Smooth varieties up to A1-homotopy and algebraic h-cobordisms
Abstract
We start to study the problem of classifying smooth proper varieties over a field k from the standpoint of A1-homotopy theory. Motivated by the topological theory of surgery, we discuss the problem of classifying up to isomorphism all smooth proper varieties having a specified A1-homotopy type. Arithmetic considerations involving the sheaf of A1-connected components lead us to introduce several different notions of connectedness in A1-homotopy theory. We provide concrete links between these notions, connectedness of points by chains of affine lines, and various rationality properties of algebraic varieties (e.g., rational connectedness). We introduce the notion of an A1-h-cobordism, an algebro-geometric analog of the topological notion of h-cobordism, and use it as a tool to produce non-trivial A1-weak equivalences of smooth proper varieties. Also, we give explicit computations of refined A1-homotopy invariants, such as the A1-fundamental sheaf of groups, for some A1-connected varieties. We observe that the A1-fundamental sheaf of groups plays a central yet mysterious role in the structure of A1-h-cobordisms. As a consequence of these observations, we completely solve the classification problem for rational smooth proper surfaces over an algebraically closed field: while there exist arbitrary dimensional moduli of such surfaces, there are only countably many A1-homotopy types, each uniquely determined by the isomorphism class of its A1-fundamental sheaf of groups.