Algebraic Differential Characters

In [5], Cheeger and Simons defined on a C∞ manifold X a group of differential characters Ĥ2n (X, ℝ/ℤ), which is an extension of the global ℝ-valued closed forms of degree 2n having ℤ periods, by the group H2n-1 (X, ℝ/ℤ). (In fact, they write Ĥ2n-1 (X, ℝ/ℤ), but the notation 2n rather than (2n-1) fits better with weights in algebraic geometry). Similarly, there is a group of (complex) differential characters Ĥ2n (X,ℂ/ℤ), presented as an extension of the global ℂ-valued closed forms of degree 2n having ℤ periods, by the group Ĥ2n-1 (X,ℂ/ℤ). The group Ĥ2n (X, ℝ/ℤ) (resp. Ĥ2n (X,ℂ/ℤ) is also presented as an extension of the Betti cohomology group Ĥ2n (X, ℤ) by global ℝ-valued (resp. ℂ-valued) differential forms of degree 2n - 1, modulo the closed ones with ℤ periods.