On Bott-Chern forms and their applications
The authors use Chern-Weil theory for Hermitian holomorphic vector bundles with canonical connections for the explicit computation of the Chern forms of trivial bundles with special non-diagonal Hermitian metrics. They prove that every \(\overline{\partial}\partial\)-exact real form of 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 \(\mathrm{Im\,} \partial + \mathrm{Im\,} \overline{\partial}\) every real form of type \((k,k)\), \(k < n\), arises as a Bott-Chern form for two Hermitian metrics of some trivial bundle over \(X\). As an application, the authors obtain an explicit formula for the Bott-Chern form of a short exact sequence of holomorphic vector bundles considered by \textit{R. Bott} and \textit{S.-S. Chern} [Acta Math. 114, 71--112 (1965; Zbl 0148.31906)], for the case when the first term is a line bundle. Theorem. For every \(\overline{\partial}\partial\)-exact form \(\omega \in\mathcal A(X, {\mathbb C})\cap \mathcal A^{\mathrm{even}}(X, \mathbb R)\) on a complex manifold \(X\) there is a trivial vector bundle \(E\) over \(X\) with two Hermitian metrics \(h_{1}\) and \(h_{2}\) such that \[ \mathrm{ch}(E,h_{1}) - \mathrm{ch}(E,h_{2}) = \omega. \]
scientific article; zbMATH DE number 6357953
scientific article