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

Bismut, Jean-Michel; Lebeau, Gilles

C. R. Acad. Sci., Paris, Sér. I 309, No. 7, 487-491 (1989).

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

Keywords:

immersion; compact complex manifolds; holomorphic vector bundle; Hermitian metrics; Quillen metric