rationality is detected by a non-trivial degree n unramified \'etale cohomology class with l-torsion coefficients. When l = 2, the varieties that are constructed are furthermore unirational and non-rationality cannot be detected by a torsion unramified \'etale cohomology class of lower degree.
Abstract:
We describe categories of equivariant vector bundles on certain toroidal spherical varieties in linear algebra terms: vector spaces equipped with filtrations, group and Lie algebra actions, and linear maps preserving these structures.
Notes: the paper referred to in this announcement is still under preparation and it is unclear when it will be finished.
Abstract:
We prove that the Voevodsky motive with Z-coefficients (resp. Q-coefficients) of a Z-
acylic (resp. Q-acyclic) smooth complex variety of dimension <= 2 is isomorphic to that of a
point, and discuss some related extensions.
Abstract:
We describe categories of equivariant vector bundles on certain toroidal spherical varieties in linear algebra terms: vector spaces equipped with filtrations, group and Lie algebra actions, and linear maps preserving these structures.
Notes: the paper referred to in this announcement is still under preparation and it is unclear when it will be finished.
Abstract:
We start to study the problem of classifying smooth proper varieties over a field k from the standpoint of A^1-homotopy theory. Motivated by the topological theory of surgery, we discuss the problem of classifying up to isomorphism all smooth proper varieties having a specified A^1-homotopy type. Arithmetic considerations involving the sheaf of A^1-connected components lead us to introduce several different notions of connectedness in A^1-homotopy theory. We provide concrete links between these notions, connectedness of points by chains of affine lines, and various rationality properties of algebraic varieties (e.g., rational connectedness).
We introduce the notion of an A^1-h-cobordism, an algebro-geometric analog of the topological notion of h-cobordism, and use it as a tool to produce non-trivial A^1-weak equivalences of smooth proper varieties. Also, we give explicit computations of refined A^1-homotopy invariants, such as the A^1-fundamental sheaf of groups, for some A^1-connected varieties. We observe that the A^1-fundamental sheaf of groups plays a central yet mysterious role in the structure of A^1-h-cobordisms. As a consequence of these observations, we completely solve the classification problem for rational smooth proper surfaces over an algebraically closed field: while there exist arbitrary dimensional moduli of such surfaces, there are only countably many A^1-homotopy types, each uniquely determined by the isomorphism class of its A^1-fundamental sheaf of groups.
Notes: there is an error in the proof of Proposition 3.1.7, which renders the proof of classification of smooth proper rational surfaces incomplete. It is possible to fix this by a different argument and details are forthcoming.
Abstract:
We prove that existence of a k-rational point can be detected by the stable A^1-homotopy category of S^1-spectra, or even a "rationalized" variant of this category.
Abstract:
We extend previous results on A^1-homotopy groups of smooth proper toric varieties to the case of smooth proper toric models, i.e., smooth proper equivariant compactifications of possibly non-split tori, in characteristic 0.
Notes: never posted on the ArXiv.
Abstract:
We study some properties of A^1-homotopy groups: geometric interpretations of connectivity, excision results, and a re-interpretation of quotients by free actions of connected solvable groups in terms of covering spaces in the sense of A^1-homotopy theory. These concepts and results are well-suited to the study of certain quotients via geometric invariant theory.
As a case study in the geometry of solvable group quotients, we investigate A^1-homotopy groups of smooth toric varieties. We give simple combinatorial conditions (in terms of fans) guaranteeing vanishing of low degree A^1-homotopy groups of smooth (proper) toric varieties. Finally, in certain cases, we can actually compute the "next" non-vanishing A^1-homotopy group (beyond \pi_1^{A^1}) of a smooth toric variety. From this point of view, A^1-homotopy theory, even with its exquisite sensitivity to algebro-geometric structure, is almost "as tractable" (in low degrees) as ordinary homotopy for large classes of interesting varieties.
Abstract:
We discuss algebraic vector bundles on smooth k-schemes X contractible from the standpoint of A^1-homotopy theory; when k = C, the smooth manifolds X(C) are contractible as topological spaces. The integral algebraic K-theory and integral motivic cohomology of such schemes are that of Spec k. One might hope that furthermore, and in analogy with the classification of topological vector bundles on manifolds, algebraic vector bundles on such schemes are all isomorphic to trivial bundles; this is almost certainly true when the scheme is affine. However, in the non-affine case this is false: we show that (essentially) every smooth A^1-contractible strictly quasi-affine scheme that admits a U-torsor whose total space is affine, for U a unipotent group, possesses a non-trivial vector bundle. Indeed we produce explicit arbitrary dimensional families of non-isomorphic such schemes, with each scheme in the family equipped with "as many" (i.e., arbitrary dimensional moduli of) non-isomorphic vector bundles, of every sufficiently large rank n, as one desires; neither the schemes nor the vector bundles on them are distinguishable by algebraic K-theory. We also discuss the triviality of vector bundles for certain smooth complex affine varieties whose underlying complex manifolds are contractible, but that are not necessarily A^1-contractible.
Abstract:
Atiyah and Bott used equivariant Morse theory applied to the Yang-Mills functional to calculate the Betti numbers of moduli spaces of vector bundles over a Riemann surface, rederiving inductive formulae obtained from an arithmetic approach which involved the Tamagawa number of SL_n. This article surveys this link between Yang-Mills theory and Tamagawa numbers, and explains how methods used over the last three decades to study the singular cohomology of moduli spaces of bundles on a smooth complex projective curve can be adapted to the setting of A^1-homotopy theory to study the motivic cohomology of these moduli spaces.
Notes: the preprint ``equivariant motivic cohomology" reference here was never made publicly available. A number of other authors have developed this theory.
Abstract:
We study quotients of quasi-affine schemes by unipotent groups over fields of characteristic 0. To do this, we introduce a notion of stability which allows us to characterize exactly when a principal bundle quotient exists and, together with a cohomological vanishing criterion, to characterize whether or not the resulting quasi-affine quotient scheme is affine. We completely analyze the case of G_a-invariant hypersurfaces in a linear G_a-representation W; here the above characterizations admit simple geometric and algebraic interpretations. As an application, we produce arbitrary dimensional families of non-isomorphic smooth quasi-affine but not affine n-dimensional varieties (n \geq 6) that are contractible in the sense of A^1-homotopy theory. Indeed, existence follows without any computation; yet explicit defining equations for the varieties depend only on knowing some linear G_a- and SL_2- invariants, which, for a sufficiently large class, we provide. Similarly, we produce infinitely many non-isomorphic examples in dimensions 4 and 5. Over C, the analytic spaces underlying these varieties are non-isomorphic, non-Stein, topologically contractible and often diffeomorphic to C^n.
Notes: the preprint version does not contain various typographical errors that were introduced during the course of typesetting.
Abstract:
We give a concrete description of the category of G-equivariant vector bundles on certain affine G-varieties (where G is a reductive linear algebraic group over an algebraically closed field of characteristic 0) in terms of linear algebra data.