## 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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### References:

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