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

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

2007-03-05

arxiv.org/abs/math/0703137v2

Int. Math. Res. Pap. 2007 Art. ID rpm005

Mathematics

Algebraic Geometry

41 pages, published version (before page proofs); some typos corrected

Scientific paper

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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