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

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

2009-02-10

arxiv.org/abs/0902.1564v2

Mathematics

Algebraic Geometry

48 pages, To appear Adv. Math, typographical and grammatical updates

Scientific paper

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 \pi_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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