Astérisque - Parutions - 328 (2009) 1-44 < Parutions < 2009 < 328

From Probability to Geometry (II). Volume in honor of the 60th birthday of Jean-Michel Bismut
Xianzhe Dai, Rémi Léandre, Xiaonan Ma, Weiping Zhang, éditeurs
Astérisque 328 (2009), xi+389 pages
Acheter l'ouvrage
Présentation, Sommaire

The signature operator on manifolds with a conical singular stratum
Jochen Brüning
Astérisque 328 (2009), 1-44

Résumé :
Opérateur de signature sur les variétés avec une strate singulière conique
Nous considèrons une variété riemannienne , qui peut être compactifiée en lui adjoignant une variété riemannienne compacte orientée, telle qu'un voisinage de la strate singulière , de codimension au moins deux, est donné par une famille de cônes métriques. Sous une hypothèse d'annulation de la cohomologie de la section du cône en dimension moitié, nous montrons qu'il existe une extension auto-adjointe naturelle de l'opérateur de Dirac agissant sur les formes qui est de spectre discret, et nous déterminons la condition sous laquelle l'opérateur de Dirac est essentiellement auto-adjoint. Nous décrivons les conditions de bord, et nous construisons une parametrix qui donne le développement asymptotique de la trace de la résolvante, comme dans un travail antérieur. Nous donnons aussi une preuve nouvelle de la formule locale pour la signature .

Mots-clefs : Opérateur de signature, extension fermée, parametrix, espace de Witt

Abstract:
We consider a Riemannian manifold, , which can be compactified by adjoining a smooth compact oriented Riemannian manifold such that a neighbourhood of the singular stratum , of codimension at least two, is given by a family of metric cones. Under the assumption that the middle cohomology of the cross-section vanishes, we show that there is a natural self-adjoint extension for the Dirac operator on forms with discrete spectrum, and we determine the condition of essential self-adjointness. We describe the boundary conditions analytically and construct a good parametrix which leads to the asymptotic expansion of a suitable resolvent trace as in our previous work. We also give a new proof of the local formula for the -signature.

Keywords: Signature operator, closed extension, parametrix, Witt space

Class. math. : 53C20

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