A Proper G a Action on C 5 Which is Not Locally Trivial

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 .

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“…All of these can be explicitly checked. Here is one such construction, which recovers the example due to Deveney and Finston [DF95].…”

Section: Spaces Of Actionssupporting

confidence: 54%

On Unipotent Quotients and some -contractible Smooth Schemes

Asok

Doran

2010

International Mathematics Research Papers

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“…Its quotient is an algebraic space that is not a scheme [5]. In particular, W, as in Seshadri's construction above, is not quasiprojective.…”

Section: Proper Actionsmentioning

confidence: 99%

“…Locally trivial actions are proper, and proper actions on C n are locally trivial provided that C[X] is a flat ring extension of C 0 [4, Theorem 2.8]. This need not always be the case as shown in [5]. On the other hand, Holmann [12] showed that any proper holomorphic action on a complex manifold admits a quotient that is a manifold, while Popp [11,Lecture 3] showed that this quotient admits the structure of an algebraic space if the action is algebraic and the manifold is a smooth variety.…”

Section: Introductionmentioning

confidence: 99%

“…On the other hand, Holmann [12] showed that any proper holomorphic action on a complex manifold admits a quotient that is a manifold, while Popp [11,Lecture 3] showed that this quotient admits the structure of an algebraic space if the action is algebraic and the manifold is a smooth variety. Based on these results, we give in section 3 a ring-theoretic criterion for a proper action on C n to be locally trivial and indicate where the hypotheses fail for the example in [5] of a nonlocally trivial proper action on C 5 .…”

Section: Introductionmentioning

confidence: 99%

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Triangular 𝐺ₐ actions on 𝐂⁴

Deveney

Finston

Rossum

2004

Proc. Amer. Math. Soc.

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“…2. An infinite class of fixed point free but nonproper G, actions on C4 were given in [5], and an example of a proper, but not stable action on C5 appears in [7]. Downloaded by [University Of Pittsburgh] at 08:31 01 December 2014 3.…”

Section: Definition 1 T H E Action a Is Said T O Be Stable I F The Kementioning

confidence: 99%