Comparing \(\mathbb{A}^1\)-\(h\)-cobordism and \(\mathbb{A}^1\)-weak equivalence.

*(English)*Zbl 1401.14058Summary: We study the problem of classifying projectivizations of rank-two vector bundles over \(\mathbb{P}^2\) up to two notions of equivalence that arise naturally in \(\mathbb{A}^1\)-homotopy theory, namely \(\mathbb{A}^1\)-weak equivalence and \(\mathbb{A}^1\)-\(h\)-cobordism.

First, we classify such varieties up to \(\mathbb{A}^1\)-weak equivalence: over algebraically closed fields having characteristic unequal to two the classification can be given in terms of characteristic classes of the underlying vector bundle. When the base field is \(\mathbb{C}\), this classification result can be compared to a corresponding topological result and we find that the algebraic and topological homotopy classifications agree.

Second, we study the problem of classifying such varieties up to \(\mathbb{A}^1\)-\(h\)-cobordism using techniques of deformation theory. To this end, we establish a deformation rigidity result for \(\mathbb{P}^1\)-bundles over \(\mathbb{P}^2\) which links \(\mathbb{A}^1\)-\(h\)-cobordism to deformations of the underlying vector bundles. Using results from the deformation theory of vector bundles we show that if \(X\) is a \(\mathbb{P}^1\)-bundles over \(\mathbb{P}^2\) and \(Y\) is the projectivization of a direct sum of line bundles on \(\mathbb{P}^2\), then if \(X\) is \(\mathbb{A}^1\)-weakly equivalent to \(Y\), \(X\) is also \(\mathbb{A}^1\)-\(h\)-cobordant to \(Y\).

Finally, we discuss some subtleties inherent in the definition of \(\mathbb{A}^1\)-\(h\)-cobordism. We show, for instance, that direct \(\mathbb{A}^1\)-\(h\)-cobordism fails to be an equivalence relation.

First, we classify such varieties up to \(\mathbb{A}^1\)-weak equivalence: over algebraically closed fields having characteristic unequal to two the classification can be given in terms of characteristic classes of the underlying vector bundle. When the base field is \(\mathbb{C}\), this classification result can be compared to a corresponding topological result and we find that the algebraic and topological homotopy classifications agree.

Second, we study the problem of classifying such varieties up to \(\mathbb{A}^1\)-\(h\)-cobordism using techniques of deformation theory. To this end, we establish a deformation rigidity result for \(\mathbb{P}^1\)-bundles over \(\mathbb{P}^2\) which links \(\mathbb{A}^1\)-\(h\)-cobordism to deformations of the underlying vector bundles. Using results from the deformation theory of vector bundles we show that if \(X\) is a \(\mathbb{P}^1\)-bundles over \(\mathbb{P}^2\) and \(Y\) is the projectivization of a direct sum of line bundles on \(\mathbb{P}^2\), then if \(X\) is \(\mathbb{A}^1\)-weakly equivalent to \(Y\), \(X\) is also \(\mathbb{A}^1\)-\(h\)-cobordant to \(Y\).

Finally, we discuss some subtleties inherent in the definition of \(\mathbb{A}^1\)-\(h\)-cobordism. We show, for instance, that direct \(\mathbb{A}^1\)-\(h\)-cobordism fails to be an equivalence relation.