Chern–Simons classes for a superconnection

In this note we define the Chern–Simons classes of a flat superconnection, D+L$D+L$, on a complex Z/2Z$\mathbb{Z}/2\mathbb{Z}$-graded vector bundle E$E$ on a manifold such that D$D$ preserves the grading and L$L$ is an odd endomorphism of E$E$. As an application, we obtain a definition of Chern–Simons classes of a (not necessarily flat) morphism between flat vector bundles on a smooth manifold. An application of Reznikov's theorem shows the triviality of these classes when the manifold is a compact Kähler manifold or a smooth complex quasi-projective variety in degrees >1$>1$.