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Bott-Chern currents and complex immersions. (English) Zbl 0697.58005
For a holomorphic immersion i: M\({}'\to M\) and a holomorphic vector bundle \(\mu\) on \(M'\), \(\xi\) a holomorphic vector bundle complex on M which provides a resolution of \(i_*{\mathcal O}_{M'}(\mu)\), for \(g^{\mu}\) and \(h^{\xi}\) hermitian metrics for \(\mu\) and \(\xi\), and for \(g^ N\) a hermitian metric for the normal bundle N of \(M'\) in M, a Bott-Chern current \(T(h^{\xi})\) on M is constructed. Compatibility assumptions between the metrics are made such that the current \(T(h^{\xi})\) satisfies the formula \[ ({\bar \partial}\partial /2\pi i)T(h^{\xi})=Td^{-1}(g^ N)ch(g^{\mu})\delta_{M'}-ch(h^{\xi}). \] The forms on the right hand side come from the Chern-Weil theory, and \(\delta_{M'}\) is the current on M given by integration over \(M'\). The case \(M'=\emptyset\), i.e. if \(\xi\) is acyclic, has been considered by Bott-Chern and by Donaldson. They introduced currents similar to \(T(h^{\xi})\).
Reviewer: A.Aeppli

MSC:
58A25 Currents in global analysis
32L05 Holomorphic bundles and generalizations
57R42 Immersions in differential topology
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