This article considers the existence and regularity of \KE metrics on a
compact \K manifold $M$ with edge singularities with cone angle $2\pi \be$
along a smooth divisor $D$. We prove existence of such metrics with negative,
zero and some positive cases for all cone angles $2\pi \be \leq 2\pi$. 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 $2\pi
\be < 2\pi$.

We investigate the singular sets of solutions of conformally covariant
elliptic operators of fractional order with the goal of developing
generalizations of some well-known properties of solutions of the singular
Yamabe problem.

Let $\Omega_0$ be a polygon in $\RR^2$, or more generally a compact surface
with piecewise smooth boundary and corners. Suppose that $\Omega_\e$ is a
family of surfaces with $\calC^\infty$ boundary which converges to $\Omega_0$
smoothly away from the corners, and in a precise way at the vertices to be
described in the paper. Fedosov \cite{Fe}, Kac \cite{K} and McKean-Singer
\cite{MS} recognized that certain heat trace coefficients, in particular the
coefficient of $t^0$, are not continuous as $\e \searrow 0$.