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Comparing $$\mathbb{A}^1$$-$$h$$-cobordism and $$\mathbb{A}^1$$-weak equivalence. (English) Zbl 1401.14058
Summary: 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.
##### MSC:
 14D20 Algebraic moduli problems, moduli of vector bundles 14F42 Motivic cohomology; motivic homotopy theory 57R22 Topology of vector bundles and fiber bundles
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