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Métriques sur les fibrés d’intersection. (Metrics on intersection fiber bundles). (French) Zbl 0706.14008

Given a smooth projective morphism \(X\to S\) of dimension d of smooth schemes over \({\mathbb{C}}\), one considers the Chern classes \(c_ k(E)\) of hermitean vector bundles E on X. Using k, E as indices, let a homogeneous polynomial m of degree \( d+1\) in the variables \(c_ k(E)\) (taken with weight k) be given. Then one has an intersection line bundle \(I_{X/S}(m)\) on S and a metric on it with which its first Chern class is the integral of m along the fibres over X/S. If among the occurring E’s is a line bundle defined by a relative divisor Y on X/S which is smooth over S, one has a restriction isomorphism and a formula for the logarithm of its norm expressed as an integral along the fibres. If a short exact sequence of vector bundles is given, as well as an integer n, and now m is of degree \(d+1-n\), one has a multiplicativity isomorphism and a similar formula for the logarithm of its norm.
Reviewer: J.H.de Boer

MSC:

14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
57R20 Characteristic classes and numbers in differential topology
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