Explicit computation of the Chern character forms

Authors: Leon A Takhtajan

Publish Date: 2015/10/15

Volume: 181, Issue: 1, Pages: 223-237

PDF Link

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 qforms 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 socalled ascent equations that start from the 2k10 component of mathrm cs k and use the Cholesky decomposition of the Hermitian metric h to represent the Chern–Simons form modulo dexact forms as a partial exact form This yields a formula for the Bott–Chern form mathrm bc k of type k1k1 such that displaystyle mathrm ch k=fracsqrt12pi barpartial partial mathrm bc k Explicit computation is presented for the cases k=2 and 3

Keywords: