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Equivariant immersions and Quillen metrics. (English) Zbl 0826.32024

The author and G. Lebeau [Publ. Math., Inst. Hautes Etud. Sci. 74, 1-297 (1991; Zbl 0784.32010)] proved an integral formula being an important step in the proof of Gillet and Soulé of the Riemann-Roch Theorem in Arakelov geometry. In the present paper the author extends the formula to the equivariant case. More exactly he constructs Quillen metrics on the equivariant determinant of the cohomology of a holomorphic vector bundle with respect to the action of a compact group \(G\) and he calculates the behaviour of the equivariant Quillen metric by immersions. The author considers his result as a new block in the construction of an Arakelov theory to be in an equivariant context. At the beginning of the paper the author describes the complications which arise in generalising the techniques and arguments from the previous paper to the present case.
The results of this paper were announced in C. R. Acad. Sci., Paris, Sér. I 316, No. 8, 827-832 (1993; Zbl 0773.53026).

MSC:

32L10 Sheaves and cohomology of sections of holomorphic vector bundles, general results
32C30 Integration on analytic sets and spaces, currents
32J18 Compact complex \(n\)-folds
32M05 Complex Lie groups, group actions on complex spaces
53C56 Other complex differential geometry
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