Abstract:
The Chow-Witt group is a kind of cohomology theory of smooth varieties which encodes information of singular cohomology of both real and complex points. J. Fasel computed the Chow-Witt group of projective spaces but that of projective bundles was still open. The main subtlety is that there is no proper definition of Chern classes on Chow-Witt rings, as in the case of real vector bundles and integral coefficients.
Recently, B. Calmès, F. Déglise and J. Fasel defined a kind of motivic theory representing the Chow-Witt groups, namely the Milnor-Witt motives. It's rationally equivalent to the motivic stable homotopy category defined by F. Morel. In this talk, we compute the MW-motive of projective bundles thus answer the question above. This in particular computes the Chow-Witt group of blow-ups.
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