On Unipotent Quotients and some -contractible Smooth Schemes

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 ≥ 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 .

Moduli of unipotent representations I: foundational topics

“…], and by g ab the abelianization: (2) . The pronilpotent completion of g is the inverse limit lim ← − g/g (n) .…”

“…Partial analogs of Mumford's theory for groups which are not reductive are currently under development in works by A. Asok, B. Doran, and F. Kirwan (c.f. [6], [2]). …”

“…The dimension of the automorphism group of a nondegenerate representation of dimension n is independent of the choice of nondegenerate representation; it thus defines an invariant of g and n which we denote by A(g, n). We illustrate the behavior of A(g, n) for low n with several examples, showing in particular that the possible triples (A(g, 2), A(g, 3), A(g, 4)) are (2, 2, 2), (2,2,3) and (2,3,4).…”

The homotopy Leray spectral sequence

“…Thus, if k is infinite and perfect, X \ x satisfies the hypotheses of the proposition and yields an "exotic motivic sphere". In dimension d ≥ 4, the examples in [AD07] or [ADF17] also satisfy the hypotheses of the theorem, at least over an infinite base field.…”

Yang-Mills theory and Tamagawa numbers: the fascination of unexpected links in mathematics

“…We remark that a Zariski locally trivial smooth morphism f : X → Y of smooth schemes with A 1 -contractible fibres (for example, affine space fibres) is an A 1 -weak equivalence. (See [56] for a precise definition of A 1 -weak equivalence and [5] for a more detailed discussion of this example.) The space X G (ρ) gives rise to an object in H(k) or H · (k).…”