Jacobian variety
The same construction for complex vector bundles shows that the Chern class defines a bijection between complex line bundles over "X" and "H"2("X"; Z), because the corresponding classifying space is P∞(C), a K(Z, 2). This isomorphism is true for topological line bundles, the obstruction to injectivity of the Chern class for algebraic vector bundles is the Jacobian variety.