tailieunhanh - Đề tài " Equivariant de Rham torsions "

The purpose of this paper is to give an explicit local formula for the difference of two natural versions of equivariant analytic torsion in de Rham theory. This difference is the sum of the integral of a Chern-Simons current and of a new invariant, the V -invariant of an odd dimensional manifold equipped with an action of a compact Lie group. | Annals of Mathematics Equivariant de Rham torsions By Jean-Michel Bismut and Sebastian Goette Annals of Mathematics 159 2004 53 216 Equivariant de Rham torsions By Jean-Michel Bismut and Sebastian Goette Abstract The purpose of this paper is to give an explicit local formula for the difference of two natural versions of equivariant analytic torsion in de Rham theory. This difference is the sum of the integral of a Chern-Simons current and of a new invariant the V-invariant of an odd dimensional manifold equipped with an action of a compact Lie group. The V-invariant localizes on the critical manifolds of invariant Morse-Bott functions. The results in this paper are shown to be compatible with results of Bunke and also our with previous results on analytic torsion forms. Contents Introduction 1. The classical equivariant de Rham torsion 2. The Chern equivariant infinitesimal analytic torsion 3. Equivariant fibrations and the classes VK M S 4. Morse-Bott functions multifibrations and the class VK M S 5. A comparison formula for the equivariant torsions 6. A fundamental closed form 7. A proof of the comparison formula 8. A proof of Theorem 9. A proof of Theorem 10. A proof of Theorem 11. A proof of Theorem 12. A proof of Theorem References Jean-Michel Bismut was supported by Institut Universitaire de France . . Sebastian Goette was supported by a research fellowship of the Deutsche Forschungsgemeinschaft . . 54 JEAN-MICHEL BISMUT AND SEBASTIAN GOETTE Introduction In a previous paper BGol we have established a comparison formula for two natural versions of the holomorphic equivariant analytic torsion. This comparison formula is related to a similar formula obtained in Go for -invariants. In this paper we establish a corresponding formula where we compare two natural versions of equivariant analytic torsion in de Rham theory. On one hand the classical equivariant version LoRo of the Ray-Singer analytic torsion RS appears. On the other hand

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