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The purpose of this paper is to establish an intersection formula in equivariant complex geometry, in the presence of an excess normal bundle. The contribution of the excess normal bundle to the formula appears through an additive genus K R. In a forthcoming paper, an infinite dimensional analogue of this formula will be shown to be the result of Bismut-Lebeau on the behaviour of Quillen metrics under complex immersions.

In this article a geometric process to compare spinors for different metrics is constructed. It makes possible the extension to spinor fields of a variant of the Lie derivative (called the metric Lie derivative), giving a geometric approach to a construction originally due to Yvette Kosmann. The comparison of spinor fields for two different Riemannian metrics makes the study of the variation of Dirac operators feasible. For this it is crucial to take into account the fact that the bundle in which the sections acted upon by the Dirac operators take their values is changing. We also give the formulas for the change in the eigenvalues of the Dirac operator. We conclude by giving a few cases in which an eigenvalue is stationary.

Using a link between Einstein-Sasakian structures and Killing spinors we prove a general construction principle of odd-dimensional Riemannian manifolds with real Killing spinors. In dimensionn=7 we classify all compact Riemannian manifolds with two or three Killing spinors. Finally we classify nonflat 7-dimensional Riemannian manifolds with parallel spinor fields.

For a Riemann surface with smooth boundaries, conformal (Weyl) invariant quantities proportional to the determinant of the scalar Laplacian operator are constructed both for Dirichlet and Neumann boundary conditions. The determinants are defined by zeta function regularization. The other quantities in the invariants are determined from metric properties of the surface. As applications explicit representations for the determinants on the flat disk and the flat annulus are derived.

On developpe un formalisme pfaffien en volume infini pour le modele d'Ising. On etablit une relation entre le formalisme pfaffien pour le modele d'Ising et le formalisme de la matrice de transfert