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Smooth varieties up to \(\mathbb A^1\)-homotopy and algebraic \(h\)-cobordisms. (English) Zbl 1255.14018
This important paper sets the stage for the classification of smooth proper varieties over a field up to \(\mathbb{A}^1\)-homotopy. The authors discuss the foundations of classification problems and determine various important \(\mathbb{A}^1\)-homotopy types. In particular, the authors provide deep discussions and calculations of the sheaf of connected components and the fundamental sheaf of groups.
The classification problem in geometric topology for compact smooth manifolds without boundary up to homotopy equivalence can be reformulated in algebraic geometry by replacing usual homotopy equivalences by an appropriate notion of \(\mathbb{A}^1\)-weak equivalences. The first task then consists in classifying algebraic varieties up to \(\mathbb{A}^1\)-weak equivalence. In the topological case, it suffices to consider connected manifolds. Unfortunately, already at this stage the algebraic theory is much more complicated than the topological one. The authors call a scheme \(X\) over a field \(k\) \(\mathbb{A}^1\)-connected if it has the same sheaf \(\pi_0^{\mathbb{A}^1}(X)\) of \(\mathbb{A}^1\)-connected components as \(k\). Though the sheaf \(\pi_0^{\mathbb{A}^1}(X)\) turns out to be a more complicated invariant than one would expect. For the canonical epimorphism \(X \to \pi_0^{\mathbb{A}^1}(X)\) is in general highly non-trivial. The authors take a lot of care to analyze \(\pi_0^{\mathbb{A}^1}(X)\). In particular, they show that a smooth proper scheme over a field is \(\mathbb{A}^1\)-connected if and only if it is \(\mathbb{A}^1\)-chain connected which is closer to the usual notion of path-connectedness.
Back in the topological case, the more difficult task to classify smooth manifolds up to diffeomorphism can be reduced in higher dimensions to computations in homotopy theory by the groundbreaking, now classical, theory of surgery and the notion of \(h\)-cobordism. The analogous classification of smooth schemes up to isomorphism seems again much more complicated. In order to attack this fundamental problem the authors introduce the notion of an \(\mathbb{A}^1\)-\(h\)-cobordism between smooth proper varieties and formulate an \(\mathbb{A}^1\)-surgery problem for motivic spaces. They provide a detailed study of \(\mathbb{A}^1\)-\(h\)-cobordisms between rational smooth proper surfaces.
The techniques and ideas introduced in this paper provide a rich inventory and a lot of inspiration for future research in motivic homotopy theory and algebraic geometry. In particular, the notion of \(\mathbb{A}^1\)-\(h\)-cobordism and a possible motivic surgery theory are very interesting new aspects. The first section provides a very nicely written and detailed introduction to the subject and points out the most important problems. The two appendices provide the fundamental definitions in motivic homotopy theory and make the paper rather self-contained.

MSC:
14F42 Motivic cohomology; motivic homotopy theory
14J10 Families, moduli, classification: algebraic theory
14L30 Group actions on varieties or schemes (quotients)
14F35 Homotopy theory and fundamental groups in algebraic geometry
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