On the KO-orientability of complex projective varieties (Q1061796)

\textit{P. Baum}, \textit{W. Fulton} and \textit{R. MacPherson} [Publ. Math., Inst. Hautes Étud. Sci. 45, 101-145 (1975; Zbl 0332.14003)] constructed a Riemann-Roch homomorphism \(\alpha_ 0: K_ 0^{alg}(X)\to K_ 0^{top}(X)\) for a complex, quasi-projective variety, X. Here \(K_ 0^{alg}\) is the Grothendieck group of coherent algebraic sheaves on X and \(K_ 0^{top}\) is topological, unitary K-homology theory. The image of the structure sheaf, \(\alpha_ 0({\mathcal O}_ X)\), gives X a KU- orientation. By studying the Bott sequence in homology \(...\to KO^{top}_{n-1}(X)\to KO_ n^{top}(X)\to K_ n^{top}(X)\to^{\gamma}KO^{top}_{n-2}(X)\) the author gives, with illustrative examples, a criterion for X to be orientable with respect to real K-theory, \(KO_ X^{top}\).