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# Communications in Analysis and Geometry

## Volume 27 (2019)

### Number 1

### Desingularization of Lie groupoids and pseudodifferential operators on singular spaces

Pages: 161 – 209

DOI: https://dx.doi.org/10.4310/CAG.2019.v27.n1.a5

#### Author

#### Abstract

We study the integral kernels of certain natural operators on desingularization (or blown-up) spaces. A useful desingularization $\Sigma (X)$ of a singular space $X$ is obtained by successively blowing up the lowest dimensional singular strata of $X$. To study integral kernel operators on $\Sigma (X)$, we introduce and study the “desingularization” $[[\mathcal{G}: L]] \rightrightarrows [M : L]$ of a Lie groupoid $\mathcal{G}\rightrightarrows M$ along an “$A(\mathcal{G})$-tame” submanifold $L$ of its space of units $M$, where $A(\mathcal{G})$ denotes the Lie algebroid of $\mathcal{G}$. An $A(\mathcal{G})$-tame submanifold $L \subset M$ is one that has, by definition, a tubular neighborhood on which $A(\mathcal{G})$ becomes the thick pull-back Lie algebroid of an algebroid on $L$. Here $[M : L]$ denotes the usual (real) blow-up of $M$ with respect to $L$ and $M$ is obtained from $X$ by a sequence of blow-ups. (In particular, $M$ is an intermediate desingularization step between $X$ and $\Sigma (X)$.) The construction of the desingularization $[[\mathcal{G}: L]]$ of $\mathcal{G}$ along $L$ is based on a canonical fibered pull-back groupoid structure result for $\mathcal{G}$ in a neighborhood of the tame $A(\mathcal{G})$-submanifold $L \subset M$ (Theorem 3.3). Technically, this local structure result is obtained by using an integration result of Moerdijk and Mrčun (*Amer. J. Math.* 2002). Locally, the desingularization $[[\mathcal{G}: L]]$ is defined using the gauge adiabatic groupoid of Debord and Skandalis (*Advances in Math.*, 2014). The space of units of the desingularization $[[\mathcal{G}: L]]$ is $[M : L]$, the blow-up of $M$ along $L$. The desingularization groupoid $[[\mathcal{G}: L]]$ is constructed using a gluing construction of Gualtieri and Li (*IMRN* 2014). The glueing construction is applied to a groupoid that is Morita equivalent to the gauge-adiabatic groupoid and to $\mathcal{G}^{M \diagdown L}_{M \diagdown L}$, the reduction of the given groupoid to the complement of $L$. We provide an explicit description of the structure of the desingularized groupoid $[[\mathcal{G}: L]]$ and we identify its Lie algebroid, which is significant in analysis applications. We also discuss a variant of our construction that is useful for analysis on asymptotically hyperbolic manifolds. We conclude with an example discussing the groupoid associated to one of the simplest singularities, namely an edge-type singularity. The resulting groupoid is related to the so called “edge pseudodifferential calculus,” which is quite important in applications. The paper also provides an introduction to Lie groupoids for applications to analysis on singular spaces.

Received 20 January 2016

Published 7 May 2019