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.