  We examine the derived heat trace asymptotics in both the real and the
complex settings for a generalized Witten perturbation. If the dimension is
even, in the real context we show the integral of the local density for the
derived heat trace asymptotics is half the Euler characteristic of the
underlying manifold. In the complex context, we assume the underlying geometry
is K\"ahler and show the integral of the local density for the derived heat
trace asymptotics defined by the Dolbeault complex is a characteristic number
of the complex tangent bundle and the twisting vector bundle. We identify this
characteristic number if the real dimension is $2$ or $4$. In both the real and
complex settings, the local density differs from the corresponding
characteristic class by a divergence term.
