arXiv subjects
Publications and source records attributed to Aravind Asok.
We study the problem of when a topological vector bundle on a smooth complex affine variety admits an algebraic structure. We prove that all rank $2$ topological complex vector bundles on smooth affine quadrics of dimension $11$ over the complex numbers admit algebraic structures.
We establish a relative version of the abstract "affine representability" theorem in ${\mathbb A}^1$--homotopy theory from Part I of this paper. We then prove some ${\mathbb A}^1$--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 ${\mathbb A}^1$--homotopy theory.
We study generically split octonion algebras over schemes using techniques of ${\mathbb A}^1$-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.
The goal of this note is to study the analog in unstable ${{\mathbb A}^1}$-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 $i \leq 3$, 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 $π_{n}^{{\mathbb A}^1}({\mathbb A}^n \setminus 0)$ for $n \geq 4$.
We establish a general "affine representability" result in ${\mathbb A}^1$-homotopy theory over a general base. We apply this result to obtain representability results for vector bundles in ${\mathbb A}^1$-homotopy theory. Our results simplify and significantly generalize F. Morel's ${\mathbb A}^1$-representability theorem for vector bundles.
We study a version of the James model for the loop space of a suspension in unstable ${\mathbb A}^1$-homotopy theory. We use this model to establish an analog of G.W. Whitehead's classical refinement of the Freudenthal suspension theorem in ${\mathbb A}^1$-homotopy theory: our result refines F. Morel's ${\mathbb A}^1$-simplicial suspension theorem. We then describe some $E_1$-differentials in the EHP sequence in ${\mathbb A}^1$-homotopy theory. These results are analogous to classical results of G.W. Whitehead's. Using these tools, we deduce some new results about unstable ${\mathbb A}^1$-homotopy sheaves of motivic spheres, including the counterpart of a classical rational non-vanishing result.
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.
We study the relationship between several notions of connectedness arising in ${\mathbb A}^1$-homotopy theory of smooth schemes over a field $k$: ${\mathbb A}^1$-connectedness, stable ${\mathbb A}^1$-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 ${\mathbb A}^1$-connectedness coincides with motivic connectedness, under suitable hypotheses on $k$. Then, we observe that there exist stably ${\mathbb A}^1$-connected smooth proper varieties over the field of complex numbers that are not ${\mathbb A}^1$-connected.
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 variant of the classical construction of vector bundles on spheres via clutching functions.
We discuss secondary (and higher) characteristic classes for algebraic vector bundles with trivial top Chern class. We show that if X is a smooth affine scheme of dimension d over a field k of finite 2-cohomological dimension (with char(k) $\neq$ 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.
We study the problem of classifying projectivizations of rank-two vector bundles over ${\mathbb P}^2$ up to various notions of equivalence that arise naturally in ${\mathbb A}^1$-homotopy theory, namely ${\mathbb A}^1$-weak equivalence and ${\mathbb A}^1$-$h$-cobordism. First, we classify such varieties up to ${\mathbb A}^1$-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 ${\mathbb C}$, 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 ${\mathbb A}^1$-$h$-cobordism using techniques of deformation theory. To this end, we establish a deformation rigidity result for ${\mathbb P}^1$-bundles over ${\mathbb P}^2$ which links ${\mathbb A}^1$-$h$-cobordisms to deformations of the underlying vector bundles. Using results from the deformation theory of vector bundles we show that if $X$ is a ${\mathbb P}^1$-bundle over ${\mathbb P}^2$ and $Y$ is the projectivization of a direct sum of line bundles on ${\mathbb P}^2$, then if $X$ is ${\mathbb A}^1$-weakly equivalent to $Y$, $X$ is also ${\mathbb A}^1$-$h$-cobordant to $Y$. Finally, we discuss some subtleties inherent in the definition of ${\mathbb A}^1$-$h$-cobordism. We show, for instance, that direct ${\mathbb A}^1$-$h$-cobordism fails to be an equivalence relation.
We discuss the relationship between the ${\mathbb A}^1$-homotopy sheaves of ${\mathbb A}^n \setminus 0$ and the problem of splitting off a trivial rank $1$ summand from a rank $n$-vector bundle. We begin by computing $π_3^{{\mathbb A}^1}({\mathbb A}^3 \setminus 0)$, and providing a host of related computations of "non-stable" ${\mathbb A}^1$-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.
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 ${\mathbb A}^1$-homotopy sheaf of the symplectic group. Second, these computations can be used in concert with F. Morel's ${\mathbb A}^1$-homotopy classification of vector bundles on smooth affine schemes and obstruction theoretic techniques (stemming from a version of the Postnikov tower in ${\mathbb A}^1$-homotopy theory) to reduce the classification results to cohomology vanishing statements. Third, we prove the required vanishing statements.
We determine the first non-stable ${\mathbb A}^1$-homotopy sheaf of $SL_n$. Using techniques of obstruction theory involving the ${\mathbb A}^1$-Postnikov tower, supported by some ideas from the theory of unimodular rows, we classify vector bundles of rank $\geq d-1$ 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 ${\mathbb A}^1$-homotopy sheaves with real and complex realization.
We study aspects of the A^1-homotopy classification problem in dimensions >= 3 and, to this end, we investigate the problem of computing A^1-homotopy groups of some A^1-connected smooth varieties of dimension >=. Using these computations, we construct pairs of A^1-connected smooth proper varieties all of whose A^1-homotopy groups are abstractly isomorphic, yet which are not A^1-weakly equivalent. The examples come from pairs of Zariski locally trivial projective space bundles over projective spaces and are of the smallest possible dimension. Projectivizations of vector bundles give rise to A^1-fiber sequences, and when the base of the fibration is an A^1-connected smooth variety, the associated long exact sequence of A^1-homotopy groups can be analyzed in detail. In the case of the projectivization of a rank 2 vector bundle, the structure of the A^1-fundamental group depends on the splitting behavior of the vector bundle via a certain obstruction class. For projective bundles of vector bundles of rank >=, the A^1-fundamental group is insensitive to the splitting behavior of the vector bundle, but the structure of higher A^1-homotopy groups is influenced by an appropriately defined higher obstruction class.
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 Lüroth 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 étale 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 étale cohomology class of lower degree.
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 étale 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.
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.