## Immersions complexes et métriques de Quillen. (Complex immersions and Quillen metrics).(French. Abridged English version)Zbl 0681.53034

Summary: Let i be an immersion of compact complex manifolds, let $$\eta$$ be a holomorphic vector bundle on Y, let ($$\xi$$,v) be a holomorphic chain complex on X which provides a resolution of the sheaf $$i_*{\mathcal O}_ Y(\eta)$$. Let $$\lambda$$ ($$\xi)$$, $$\lambda$$ ($$\eta)$$ be the inverses of the determinants of the cohomology of $$\xi$$, $$\eta$$, and let $$\sigma \in \lambda (\eta)^{-1}\otimes \lambda (\xi)$$ be the canonical section which identifies $$\lambda$$ ($$\eta)$$ to $$\lambda$$ ($$\xi)$$. When X, Y, $$\xi$$, $$\eta$$ are equipped with Hermitian metrics, we calculate the norm of $$\sigma$$ with respect to the corresponding Quillen metric.

### MSC:

 53C55 Global differential geometry of Hermitian and Kählerian manifolds 32Q99 Complex manifolds