Explicit computation of the Chern character forms

2014/02/25 by Leon A Takhtajan, Takhtajan, Leon A
Mathematics · Physics and Astronomy · #32L05 #32Q #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #hep-th #math.DG #msc:32L05 #msc:32Q

paper · pdf · doi:10.48550/arxiv.1402.6279

14 pages, reference added, typos corrected. New remark on Bott-Chern forms for bundles with upper-triangular transition functions added

arxiv created 2015/06/26 · arxiv updated 2015/06/29

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 explicit degree decomposition of the Chern-Simons form cs_k associated to the Chern character form ch_k of (E,h). Second, we introduce the `ascent' equations that start from the (2k-1,0) component of cs_k, and use 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 bc_k of type (k-1,k-1) such that ch_k=(√(-1))/(2π)∂∂ bc_k. Explicit computation is presented for the cases k=2 and 3.

Related

• Cech - de Rham Chern character on the stack of holomorphic vector bundles
• Chern connections and Chern curvature of the tangent bundle of almost complex manifolds
• Bott-Chern Forms and Arithmetic Intersections
• Hodge classes of Chern character forms on compact Kähler manifolds
• A note on the residue Chern character