arXiv subjects

Aravind Asok

Publications and source records attributed to Aravind Asok.

44 records · Page 3Linked to original sources

arXiv2011

Smooth varieties up to A^1-homotopy and algebraic h-cobordisms

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.

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arXiv2011

Stable A^1-homotopy and R-equivalence

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.

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arXiv2009

Equivariant sheaves on some spherical varieties

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.

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arXiv2009

A^1-homotopy groups, excision, and solvable quotients

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 π_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.

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arXiv2008

Yang-Mills theory and Tamagawa numbers

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.

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arXiv2007

Vector bundles on contractible smooth schemes

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.

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arXiv2007

On unipotent quotients and some A^1-contractible smooth schemes

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.

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arXiv2006

Equivariant Vector Bundles on Certain Affine G-Varieties

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.

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