On Bott-Chern forms and their applications

TL;DR

This paper develops explicit formulas for Bott-Chern forms and Chern forms of trivial bundles with special metrics, demonstrating their applications in complex geometry and providing new insights into their structure.

Contribution

It introduces explicit computations of Bott-Chern forms for trivial bundles with special metrics and shows their relation to exact real forms, extending classical results in complex geometry.

Findings

Explicit formula for Bott-Chern form of a short exact sequence with a line bundle

Representation of (k,k) forms as differences of Chern character forms

Simple formula for total Chern form of a hypersurface in projective space

Abstract

We use Chern-Weil theory for Hermitian holomorphic vector bundles with canonical connections for explicit computation of the Chern forms of trivial bundles with special non-diagonal Hermitian metrics. We prove that every del-dellbar exact real form of the type (k,k) on an n-dimensional complex manifold X arises as a difference of the Chern character forms of trivial Hermitian vector bundles with canonical connections, and that (modulo the image of del and delbar) every real form of type (k,k), k<n, arises as a Bott-Chern form for two Hermitian metrics on some trivial vector bundle over X. The latter result is a complex manifold analogue of Proposition 2.6 in the paper arXiv: 0810.4935 by J. Simons and D. Sullivan. As an application, we obtain an explicit formula for the Bott-Chern form of a short exact sequence of holomorphic vector bundles, considered by Bott and Chern in classic 1965 paper, for the case when the first term is a line bundle. We also present a very simple explicit formula for the total Chern form of a hypersurface in the complex projective space.