Under cohomological assumptions on the real locus, vector bundle classification on smooth real affine surfaces and threefolds mirrors the algebraically closed case, including the first example of a non-stably-free projective module with trivial Chern classes over a 3-dimensional real affine algebra.
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On the cohomological classification of vector bundles on smooth real affine surfaces and threefolds
Under cohomological assumptions on the real locus, vector bundle classification on smooth real affine surfaces and threefolds mirrors the algebraically closed case, including the first example of a non-stably-free projective module with trivial Chern classes over a 3-dimensional real affine algebra.