We study the problem of classifying projectivizations of rank-two vector bundles over ${ \mathbb P}^2$ up to various 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 ${ \mathbb A}^1$-$h$-cobordisms to deformations of the underlying vector bundles. Using results from the deformation theory of vector bundles we show that if $X$ is a ${ \mathbb P}^1$-bundle 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.
Categories: math.AG math.AT math.KT