The signature of f(x, y) + z^{N}

by András Némethi ^{1} The Ohio State University, Dept. of Mathematics 231 West 18th Avenue, Columbus, OH 43210, USA.

Dedicated to C. T. C. Wall on the occasion of his 60th birthday

The basic object of this note is a surface singularity g = f(x, y) + z^{N} : (C^{3}, 0) → (C, 0), where f is an isolated curve singularity. Since its Milnor fiber is a 4–dimensional manifold, its signature is well–defined. In this note we collected some results about this signature, namely precise formulas, estimations, and some of its connections with different mathematical objects. One of them is the (real) link of g. In the appendix, we give an algorithm for the computation of its plumbing diagram (or equivalently, of the resolution graph of ({g = 0}, 0)). A general reference for the terminology is [3].

§1. The signature of f + z^{N} in terms of f and N.

Similarly to some other invariants of hypersurface germs, we would like to compute the signature of f + z^{N} in terms of some invariants of f and the integer N. For example, the Milnor number satisfies μ(f +z^{N}) = μ(f)·(N − 1), the classical Sebastiani–Thom theorem [45] for the monodromy operators gives h(f + z^{N}) = h(f) ⊗ h(z → z^{N}), and the generalized Sebastiani–Thom type theorems provide similar relations for the Seifert form (or variation map) [42], or for the spectrum and spectral pairs [43, 51]. Unfortunately, the signature is a more complicated invariant, in the sense that there is no formula which provides the signature of f + z^{N} in terms of the signature of f and a universal object depending only on N. The main reason for this is that the signature of manifolds with boundary is not additive with respect to cutting and glueing.

On the other hand, obviously f has such invariants, which together with the integer N do provide the signature σ(f + z^{N}) (e.g. since the signature is computable from the Hodge numbers [47], the Sebastiani–Thom formula for Hodge numbers [43, 51] provides such a formula). In the next subsections we will present such relations. Each of them has its own flavour.

1Partially supported by NSF Grant No. DMS-9622724 and OSU Seed Grant

1.1. The signature σ(f + z^{N}) in terms of the eta–invariant (differential topological, index theoretical point of view).

Let F be the Milnor fiber of f, H = H_{1}(F, C); b : H → H^{∗} corresponds to the intersection form ⟨, ⟩ : H ⊗ H → C via b(x)(y) = ⟨x, y⟩; h : H → H is the monodromy operator and V : H^{∗} → H is the variation map (here H_{1}(F, ∂F) is identified with the dual space H^{∗} via the perfect pair H_{1}(F) ⊗ H_{1}(F, ∂F) → C). For any natural number N we define V (N) := (I+h+⋯+ h^{N−1}) ∘ V. Then the system (H; b, h^{N}, V (N)) has a spectral decomposition ⊕_{χ}(H_{χ}; b_{χ}, (h^{N})_{χ}, V (N)_{χ}) with respect to the automorphism h^{N} (i.e. H_{χ} is the χ–generalized eigenspace of h^{N}), and the spectral decomposition is compatible with the extra-structure (b, V (N)).

For any χ we define:

η(f; N)_{χ} = \{ \begin{matrix}(1 − 2c) · signature(ib_{χ}) & if χ = e^{2πic}, 0 < c < 1; \\ −signature[(1 + (h^{N})_{χ}^{−1})V (N)_{χ}] & if χ = 1. \end{matrix}

Now, for any N > 0, the eta–invariant of f is defined by: η(f; N) = ∑_{χ} η(f; N)_{χ}. In [22, 23] it is proved that:

Theorem. With the notation introduced above:

σ(f + z^{N}) = η(f; N) − N · η(f; 1).

This relation has an index theoretical flavour: in the case of f with nondegenerate Milnor lattice, the eta–invariant defined above is exactly the eta– invariant (in the sense of Atiyah-Patodi-Singer [6]) of the signature operator of the unit circle with the flat bundle provided by the monodromy of f. This, and also its connection with “Meyer’s cocycles” and “Wall’s non–additivity formula of signature” is explained in [22, 23].

1.2. σ(f + z^{N}) in terms of the imbedded resolution graph of f (algebraic geometrical point of view).

Let f : (C^{2}, 0) → (C, 0) be as above. Consider an embedded resolution ϕ : (Y, D) → (C^{2}, f^{−1}(0)) of (f^{−1}(0), 0) ⊂ (C^{2}, 0) (here D = ϕ^{−1}(f^{−1}(0))). Let E = ϕ^{−1}(0) be the exceptional divisor and E = ∪_{w∈W}E_{w} be its decomposition in irreducible divisors. If f = ∏_{a∈A} f_{a} is the irreducible decomposition of f, then D = E ∪ ∪_{a∈A}S_{a}, where S_{a} is the strict transform of f_{a}^{−1}(0). Let G_{f} be the resolution graph of f, i.e. its vertices V = W ∐ A consist of the nonarrowhead vertices W (corresponding to the irreducible exceptional divisors), and arrowhead vertices A (correponding to the strict transform divisors of D). We will assume that no irreducible exceptional divisor has a self intersection and W ≠ 0. If two irreducible divisors corresponding to v_{1}, v_{2} ∈ V have an intersection point then (v_{1}, v_{2}) (= (v_{2}, v_{1})) is an edge of G_{f}. The set of edges is denoted by E.

For any w ∈ W, we denote by V_{w} the set of vertices v ∈ V adjacent to w. Set δ_{w} = #V_{w} for any w ∈ W. If δ_{w} > 2, then w ∈ W is called “rupture point”. The set of rupture points is denoted by R.

The graph G_{f} is decorated by the self intersection numbers e_{w} := E_{w} ·E_{w} for any w ∈ W. For any v ∈ V let m_{v} be the multiplicity of f ∘ ϕ along the irreducible divisor corresponding to v. For any e = (v_{1}, v_{2}) ∈ E, we define m_{e} := g.c.d.(m_{v_{1}}, m_{v_{2}}), and for any w ∈ W, we define M_{w} := g.c.d.(m_{v}|v ∈ V_{w} ∪ {w}).

In [25], the eta-invariant of f is computed in terms of its embedded resolution graph G_{f}. This gives:

Theorem. Let f be as above. Then: σ(f +z^{N}) = η(f; N)−N ·η(f; 1), where

η(f; K) = #A − 1 + ∑_{e∈E}((K, m_{e}) − 1) − ∑_{w∈W}((K, M_{w}) − 1)+

+4 · ∑_{w∈R} ∑_{v∈V_{w}} ∑_{k=1}^{m_{w}} ((\frac{km_{v}}{m_{w}})) · ((\frac{kK}{m_{w}})).

If the link of f + z^{N} is an integer homology sphere (equivalently, if f is irreducible, and N is relatively prime with all the Newton pairs of f), then the above theorem was proved by W. Neumann and J. Wahl (in a slightly different form) in [35].

1.3. σ(f + z^{N}) in terms of generalized Dedekind sums (arithmetical point of view).

For arbitrary non–zero integers a, b and c, the generalized Dedekind sum [39, 57] is defined as follows:

s(b, c; a) = ∑_{k=1}^{a−1}((\frac{kb}{a}))((\frac{kc}{a})),

where ((x)) is defined via the fractional part {x} as: ((x)) = {x} − 1/2 if x ∉ Z, and = 0 otherwise. Therefore, the last expression in the formula (1.2) can be replaced by:

∑_{w∈R} ∑_{v∈V_{w}} ∑_{k=1}^{m_{w}} ((\frac{km_{v}}{m_{w}})) · ((\frac{kN}{m_{w}})) = ∑_{w∈R} ∑_{v∈V_{w}} s(m_{v}, N; m_{w}).

The formula (1.2) shows that the function N ↦ σ(f + z^{N}) is a sum of a periodic function (namely η(f; N)) and of a linear function (namely −N · η(f; 1)). This fact was conjectured by A. Durfee, E. Brieskorn and D. Zagier, and first proved by W. Neumann.

Now, the famous generalization of the reciprocity law of Dedekind given by Rademacher [39] asserts that: if a, b, c are strict positive, mutually coprime integers, then: s(b, c; a) + s(c, a; b) + s(b, a; c) = −1/4 + (a^{2} + b^{2} + c^{2})/12abc. This relation for arbitrary, strict positive integers reads as:

s(b, c; a) + s(c, a; b) + s(b, a; c) = −\frac{(a, b, c)}{4} + \frac{a^{2}(b, c)^{2} + b^{2}(a, c)^{2} + c^{2}(b, a)^{2}}{12abc} .

This applied in our case (after some computation) gives the following ([26]). Theorem. For any a ∈ A, let w_{a} ∈ W be the unique vertex such that (a, w_{a}) ∈ E. Then:

σ(f + z^{N}) = (N − 1) · #W − ∑_{w∈W}((N, M_{w}) − 1) − 4 ∑_{(u,v)∈E} s(m_{u}, m_{v}; N)

+\frac{1}{3N} · ∑_{w∈W}(−e_{w})[(N, m_{w})^{2} − N^{2}] + \frac{1 − N^{2}}{3N} ∑_{a∈A} m_{w_{a}}.

Example. Assume that N = 2. Then:

σ(f + z^{2}) = #\{w ∈ W : 2 |̸M_{w}\} − \frac{1}{2}[ ∑_{{w∈W \atop 2 |̸mw}} (−e_{w}) + ∑_{a∈A} m_{w_{a}}].

1.4. σ(f + z^{N}) in terms of the spectral pairs of f (Hodge theoretical point of view).

The set of spectral pairs codifies the equivariant Hodge numbers of the mixed Hodge structure of the vanishing cohomology [47, 43]. It determines the signature (actually all the equivariant signatures) by a result of J. Steenbrink [47]. On the other hand, the set of spectral numbers is an easier invariant to handle, but unfortunately it does not determine the signature (as it is explained in [22, 24]). In the next theorem we will separate two parts, one which depends only on the spectral numbers of f, and another which involves some equivariant Hodge numbers. The second expression will vanish if f has finite monodromy.

Following [22], we set: Σp_{λ,±}(f) =

#{c| c is a spectral number of f with e^{−2πic} = λ, and (−1)^{[c]} = ±1}.

Then the general result (5.21) [22], in the plane singularity case reads as:

Theorem. σ(f + z^{N}) = η(f; N) − N · η(f; 1), where:

η(f; K) = ∑_{{λ^{K} =1 \atop λ≠1}} h_{λ}^{11}(f) − ∑_{{λ^{K} ≠1 or λ=1 \atop λ=e^{−2πic};0≤c<1}} (1 − 2\{Kc\})(Σp_{λ,−}(f) − Σp_{λ,+}(f)).

This gives η(f; K) explicitly in terms of the spectral pairs of f. The connection with the resolution graph is given by [44], where the spectral pairs are computed in terms of the embedded resolution graph G_{f}.

Notice that the expression

(1.4.1)  ∑_{e∈E}((N, m_{e}) − 1) − ∑_{w∈W}((N, M_{w}) − 1)

(used in 1.2-1.3) is exactly ∑_{λ^{N} =1,λ≠1} h_{λ}^{11}(f). But for λ ≠ 1, the Hodge number h_{λ}^{11}(f) is exactly the number of Jordan blocks of the monodromy operator h of f with eigenvalue λ and size two (in the sequel denoted by #_{2}h_{λ}(f)). If f is irreducible then this number is zero by a result of Lê [18]. The fact that these numbers provide the Jordan block structure of the monodromy of f was first proved by W. Neumann [33, 34] (see also [10]).

1.5. σ(f + z^{N}) in terms of the equivariant signatures of f (topological point of view).

In the spirit of Sebastiani–Thom type theorems, the most natural result would be to express σ(f + z^{N}) in terms of the signature of f. As we already noticed, this is not possible, but it is possible to express it (mostly) in terms of the equivariant signatures σ_{λ}(f) of f. Let r_{f} be the number of irreducible components of f (= #A). Theorem. [29]

σ(f + z^{N}) = −(r_{f} − 1)(N − 1) + ∑_{λ^{N} =1,λ≠1} #_{2}h_{λ}(f)+

−2 ∑_{{λ≠1 \atop λ=e^{2πiα}}} σ_{λ}(f) · (((Nα)) − N · ((α))).

The connection between this theorem and the previous ones is realized by Neumann’s formula of σ_{λ}(f) in terms of G_{f} [33].

Example. Let g = x^{2} + y^{3}. Then r_{f} = 1, and σ_{λ}(f) = ±1 if λ = exp(±2πi/6). Therefore:

σ(x^{2} + y^{3} + z^{N}) = −4 · (((N/6)) − N((1/6))).

1.6. The particular case of irreducible germs f.

If f is irreducible in the above relations we can make some simplifications: #A = 1, and #_{2}h_{λ}(f) = h_{λ}^{11}(f) = 0 for any eigenvalue λ. But what is really surprising, is that the signature of f + z^{N} is a sum of signatures of Brieskorn singularities. Theorem. [25] Assume that f is irreducible with Newton pairs (p_{i}, q_{i})_{i=1}^{s}. Define the integers {a_{i}}_{i=1}^{s} by:

a_{1} = q_{1}, and a_{i+1} = q_{i+1} + p_{i+1}p_{i}a_{i} if i ≥ 1.

Additionally, set d_{i} = (N, p_{i+1} ⋯ p_{s}) for 1 ≤ i < s and d_{s} = 1. Then:

σ(f + z^{N}) = ∑_{i=1}^{s} d_{i} · σ(a_{i}, p_{i}, N/d_{i}),

where σ(a, b, c) is the signature of x^{a} + y^{b} + z^{c}.

It would be very nice (and very helpful) to have a similar expression in the reducible case as well (but the author knows no such a relation in general).

The case when the link of f is an integer homology sphere was considered by Neumann and Wahl in [35]. In that case d_{i} = 1, and σ(f + z^{N}) = ∑_{i} σ(a_{i}, p_{i}, N).

1.7. The particular case of Brieskorn singularities.

For germs g = ∑_{i=1}^{3} z_{i}^{a_{i}}, E. Brieskorn’s beautiful result [7] reads as: Theorem.

σ(∑_{i=1}^{3} z_{i}^{a_{i}}) = ∑_{k_{1}=1}^{a_{1}−1} ∑_{k_{2}=1}^{a_{2}−1} ∑_{k_{3}=1}^{a_{3}−1} sign sin(π ∑_{j=1}^{3} k_{j}/a_{j}).

Hence σ is connected to the following lattice point problem. For t = 0, 1, 2 set:

S_{t} := #{k ∈ Z^{3} : 1 ≤ k_{j} ≤ a_{j} − 1, (1 ≤ j ≤ 3), t < ∑_{j=1}^{3} k_{j}/a_{j} < t + 1}.

Then: σ(∑_{j} z_{j}^{a_{j}}) = ∑_{t}(−1)^{t}S_{t}. Actually, this formula is much simpler. First notice that #{(k_{1}, k_{2}, k_{3}) ∈ Z^{3}; ∑_{j} k_{j}/a_{j} ∈ Z} = μ_{1}, where μ_{1} is the dimension of the 1–eigenspace of H_{2}(∑ z_{j}^{a_{j}}), hence by [19],

μ_{1} = \frac{(a_{1}, a_{2})(a_{2}, a_{3})(a_{3}, a_{1})}{(a_{1}, a_{2}, a_{3})} − (a_{1}, a_{2}) − (a_{2}, a_{3}) − (a_{3}, a_{1}) + 2.

Since S_{0} = S_{2} by the symmetry of the lattice points, S_{0}+S_{1}+S_{2} = μ_{≠1}(g) = (a_{1} − 1)(a_{2} − 1)(a_{3} − 1) − μ_{1}, and S_{0} − S_{1} + S_{2} = σ(g), one has:

(1.7.1) 4 · S_{0} = σ(g) + (a_{1} − 1)(a_{2} − 1)(a_{3} − 1) − μ_{1}.

Therefore, the computation of the signature is equivalent to the counting of the lattice points in the open tetrahedron T (a_{1}, a_{2}, a_{3}) with vertices (0, 0, 0), (a_{1}, 0, 0), (0, a_{2}, 0), and (0, 0, a_{3}). The latter was a famous classical problem, it was searching for a precise formula of the number of the lattice points in T (a_{1}, a_{2}, a_{3}) in terms of the numbers a_{1}, a_{2} and a_{3}. The problem was solved by L. J. Mordell [20] in the case when the numbers a_{j}’s are mutually coprime integers, and by Pommersheim [38] in general. Their formula provides S_{0} in terms of some Dedekind sums involving the a_{j}’s.

Now, notice that the results presented in (1.2) or in (1.3), applied for f = x^{a_{1}} + y^{a_{2}} and N = a_{3}, give a new proof of Mordell’s and Pommersheim’s lattice point formula.

1.8. Final remarks.

At the end of this section we mention briefly some more connections, remarks about the signature. Steenbrink computed the signature of any isolated hypersurface germ in terms of their Hodge numbers [47], and the signature of any weighted homogeneous isolated hypersurface singularity in terms of weights [48]. Brieskorn [7], and Hirzebruch-Meyer [15] present the signature as the crucial invariant which detects the exotic structures on the links of isolated singularities. We invite the reader to read Hirzebruch’s beautiful paper [14] about a global picture of the equivariant index theorems, signatures, Dedekind sums and their geometrical-arithmetical symbiosis.

A different formula for σ(f + z^{N}) is given by Ashikaga [5] who uses globalization methods, and an inductive procedure with respect to the graph G_{f}.

In the case of g = f + z^{2}, the intersection pairing of the Milnor fiber of g is the same as the symmetrized Seifert matrix of the compound torus link {f = 0} ∩ S_{ϵ}^{3} ⊂ S_{ϵ}^{3}. In [46] Shinohara presents a formula for the signature of the symmetrized Seifert matrix of a compound link of one component, hence if f is irreducible, then this provides σ(f + z^{2}) in terms of the Puiseux pairs of f. If f is not irreducible, then still exists a (tedious) process for the signature of the symmetrized Seifert form given by Murasugi [21].

§2. The signature σ(f + z^{N}) and the zero set f + z^{N} = 0.

In this section we would like to connect the smoothing invariants σ(f+z^{N}) and μ(f + z^{N}) (i.e. invariats of the Milnor fiber of f + z^{N}) to the analytic germ (X_{f,N}, 0) = ({f + z^{N} = 0}, 0). Basically, we are interested in the real link K_{f,N} of (X_{f,N}, 0) (as a topological invariant of (X_{f,N}, 0)), and in the geometric genus p_{g}(X_{f,N}) (as an analytic invariant). We present in (2.4) a formula for p_{g}. In order to give a more complete picture about g = f + z^{N} type germs, we provide the construction of the plumbing graph of K_{f,N} (or the resolution graph of X_{f,N}) in the appendix.

First notice that if g : (C^{3}, 0) → (C, 0) is an isolated singularity, then its real link K_{g}, in general, does not determine the smoothing invariants σ(g) and μ(g), nor the geometric genus p_{g}(g). E.g.: if g_{1} = x^{2} + y^{7} + z^{14} and g_{2} = x^{3} + y^{4} + z^{12}, then their minimal resolution graphs are the same (namely, it has only one vertex with self intersection −1 and genus 3; see e.g.

[16] or use §5), in particular their links are diffeomorphic. But, (μ, σ, p_{g}) in the first case is (78, −48, 9), and in the second case (66, −40, 8).

On the other hand, these numerical invariants are not independent. Let g be an arbitrary germ as above, and consider the resolution π : (Y, D) → (g^{−1}(0), 0). Let W(g) be the set of irreducible exceptional divisors. Then the canonical class (associated with π) is an element K = ∑ r_{w}E_{w} ∈ H_{2}(Y, Z) defined by the adjunction formula:

−K · E_{w} = E_{w}^{2} + 2 − 2 · genus(E_{w}) for any w ∈ W(g).

Since the intersection matrix (E_{α}·E_{β}) is negative definite, the above relations determine K. Notice that K depends on the choice of the resolution π, but the numerical invariant K · K + #W(g) is independent of π, and depends only on the link K_{g}. 2.1. Theorem. Let g : (C^{3}, 0) → (C, 0) be an isolated singularity and define the numerical invariant K · K + #W(g) from one of the resolutions of ({g = 0}, 0). Then:

(a) [Durfee] [8] 3σ(g) + 2μ(g) + 2 · dim H_{1}(K_{g}) + K · K + #W(g) = 0.

(b) [Laufer] [16] μ(g) = 12 · p_{g}(g) − dim H_{1}(K_{g}) + K · K + #W(g).

In particular, modulo the link K_{g}, the invariants μ(g), σ(g) and p_{g}(g) are equivalent.

The above theorem implies the following:

(2.2) 4p_{g}(g) = μ(g) + σ(g) + dim H_{1}(K_{g}).

This is equivalent to Durfee’s formula: 2p_{g}(g) = μ_{0}(g) + μ_{+}(g) [8] (because of dim H_{1}(K_{g}) = μ_{0}(g)).

Now, assume that g(x, y, z) = f(x, y) + z^{N}. Then μ(g) = μ(f)(N − 1), and σ(g) is given in §1 in terms of the embedded resolution graph G_{f} of f. In the appendix, we present a construction of a resolution graph of (X_{f,N}, 0) in terms of G_{f} and N. This will provide dim H_{1}(K_{g}) as well. But, even without the knowledge of this graph, we can compute dim H_{1}(K_{g}) as follows.

Fix a resolution π of (X_{f,N}, 0) as above, and set g′ = ∑_{w∈W(g)} genus(E_{w}).

2.3. Proposition. Assume that g = f + z^{N}. Then:

(a) dim H_{1}(K_{g}) = dim H_{1}(Y) = 2g′ + c_{π}, where c_{π} is the number of independent cycles of the resolution graph of π.

(b) In fact c_{π} does not depend on the choice of π: c_{π} is equal to the number of 2 × 2–Jordan blocks with eigenvalue 1 of the monodromy h(z) of z : (X_{f,N}, 0) → (C, 0); (x, y, z) ↦ z. Therefore (since h(z) = h(f)^{N}, where h(f) is the monodromy of f), the number in (1.4.1), given in terms of the graph G_{f} of f, is exactly c_{π}, i.e.:

c_{π} = ∑_{e∈E}((N, m_{e}) − 1) − ∑_{w∈W}((N, M_{w}) − 1).

(c) dim ker(h(f)^{N} − 1) = 2g′ + c_{π} + r_{f} − 1, and

(d) the generalized 1–eigenspace of h(f)^{N} is 2g′ + 2c_{π} + r_{f} − 1.

For the proof, see e.g. [30]. Notice also that the above identities give another theorem of Durfee [9], which says that the monodromy of f is finite if and only if the resolution of any cyclic covering of X_{f,N} contains no cycles.

Now, the number in (d) can be computed from the zeta function of f +z^{N} (which follows from results of Sebastiani–Thom [45] and of A’Campo’s [1]). It is 1 + ∑_{w∈W}(N, m_{w})(δ_{w} − 2). Since (b) gives c_{π}, by (a) one can determine dim H_{1}(K_{g}) in terms of the embedded resolution graph G_{f} of f. Finally, by (2.1), p_{g} can be determined from G_{f}. More precisely:

2.4. Theorem. The geometric genus p_{g} of f + z^{N}, in terms of the embedded resolution graph G_{f} of f and the integer N, is given as follows:

4p_{g} = N(2 − r_{f}) + ∑_{w∈W}(m_{w}(N − 1) + (N, m_{w}))(δ_{w} − 2)

+4 ∑_{w∈R} ∑_{v∈V_{w}}( s(m_{v}, N; m_{w}) − N · s(m_{v}, 1; m_{w}) ).

Example. If f = x^{2} + y^{3}, then p_{g}(x^{2} + y^{3} + z^{N}) = [N/6] (where [x] is the integer part of x).

The geometric genus p_{g}(f + z^{N}) can be computed in the following way as well (but the proof is not simpler). By [49, 41], p_{g}(g) =the number of spectral numbers of g situated in the interval (−1, 0]. Now, compute this using the Sebastiani–Thom type result for the spectra [43, 51], and the formula of the spectral numbers in terms of G_{f} [44].

Notice that p_{g} and μ(g) depend only on the spectral numbers of f and the integer N, but σ(g) and dim H_{1}(K_{g}) depend also on the weight filtration of the mixed Hodge structure of f. In both terms, the “guilty contribution” is c_{π} (cf. 1.4 and 2.4). Actually, c_{π} has some other interpretations too: it is the equivariant signature σ_{1}(g) and the equivariant Hodge number h_{1}^{22}(g). 2.5. Now, it is a fascinating (but difficult) problem to find special families of germs, such that for each germ g of the family its link K_{g} does determine all of the numbers μ(g), σ(g) and p_{g}(g). A very interesting example is the following.

Casson [2] defined an invariant λ(Σ) for any integer homology 3–manifold Σ (a fact generalized for rational homology spheres by Walker [53]).

2.6. Theorem. Let g be as before, and assume that K_{g} is an integer homology sphere. Then 8 · λ(K_{g}) = σ(g) in the following cases:

(a) if g is a Brieskorn singularity [Fintushel–Stern] [11];

(b) if g(x, y, z) = f(x, y) + z^{N} [Neumann–Wahl] [35].

In particular, for these singularities, the analytic invariant p_{g}(g), and the smoothing invariants μ(g) and σ(g) are determined completely by the link K_{g} only !

Actually, Neumann and Wahl in [35] characterized those pairs (f, N) for which K_{f+z^{N}} is a Z–homology sphere: f must be irreducible and (N, l) = 1 for any l ∈ {p_{1}, …, p_{s}, a_{1}, …, a_{s}}, where the integers {a_{i}}_{i} are defined in terms of the Newton pairs (p_{i}, q_{i})_{i=1}^{s} of f as in (1.6).

By the above theorem, the signature of g has a very subtle connection with the link. The starting point of the above result consisted of the following remark. The SU(2) representations of π_{1}(K(∑_{i=1}^{3} z_{i}^{a_{i}})) (where (a_{i}, a_{j}) = 1 for i ≠ j) is N/4, where N is the number of lattice points in the open tetrahedron with vertices (a_{1}, 0, 0), (0, a_{2}, 0), (0, 0, a_{3}) and (a_{1}, a_{2}, a_{3}) [12]. But N = μ(g) − 4S_{0} (see 1.7 for the notation), hence by (1.7.1) N = −σ(∑_{i=1}^{3} z_{i}^{a_{i}}).

Actually, a modulo 16 connection between the link and the signature works not only for germs g = f + z^{N}. We recall that for a Z_{2}–homology sphere Σ (i.e. H_{∗}(Σ, Z_{2}) = H_{∗}(S^{3}, Z_{2})) its μ–invariant (or Rohlin invariant) is by definition the signature σ(W^{4}) modulo 16 of any smooth compact parallelizable 4–manifold W^{4} with boundary ∂W^{4} = Σ. For an isolated germ g : (C^{3}, 0) → (C, 0), if K_{g} is a Z–homology sphere, then the intersection lattice of g is even and unimodular, hence 8|σ(g). Hence σ(g) (mod 16) codifies exactly the μ–invariant of K_{g} in 8Z/16Z (take W^{4} =Milnor fiber of g). Moreover, it is well–known that for Z–homology spheres Σ, the class of the Casson invariant λ(Σ) in Z/2Z is exactly the μ− (Rohlin) invariant of Σ.

But, even if K_{g} is only a Z_{2}–homology sphere, still 2|σ(g) (see e.g. [32]), and the μ–invariant of K_{g} considered in 2Z/16Z is the class of σ(g) in 2Z/16Z (see [32]).

§3. Inequalities.

3.1. In this section we present some inequalities satisfied by the numerical invariants σ, μ and p_{g}. We are interested basically in inequalities of type σ ≤ c_{1} · μ or p_{g} ≤ c_{2} · μ, where −c_{1} and c_{2} are some (positive, rational) numbers.

First notice that using Durfee’s formula 2p_{g} = μ_{0} + μ_{+}, the inequality p_{g} ≤ c_{2} · μ is equivalent to σ ≤ (4c_{2} − 1)μ − μ_{0}. For example, if c_{2} = 1/6, then 4c_{2} − 1 = −1/3, or if c_{2} = 1/8 then 4c_{2} − 1 = −1/2. The leading inequalities are formulated by two conjectures of Durfee [8], namely for any isolated singularity g : (C^{3}, 0) → (C, 0) (or even for any icis of dimension 2) conjecturally σ(g) ≤ 0, or an even stronger estimate holds: p_{g} ≤ μ/6 (i.e. σ ≤ −μ/3 − μ_{0}). Notice that, in general, the coefficient c_{2} = 1/6 (or c_{1} = −1/3) is the optimal one. To see this, consider the example g = ∑_{i=1}^{3} z_{i}^{d}, for which −3σ = (d − 1)(d^{2} + d − 3), μ = (d − 1)^{3}, μ_{0} = (d − 1)(d − 2). Unfortunately, we know very little about these inequalities in the general context.

On the other hand, for some particular cases we do know some nice estimates. For example, if the germ g is simple (i.e. of type A-D-E), then σ = −μ and p_{g} = 0. Moreover, if g is at the beginning of the classification list, then even its Dynkin diagram is known, hence its signature (hence p_{g} too) is explicitely computed (for some examples, see Ebeling’s book).

For weakly elliptic singularities, Xu and S. S.-T. Yau [54] proved that 12p_{g} ≤ μ + 4. For g = f(x, y) + z^{2}, Tomari proved that 8p_{g} ≤ μ [50]. For weighted homogeneous singularities with multiplicity ν, Xu and S. S.-T. Yau [55] verified (by a rather long computation) Durfee’s conjectures, actually they proved that 6p_{g} ≤ μ − ν + 1.

If g = f(x, y) + z^{N} then the negativity of the signature σ(g) is proved very shortly in [27].

3.2. Brieskorn singularities.

In this subsection we present in more details some results about the signature σ(a, b, c) of the germ x^{a} + y^{b} + z^{c}.

If a = b = c, then −3σ(a, a, a) = (a − 1)(a^{2} + a − 3). But already for a = b ≠ c, the signature is not very simple:

σ(a, a, c) = 4a · s(c, 1; a) + a − 1 − c(a^{2} − 1)/3.

(For the proof use §1.) The general inequality (for a > 0):

(3.2.1) |s(c, 1; a)| ≤ (a − 1)(a − 2)/(12a)

gives the estimate:

|σ(a, a, c) + c(a^{2} − 1)/3 − (a − 1)| ≤ (a − 1)(a − 2)/3.

If a >> c, then (3.2.1) is not very sharp, hence we replace s(c, 1; a) with s(a, 1; c) via the Dedekind reciprocity law, and we apply (3.2.1) for the latter. This gives:

|σ(a, a, c) + 1 − (2c^{2} + a^{2} + (a, c)^{2} − c^{2}a^{2})/(3c)| ≤ a(c − 1)(c − 2)/(3c).

Now, we will consider the general case of x^{a} + y^{b} + z^{c}.

Xu and S. S.-T. Yau in [56] proved for 2 ≤ a ≤ b ≤ c that:

σ ≤ −μ/3 − μ_{0} − 2(a − 1)/3

or equivalently:

6p_{g} ≤ μ − a + 1.

The author in [25] proved for 2 ≤ a ≤ b ≤ c that:

(3.2.2) σ ≤ −\frac{μ}{3} · \frac{a + 1}{a} − \frac{(c − 1)(a − 1)}{3a}

or equivalently:

(3.2.3) p_{g} ≤ \frac{1}{6}(1 − \frac{1}{2a})μ + \frac{1}{4}μ_{0} − \frac{(c − 1)(a − 1)}{12a}.

Notice that the coefficient of μ in (3.2.3) (or in 3.2.2) is better than the coefficient in Durfee’s conjecture, and for a = 2 it is exactly 1/8 as in Tomari’s result p_{g} ≤ μ/8 [50].

It is not hard to see, that all these relations imply Durfee’s conjecture p_{g} ≤ μ/6.

3.3. The g = f(x, y) + z^{N} case.

It is convenient to use the following notation: if two plane curve singularities f_{1} and f_{2} have the same topological type, then we write f_{1} ∼ f_{2}.

Let ϵ_{f} be 1 if f ∼ x^{2}+y^{n+1} (n ≥ 1), f ∼ y(x^{2}+y^{2k+1}) (k ≥ 1), f ∼ x^{3}+y^{4}, f ∼ (x^{2} + y^{2k+1})(y^{2} + x^{2l+1}) (k ≥ 1, l ≥ 1); otherwise ϵ_{f} = 0.

Theorem. [26] Assume that G_{f} is a minimal embedded resolution graph of f. Then the following inequalities hold. (For the notations, see (1.2).)

(a) If f is irreducible then

p_{g}(f + z^{N}) ≤ μ(f + z^{N})/6.

(b)

σ(f + z^{2}) ≤ −\frac{1}{2}(μ_{f} + #A − 1).

(c) For any f and N:

σ(f + z^{N}) ≤ \frac{1 − N}{N} #E + \frac{1 − N^{2}}{3N} (μ_{f} − #W − 2).

(d) There exists a constant B(f), given in terms of G_{f} (for details see [26] (3.11.b)), such that for N > B(f), one has:

σ(f + z^{N}) ≤ \frac{1 − N^{2}}{3N} (μ_{f} − 3ϵ_{f}).

(e) If (N, m_{w}) = 1 for any w ∈ W(G_{f}), then:

σ(f + z^{N}) ≤ \frac{1 − N}{N} (#W + #E) + \frac{1 − N^{2}}{3N} (μ_{f} − 3ϵ_{f});

and with the same assumption:

σ(f + z^{N}) ≤ −μ(f + z^{N})/3 or, equivalently: p_{g}(f + z^{N}) ≤ μ(f + z^{N})/6.

In some cases it is helpful to compare the signature with the multiplicity ν_{f} of f. For example, in [26] it is proved (for arbitrary f and N) that:

(3.4) σ(f + z^{N}) ≤ \frac{1 − N}{N} #E + \frac{1 − N^{2}}{3N} (ν_{f}^{2} − 2ν_{f} − 2 − #A).

Similar inequality can be found in Ashikaga’s paper [5].

Sometimes we need only the asymptotic behaviour of N ↦ σ(f + z^{N}). 3.5. Theorem. [26] The limit lim_{N→∞}(−σ_{N})/N = η(f; 1) satisfies:

η(f; 1) ≥ (μ_{f} + #W + #A − 3 + B)/3,

where B := ∑_{w∈W}(−e_{w}) − 2#W ≥ −1; and also:

η(f; 1) ≥ (3#W + #A − 8)/3.

3.6. Final remarks. All these inequalities are in subtle connection with the distribution of spectral numbers of singularities (see e.g. [40] and [27]). Actually, K. Saito in [40] presents a lot of very interesting conjectures about this distribution problem.

We mention here, that Durfee’s conjectures are not true for arbitrary smoothing of normal surface singularities [52], (and also the distribution of the spectral numbers has “pathologies” [27]).

§4. Conjectures/Open problems.

1. Generalize Theorem (1.6) for reducible germs f.

2. Generalize Theorem (2.6) for rational homology spheres.

For any f and N conjecturally the following inequalities hold: 3.

σ(f + z^{N}) ≤ −\frac{N + 1}{3N}μ(f + z^{N}).

4.

η(f; 1) + η(f; N) − η(f; N + 1) ≥ 0

(which would imply the monotonity: σ(f + z^{N+1}) ≤ σ(f + z^{N}).) 5.

η(f; 1) + η(f; N) − η(f; N + 1) ≥ μ_{f}/3,

(which would imply, for example, the inequality σ ≤ −μ/3 for f + z^{N}).

§5. Appendix.

The resolution graph G(V_{g}) of V_{g} := ({g = 0}, 0) (where g(x, y, z) = f(x, y) + z^{N}, and f is an isolated plane curve singularity) in terms of the embedded resolution graph G_{f} of f and the integer N.

I guess that the result of this appendix is known for specialists, but I have never seen it written down in this form. The idea and the method is already in the book of Laufer [17] (but actually, it is already in the “Jungian strategy”). Particular cases can be found in many places, e.g. [36, 37, 13, 4]. The proof, together with some applications, will be given in [31].

Actually, we will consider the germ z : (V_{g}, 0) → (C, 0) (induced by the projection (x, y, z) ↦ z), and we will determine the embedded resolution graph G_{z} of this germ z. Obviously, G(V_{g}) is the graph G_{z} without arrowheads and multiplicities, so it is simpler, but computing G_{z} has an advantage. Sometimes, it is easier to compute multiplicities than self intersection numbers. So, computing G_{z}, first we compute all the multiplicities of the germ z along the exceptional divisors, then (some of) the self intersection numbers are more easily computed using the relation:

(A0) e_{w}m_{w} + ∑_{v∈V_{w}(g)} m_{v} = 0 for any w ∈ W(g).

1. For any three positive integers u, v and N, with (u, v, N) = 1, we consider the unique 0 ≤ x_{1} < N/(u, N) and m_{1} ∈ N with:

(A1) v + x_{1} · \frac{u}{(u, N)} = m_{1} · \frac{N}{(u, N)}.

If x_{1} ≠ 0, then consider the continuous fraction:

\frac{N/(u, N)}{x_{1}} = k_{1} − \frac{1}{k_{2} − \frac{1}{... − \frac{1}{k_{s}}}} , k_{1}, …, k_{s} ≥ 2.

Consider the “string”:

[figure: G(u, v, N) :; (\frac{u}{(u,N)}); (\frac{v}{(v,N)}); −k_{1}; −k_{2}; −k_{s}; (m_{1}); (m_{2}); (m_{s}); ⋯]

where all vertices have genus g = 0 (i.e. they represent rational

irreducible exceptional divisors), their self intersection numbers are −k_{1}, …, −k_{s} respectively. The arrowheads have multiplicities u/(u, N) and v/(v, N), and the first vertex has multiplicity m_{1} given by (A1). Therefore, m_{2}, …, m_{s} can be easily computed using (A0), namely

−k_{1}m_{1} + \frac{u}{(u, N)} + m_{2} = 0, and − k_{i}m_{i} + m_{i−1} + m_{i+1} = 0 for i ≥ 2.

If x_{1} = 0, then the string G(u, v, N) has no vertices, it is only an edge.

2. Now, consider the embedded resolution graph G_{f} of f. (For the notations, see 1.2.)

The graph G_{z} can be considered as a “covering” q : G_{z} → G_{f}.

(a) Above w ∈ W(G_{f}) there are (M_{w}, N) vertices of G_{z}, each with multiplicity m_{w}/(m_{w}, N) and genus g̃, where:

2 − 2g̃ = \frac{(2 − δ_{w})(m_{w}, N) + ∑_{v∈V_{w}} (m_{w}, m_{v}, N)}{(M_{w}, N)}.

The vertices in q^{−1}(w) can be indexed by the group Z_{(M_{w},N)}.

(b) An edge e = (w_{1}, w_{2}) of G_{f}

[figure: (m_{w_{1}}); (m_{w_{2}})]

is covered by (m_{e}, N) copies of strings in G_{z}, each of type

G(m_{w_{1}}/(m_{e}, N), m_{w_{2}}/(m_{e}, N), N/(m_{e}, N)).

These strings can be indexed by the group Z_{(m_{e},N)}. The arrowheads of the strings are identified with the vertices q^{−1}(w_{1}), respectively q^{−1}(w_{2}), via the natural morphisms Z_{(m_{e},N)} → Z_{(M_{w_{1}},N)}, respectively Z_{(m_{e},N)} → Z_{(M_{w_{2}},N)}.

(c) An arrowhead of G_{f}

[figure: (m_{w}); (1)]

is covered by one string of type G(m_{w}, 1, N), whose right arrowhead will remain an arrowhead of G_{z} with multiplicity 1, and its left arrowhead is identified with the unique vertex above w.

(d) In this way, we obtain all the vertices, edges and arrowheads of G_{z}, and all the multiplicities (of z ∘ π along the corresponding exceptional divisors and strict transforms). Moreover, by the description of the strings (cf. part 1), one has all the self intersection numbers of the vertices which are situated on the new strings. Now, the self intersection numbers of the vertices q^{−1}(w) (w ∈ W(G_{f})) can be computed using (A0).

3. If we drop the arrowheads and multiplicities of G_{z}, we obtain G(V_{g}). The graphs G_{z} and G(V_{g}), in general, are not minimal. They can be simplified by blowing down the (−1)–rational curves with δ_{w} ≤ 2.

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