\(\mathbb A^1\)-homotopy groups, excision, and solvable quotients (Q1019754)

The \(\mathbb A^1\)-homotopy category for smooth schemes of finite type over an arbitrary ground field \(k\) has been introduced by \textit{P. Morel} and \textit{V. Voevodsky}[Publ. Math., Inst. Hautes Étud. Sci. 90, 45--143 (1999; Zbl 0983.14007) about ten years ago. Actually, this \(\mathbb A^1\)-homotopy category \({\mathcal H}(k)\) is constructed in the context of model categories by a two-stage categorical localization process from the category of simplicial Nisnevich sheaves of sets on the category of smooth schemes of finite type over \(k\). Viewing such schemes as objects in \({\mathcal H}(k)\), and mimicking constructions of classical homotopy theory in a suitable way, Morel and Voevodsky defined the concepts of \(\mathbb A^1\)-homotopy groups, \(\mathbb A^1\)-covering spaces, \(\mathbb A^1\)-connectedness, and others for them, thereby introducing new invariants of fundamental geometric and arithmetic significance. However, one of the main problems in \(\mathbb A^1\)-homotopy theory is to determine concretely, namely in terms of explicit algebraic and combinatorial data, the \(\mathbb A^1\)-homotopy types of smooth algebraic varieties over \(k\), and to compute their \(\mathbb A^1\)-homotopy groups effectively. In the paper under review, the authors provide some steps forward in this direction. More precisely, they develop a number of new techniques for computing \(\mathbb A^1\)-homotopy groups of special smooth projective varieties, with a particular emphasis on geometric quotients obtained from group actions on certain varieties. After a very lucid and detailed introduction to the paper, Section 2 reviews the basics of \(\mathbb A^1\)-homotopy theory for the sake of self-containedness, on the one hand, and establishes an analogue of the topological notion of path-connectedness, which is called ``\(\mathbb A^1\)-chain connectedness'' in \({\mathcal H}(k)\) on the other hand. Section 3 is devoted to higher \(\mathbb A^1\)-homotopy and \(\mathbb A^1\)-homology groups, including the novelty of a proof of the existence of Mayer-Vietoris sequences for \(\mathbb A^1\)-homology. Along the way, \(\mathbb A^1\)-covering spaces are discussed, together with the computation of some \(\mathbb A^1\)-homotopy groups of \(\mathbb A^1\setminus\{0\}\) and \(\mathbb P^n\). Section 4 contains the first main result of the paper, which may be interpreted as a certain ``excision theorem'' for \(\mathbb A^1\)-homotopy groups refining F. Morel's earlier approach [cf. \textit{F. Morel}, \(K\)-Theory 35, No.~1--2, 1--68 (2005; Zbl 1117.14023)]. Chapter 5 turns to the study of the geometry of toric varieties and quotients by free solvable group actions in the context of \(\mathbb A^1\)-covering spaces. Using D. A. Cox's presentation of a smooth proper toric variety as a quotient space, the authors link toric varieties to Galois \(\mathbb A^1\)-covering spaces. This, together with a new combinatorial result on toric varieties, provides the necessary ingredients for applying the \(\mathbb A^1\)-excision result (from Section 4) to the study of \(\mathbb A^1\)-homotopy groups of smooth proper toric varieties. In fact, the concluding Section 6 delivers some vanishing and non-vanishing results for low-degree \(\mathbb A^1\)-homotopy groups of smooth toric varieties. Finally, this section provides some sample computations of \(\mathbb A^1\)-homotopy groups illustrating the powerful techniques developed in the present paper. As the authors point out, related, results on the \(\mathbb A^1\)-fundamental group of a smooth toric variety were independently proved by \textit{M. Wendt} [cf.: Adv. Math. 223, No.~1, 352--378 (2010; Zbl 1276.14035)], mainly by using generalizations of the van Kampen theorem in \(\mathbb A^1\)-homotopy theory. Overall, the paper under review must be seen as a highly significant contribution to the field of \(\mathbb A^1\)-topology of algebraic varieties. Written in an utmost lucid, detailed and nearly self-contained style, this work provides new efficient techniques and results of propelling impact in current algebraic geometry.