No abstract

This article considers the existence and regularity of Kähler-Einstein metrics on a compact Kähler manifold M with edge singularities with cone angle 2πβ along a smooth divisor D. We prove existence of such metrics with negative, zero and some positive cases for all cone angles 2πβ ≤ 2π. The results in the positive case parallel those in the smooth case. We also establish that solutions of this problem are polyhomogeneous, i.e., have a complete asymptotic expansion with smooth coefficients along D for all 2πβ < 2π.

Respectfully dedicate to Professor M. Sato on the occasion of his 70th birthday Abstract. Let X be a compact manifold with boundary. Suppose that the boundary is fibred, φ : ∂X −→ Y, and let x ∈ C ∞ (X) be a boundary defining function. This data fixes the space of 'fibred cusp' vector fields, consisting of those vector fields V on X satisfying V x = O(x 2 ) and which are tangent to the fibres of φ; it is a Lie algebra and C ∞ (X) module. This Lie algebra is quantized to the 'small calculus' of pseudodifferential operators Ψ * Φ (X). Mapping properties including boundedness, regularity, Fredholm condition and symbolic maps are discussed for this calculus. The spectrum of the Laplacian of an 'exact fibred cusp' metric is analyzed as is the wavefront set associated to the calculus.

We study the space of L 2 harmonic forms on complete manifolds with metrics of fibred boundary or fibred cusp type. These metrics generalize the geometric structures at infinity of several different well-known classes of metrics, including asymptotically locally Euclidean manifolds, the (known types of) gravitational instantons, and also Poincaré metrics on Qrank 1 ends of locally symmetric spaces and on the complements of smooth divisors in Kähler manifolds. The answer in all cases is given in terms of intersection cohomology of a stratified compactification of the manifold. The L 2 signature formula implied by our result is closely related to the one proved by Dai [25] and more generally by Vaillant [67], and identifies Dai's τ invariant directly in terms of intersection cohomology of differing perversities. This work is also closely related to a recent paper of Carron [12] and the forthcoming paper of Cheeger and Dai [17]. We apply our results to a number of examples, gravitational instantons among them, arising in predictions about L 2 harmonic forms in duality theories in string theory.

A. -In this paper we prove a variety of results about the signature operator on Witt spaces. First, we give a parametrix construction for the signature operator on any compact, oriented, stratified pseudomanifold X which satisfies the Witt condition. This construction, which is inductive over the 'depth' of the singularity, is then used to show that the signature operator is essentially selfadjoint and has discrete spectrum of finite multiplicity, so that its index-the analytic signature of X--is well-defined. This provides an alternate approach to some well-known results due to Cheeger. We then prove some new results. By coupling this parametrix construction to a C * r Γ Mishchenko bundle associated to any Galois covering of X with covering group Γ, we prove analogues of the same analytic results, from which it follows that one may define an analytic signature index class as an element of the K-theory of C * r Γ. We go on to establish in this setting and for this class the full range of conclusions which sometimes goes by the name of the signature package. In particular, we prove a new and purely topological theorem, asserting the stratified homotopy invariance of the higher signatures of X, defined through the homology L-class of X, whenever the rational assembly mapR. -Dans cet article nous prouvons plusieurs résultats pour l'opérateur de la signature sur un espace de Witt X compact orienté quelconque. Nous construisons une paramétrix de l'opérateur de la signature de X en raisonnant par récurrence sur la profondeur de X et en utilisant une analyse très fine de l'opérateur normal (près d'une strate). Ceci nous permet de montrer que le domaine maximal de l'opérateur de la signature est compactement inclus dans l'espace L 2 correspondant. On peut alors (re)démontrer que l'opérateur de la signature est essentiellement self-adjoint et a un spectre L 2 discret de multiplicité finie de sorte que son indice est bien défini. Nous donnons donc une nouvelle démonstration de certains résultats dus à Jeff Cheeger. Nous considérons ensuite le cas où X est muni d'un revêtement galoisien de groupe Γ. Nous utilisons alors nos constructions pour définir la classe d'indice de signature analytique à valeurs dans le groupe de K-théorie K * (C * r Γ). Nous généralisons dans cette situation singulière la plupart des résultats connus dans le cas où X est lisse. C'est ce qu'on appelle le « forfait signature ». En particulier, nous prouvons un nouveau théorème, purement topologique, qui permet de prouver l'invariance par homotopie stratifiée des hautes signatures de X (définies à l'aide de la L−classe homologique de X) pourvu que l'application d'assemblement rationnelle K * (BΓ) ⊗ Q → K * (C * r Γ) ⊗ Q soit injective.