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 the explicit degree decomposition of the Chern–Simons form csk associated to the Chern character form chk of (E, h). Second, we introduce the so-called ascent equations that start from the (2k − 1, 0) component of csk , and use the Cholesky decomposition of the Hermitian metric h to represent the Chern–Simons form, modulo d-exact forms, as a ∂-exact form. This yields a formula for the Bott–Chern form bck of type (k − 1, k − 1) such that chk = √−1 2π ∂̄∂bck . Explicit computation is presented for the cases k = 2 and 3.