We can use this detailed description of the link $Y$, in the three-dimensional case to get a more precise understanding of the “topological obstruction” of Section 3.2.2. A representation $\alpha:\pi_{1}(Y\setminus\Sigma)\rightarrow S^{1}$ defines a covering of $Y\setminus\Sigma$ and it is clear that the metric completion of this is again an orbifold $\tilde{Y}$ with a metric of Ricci curvature $(2n-1)$. It is clear then that the usual proof of Myers Theorem extends to show that $\tilde{Y}$ is compact, so the representation maps to a finite group. Thus $\pi_{1}(Y\setminus\Sigma)$ is also finite and the torus $T$ in the discussion of 3.2.2 is in this case trivial. (Of course the set $Y_{\epsilon}$ can be assumed to be homotopy equivalent to $Y\setminus\Sigma$). Moreover it is also clear that the usual proof of the Bishop Theorem extends to this case to show that the volume of $\tilde{Y}$ cannot exceed that of $S^{2n-1}$. Hence the order of the cover, is bounded by $\kappa^{-1}$ where $\kappa$ is the volume ratio, and hence by $c^{-1}$. Let $D=D(c)$ be the least integer such that all integers less than or equal to $c^{-1}$ divide $D$. Then we see that the power $\alpha^{D}$ of any such representation must be trivial. Thus if, from the beginning of the discussion in Section 3, we consider powers $L^{Dk}$ we never encounter the topological obstruction. The point here of course is that $D$ is determined in a simple explicit way by $c$ which in turn, in the Fano case, is known explicitly. In many practical cases of interest $D$ is not too large.

We expect that in fact the same will be true in higher dimensions (with the same $D(c)$). Of course we do not expect that the singularities will always be of orbifold type, but it seems likely that the Bishop theorem can still be extended to the metric completion of a covering, as above. There is a slightly weaker statement which should be easier to prove. Let $y$ be a point in the singular set $\Sigma_{Y}$ of a $(2n-1)$-dimensional link $Y$. Let $B$ be a sufficiently small ball about $y$ and $B^{{\rm reg}}\subset B$ the regular set. Suppose that we have found a number $E$ such that for all such points (in all tangent cones of all limits of manifolds in ${\mathcal{K}}(n,c,V)$) the homology group $H_{1}(B^{{\rm reg}},{\mathbb{Z}})$ has order bounded by $E$. Let $\alpha$ be a representation of $\pi_{1}(Y\setminus\Sigma)$ as above. Then in the covering defined by $\alpha^{E}$ the pre-image of $B^{{\rm reg}}$ is a disjoint union of copies of $B^{{\rm reg}}$. In this situation it is straightforward to apply recent results of Colding and Naber [7] to show that the regular set in the metric completion $\tilde{Y}$ is geodesically convex, and then to extend the Bishop argument to this case. Then we see that if, from the beginning of the discussion in Section 3, we consider powers $L^{DEk}$ then we never encounter the topological obstruction. Arguing by induction on dimension it seems likely that in fact the number $E=D^{n-2}$ will have the property stated above so, for this weaker statement, we would consider powers $L^{D^{n-1}k}$. But, in fact it seems to us most likely that these higher powers of $D$ are not required.

In this direction we make the following conjecture, which (if true) would be a substantial sharpening of Theorem 1.1.

To put this in context, recall that for a fixed $X$ the standard asymptotics is $\rho_{k,X}\sim(2\pi)^{-n}k^{n}$ as $k\rightarrow\infty$. This essentially follows from the fact that on $\mbox{${\mathbb{C}}$}^{n}$ we have $\rho=(2\pi)^{-n}$. The conjectural lower bound here is a uniform version of this over ${\mathcal{K}}(n,c,V)$, provided we work over multiples of $D$. On the other hand the corresponding upper bound—$\rho_{kD,X}\leq\eta^{-1}(2\pi)^{-n}(kD)^{n}$—almost certainly fails, because at the vertex of a cone $C(Y)$ we have $\rho=\kappa^{-1}(2\pi)^{-n}$ where $\kappa\geq c$ is the volume ratio. This is why we believe that the plausible upper bound should include the extra factor $c^{-1}$. In a similar way, if in fact we do encounter the topological obstruction of 3.2.2 in some limit space, then it seems it would not be true that there is a lower bound on $\rho_{k,X}$ for all sufficiently large $k$, since the twisting of the line bundle will force $\rho$ to be small as we approach the singularity. This phenomenon—that near to a singularity $\rho$ gets larger or smaller depending on divisibility—is similar to the orbifold situation considered by Ross and Thomas in [17].