Explicit computation of the Chern character forms

Abstract

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 the 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 so-called ascent equations that start from the $(2k-1,0)$ component of ${\mathrm {cs}}_{k}$, and use the 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 $\displaystyle {{\mathrm {ch}}_{k}=\frac{\sqrt{-1}}{2\pi }\bar{\partial }\partial {\mathrm {bc}}_{k}}$. Explicit computation is presented for the cases $k=2$ and 3.