Explicit computation of the Chern character forms

We propose a method for explicit computation of the Chern character form of a holomorphic Hermitian vector bundle $(E,h)$ over a complex manifold $X$ in a local holomorphic frame. First, we use the descent equations arising in the double complex of $(p,q)$-forms on $X$ and find explicit degree decomposition of the Chern-Simons form $\mathrm{cs}_{k}$ associated to the Chern character form $\mathrm{ch}_{k}$ of $(E,h)$. Second, we introduce the `ascent' equations that start from the $(2k-1,0)$ component of $\mathrm{cs}_{k}$, and use Cholesky decomposition of the Hermitian metric $h$ to represent the Chern-Simons form, modulo $d$-exact forms, as a $\partial$-exact form. This yields a formula for the Bott-Chern form $\mathrm{bc}_{k}$ of type $(k-1,k-1)$ such that $\mathrm{ch}_{k}=\frac{\sqrt{-1}}{2\pi}\bar{\partial}\partial\,\mathrm{bc}_{k}$. Explicit computation is presented for the cases $k=2$ and $3$.