We show that a weighted homogeneous complex surface singularity is metrically conical (i.e., bi-Lipschitz equivalent to a metric cone) only if its two lowest weights are equal. We also give an example of a pair of weighted homogeneous complex surface singularities that are topologically equivalent but not bi-Lipschitz equivalent.
This research explores the key findings and methodology presented in the paper: Bi-Lipschitz geometry of weighted homogeneous surface singularities.
We show that a weighted homogeneous complex surface singularity is metrically conical (i.e., bi-Lipschitz equivalent to a metric cone) only if its two lowest weights are equal. We also give an example of a pair of weighted homogeneous complex surface singularities that are topologically equivalent but not bi-Lipschitz equivalent.
A natural question of metric theory of singularities is the existence of a metrically conical structure near a singular point of an algebraic set. For example, complex algebraic curves, equipped with the inner metric induced from an embedding in C N , always have metrically conical singularities. It was discovered recently (see [1]) that weighted homogeneous complex surface singularities are not necessarily metrically conical. In this paper we show that they are rarely metrically conical.
Let (V, p) be a normal complex surface singularity germ. Any set z 1 , . . . , z N of generators for O (V,p) induces an embedding of germs (V, p) → (C N , 0). The Riemannian metric on V -{p} induced by the standard metric on C N then gives a metric space structure on the germ (V, p). This metric space structure, in which distance is given by arclength within V , is called the inner metric (as opposed to outer metric in which distance between points of V is distance in C N ).
It is easy to see that, up to bi-Lipschitz equivalence, this inner metric is independent of choices. It depends strongly on the analytic structure, however, and may not be what one first expects. For example, we shall see that if (V, p) is a quotient singularity (V, p) = (C 2 /G, 0), with G ⊂ U (2) finite acting freely, then this metric is usually not bi-Lipschitz equivalent to the conical metric induced by the standard metric on C 2 .
If M is a smooth compact manifold then a cone on M will mean the cone on M with a standard Riemannian metric off the cone point. This is the metric completion of the Riemannian manifold R + × M with metric given (in terms of an element of arc length) by ds 2 = dt 2 + t 2 ds 2 M where t is the coordinate on R + and ds M is given by any Riemannian metric on M . It is easy to see that this metric completion simply adds a single point at t = 0 and, up to bi-Lipschitz equivalence, the metric on the cone is independent of choice of metric on M .
If M is the link of an isolated complex singularity (V, p) then the germ (V, p) is homeomorphic to the germ of the cone point in a cone CM . If this homeomorphism can be chosen to be bi-Lipschitz we say, following [3], that the germ (V, p) is metrically conical. In [3] the approach taken is to consider a semialgebraic triangulation of V and consider the star of p according to this triangulation. The point p is metrically conical if the intersection V ∩ B ǫ [p] is bi-Lipschitz homeomorphic to the star of p, considered with the standard metric of the simplicial complex.
Suppose now that (V, p) is weighted homogeneous. That is, V admits a good C * -action (a holomorphic action with positive weights: each orbit {λx | λ ∈ C * } approaches zero as λ → 0). The weights v 1 , . . . , v r of a minimal set of homogeneous generators of the graded ring of V are called the weights of V . We shall order them by size, v 1 ≥ • • • ≥ v r , so v r-1 and v r are the two lowest weights.
If (V, p) is a cyclic quotient singularity V = C 2 /µ n (where µ n denotes the n-th roots of unity) then it does not have a unique C * -action. In this case we use the C * -action induced by the diagonal action on C 2 .
If (V, p) is homogeneous, that is, the weights v 1 , . . . , v r are all equal, then it is easy to see that (V, p) is metrically conical. Theorem 1. If the two lowest weights of V are unequal then (V, p) is not metrically conical.
The diagonal action of C * on C 2 induces an action on C 2 /G, so they are weighted homogeneous. They are the weighted homogeneous hypersurface singularities:
x 2 + y 3 + z 4 = 0 (6, 4, 3) E 7 :
x 2 + y 3 + yz 3 = 0 (9, 6, 4) E 8 :
x 2 + y 3 + z 5 = 0 (15, 10, 6) By the theorem, none of them is metrically conical except for the quadric A 1 and possibly1 the quaternion group quotient D 4 .
The general cyclic quotient singularity is of the form V = C 2 /µ n where the n-th roots of unity act on C 2 by ξ(u 1 , u 2 ) = (ξ q u 1 , ξu 2 ) for some q prime to n with 0 < q < n; the link of this singularity is the lens space L(n, q). It is homogeneous if and only if q = 1.
Many non-homogeneous cyclic quotient singularities have their two lowest weights equal, so the converse to Theorem 1 is not generally true.
We can also sometimes distinguish weighted homogeneous singularities with the same topology from each other. Theorem 3. Let (V, p) and (W, q) be two weighted homogeneous normal surface singularities, with weights v 1 ≥ v 2 ≥ • • • ≥ v r and w 1 ≥ w 2 ≥ • • • ≥ w s respectively. If either vr-1 vr > w1 ws or ws-1 ws > v1 vr then (V, p) and (W, q) are not bi-Lipschitz homeomorphic.
We have a tentative proof that the quaternion quotient is metrically conical, see[2].
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