by Jonathan Wahl
It has been my great pleasure to collaborate on many papers with Walter Neumann, a terrific mathematician and friend. I’ll try to present an overall account of some of our work, primarily on splice diagrams and splice-quotient singularities. Our goal was to be able to write down explicit equations of certain kinds of normal surface singularities, given only the topology of their links. I mention our motivations, and why and how we were led to certain kinds of questions, leaving detailed proofs (and careful definitions) to the original papers. One can also consult interesting more recent work of others, such as [e13], which uses a tropical geometry approach.
1. Provocative early work of Walter
While I first became friendly with Walter in 1972 at the Institute for Advanced Study, our collaboration did not begin until 1987, well after he had written two papers about normal surface singularities which I found very provocative.
Recall that if
In fact, with the exception of lens spaces (from cyclic quotient
singularities) and some two-torus bundles over the circle (from cusp
singularities), already
The theorem was proved by a calculus for general plumbing diagrams
(not necessarily negative-definite), and implied other basic results
(e.g.,
A later result of Walter’s involved certain singularities with a
rational homology sphere (or
A weighted-homogeneous singularity with
Here, the continued fraction expansion
(We shall assume that
The matrix
I found this a very surprising result! It is easy to show that
on and then to
prove that (for judiciously chosen
Attempts to find a natural general statement which includes this theorem led years later to our definition of splice-quotient singularities in [6] (see Section 5).
2. Casson invariant and splice diagrams
In 1987 I attended a
lecture given by
Sir Michael Atiyah
at the Weyl Centennial Conference
at Duke University, entitled “New invariants of three and
four-manifolds”
[e6].
Andrew Casson
had discovered a new integer
invariant
First,
- The Brieskorn–Pham complete
intersection
, with pairwise relatively prime . - The singular germ
, where is an irreducible plane curve singularity and is relatively prime to all the numbers among the Puiseux pairs. - The complete intersection singularity in
defined by
The method was correct but mathematically disappointing: we compute
and compare the Casson invariant and signature in each case. Walter
realized that the key was the use of splice diagrams,
introduced earlier by
Siebenmann,
and extensively studied by
Eisenbud–Neumann
[3].
Splicing refers to a different
topological construction than plumbing, allowing one to make a new
A splice diagram
The next example is the diagram from splicing
For an edge connecting two nodes in a splice diagram the edge
determinant is the product of the two weights on the edge minus the
product of the weights adjacent to the edge. In the above example, the
one edge connecting two nodes has edge determinant
- The weights around a node are positive and pairwise coprime.
- The weight on an edge ending in a leaf is greater than 1.
- All edge determinants are positive.
From the resolution diagram for a homology sphere singularity link, one associates a splice diagram as follows: disregard all vertices of valency 2; at each node and outgoing edge, take the absolute value of the determinant of the outer diagram. For example, the splice diagram above arises from the resolution diagram shown in Figure 5.
There are methods to compute the resolution diagram from the splice diagram; see a general discussion of plumbing versus splicing in the Appendix to [6].
One reason to consider splice diagrams for the CIC is that the Casson
invariant is known to be additive under splicing. All
One consequence of the CIC would be that for a complete intersection
singularity with
3. Singularities of splice type
After the completion in 1989 of our paper on the Casson invariant, Walter (and I) pursued other topics in the 1990s, he working primarily on hyperbolic 3-manifolds and geometric group theory. But in 1998 Walter spent a semester at Duke University, at which time we resumed our discussion of two big questions (beyond the CIC) left open for us from [4].
The first was the dearth of explicit examples of hypersurface and
complete intersection singularities with
Second, since the Casson invariant adds under splicing, perhaps the CIC is true because one can “splice” singularities and their Milnor fibers, matching splicing on the link level, and so that on the Milnor fiber level the signatures add. Both of these issues were eventually discussed in [6] and [7].
A splice diagram with a single node and
Consider now the two-node splice diagram in Figure 6, where we have already added a variable to each leaf.
Recall that
Since
In analogy with the one-node case, we have first assigned a variable
to each leaf. Next, to each node we associate a weight which is the
product of the weights of the surrounding edges (so,
Thus, with the weights from the left node, the first equation is
weighted homogeneous, while the second is not. Still, the associated
graded is an integral domain, and its normalization is the singularity
One can similarly write down a system of equations for an arbitrary
splice diagram
One calls these singularities of splice type. The proof is by induction on the number of nodes; the key step starts with an “end node” (all but one of whose emanating edges is a leaf), takes a weighted blow-up, and uses inductively the result for a subdiagram. Again, setting a variable equal to 0 gives a monomial curve corresponding to a knot in the link.
How general are singularities of splice type among singularities with
- There exists a Gorenstein singularity, not a complete intersection,
whose link is the Brieskorn sphere
. - There exists a Gorenstein singularity, not a complete intersection, whose link is a homology sphere but which does not satisfy the semigroup conditions.
An analytic condition distinguishes singularities of splice type
from other singularities with the same link. Each leaf of the splice
(or resolution) diagram gives a knot in
satisfies the semigroup condition; is a complete intersection of embedding dimension less than or equal to t; generate the maximal ideal of the local ring of at 0, and is a complete intersection of splice type with respect to these generators.
Thus, some natural open questions are as follows:
- Does there exist a complete intersection singularity for the two-node splice diagram based on
, where the semigroup condition fails? - Is every complete intersection with
HS link of splice type? - Assuming there are complete intersections with
HS link that are not of splice type, do they satisfy the Casson Invariant Conjecture?
4. Casson Invariant and Milnor Fiber Conjectures
What about the CIC for these singularities of splice type? It was
realized early on that it is easier to restate it in terms of the
geometric genus
As indicated previously, the validity of the CIC for singularities of splice type suggests that there is a construction of “splicing Milnor fibers” for which the additivity of the signatures is geometrically clear.
Suppose the equations
Let
For the knot
Denote
The construction of
In
[7],
Theorem 8.2,
the Milnor Fiber Conjecture is verified
for the previously discussed singularities
5. Splice-quotient singularities
In 2000, as we worked through the basics of singularities of splice
type, we began to look for generalizations of the unexpected result of
Theorem 1.2, that the UAC of a weighted-homogeneous singularity with
From the point of view of classification of singularities, the easiest
examples not covered by Theorem 1.2 are the quotient-cusps.
These are singularities whose resolution graphs have the form
shown in Figure 7 for
They are rational singularities, log-canonical, and taut, i.e, the topology of the link determines the analytic type. The above quotient-cusp is double-covered by the cusp singularity whose resolution graph is shown in Figure 8.
It follows that the universal abelian cover is also a cusp.
It is easy to determine, given the resolution graph, when a cusp is a complete intersection; and, this was the case for the UAC for the few examples we checked. However, it required a detailed and lengthy analysis of the unimodular matrix classifying the quotient-cusp to prove the following.
Though we wrote down equations and group action for the UAC, at the time (2001) we’d not yet formulated a general method for doing so, and our result seemed ad hoc.
Given the resolution diagram
Nonetheless, the same approach as above in the in where
When
A difficulty is that splice diagram equations depend only on the
splice diagram
- The homomorphism
defined by induces an injection - Exponentiating each
via provides a faithful diagonal representation .
Now, assume
Clearly, the semigroup plus congruence conditions together mean that
the discriminant group acts on appropriate splice diagram complete
intersection singularities (higher order terms must also be
- Splice diagram equations for
define an isolated complete intersection singularity . - The discriminant group
acts freely on a punctured neighborhood of 0 in . has an isolated normal surface singularity, and a good resolution whose associated dual graph is . is the universal abelian covering. maps the curve to an irreducible curve, whose proper transform on the good resolution of is smooth and intersects the exceptional curve transversally, along . In fact the function , which is -invariant and hence defined on , vanishes to order on this curve.
A singularity
Of course, there are many things to be checked! The proof is by induction on the number of nodes, and one needs to consider certain nonminimal resolutions.
We conjectured that rational surface singularities were splice quotients; but the first proof of this fact was due to T. Okuma, who combined the semigroup and congruence conditions in a novel way.
Eventually, we were able to prove a general
result, which included
Okuma’s, identifying the crucial analytic property of a
splice-quotient
We close with a caveat: As indicated in Theorem 10.1 of
[6],
the general splice type equation can be found using any choice of
admissible monomials, as long as one allows the addition of higher
order equivariant terms. This implies that in the
Jonathan Wahl is Emeritus Professor of Mathematics at the University of North Carolina at Chapel Hill. His collaboration with Walter Neumann for over 30 years resulted in seven published papers.