THE COHOMOLOGY OF LINE BUNDLES OF SPLICE-QUOTIENT SINGULARITIES

ANDRÁS NÉMETHI

Abstract. We provide several results on splice-quotient singularities: (1) a combinatorial expression of the dimension of the first cohomology of all ‘natural’ line bundles (involving the Seiberg-Witten invariants of the singularity link); (2) an equivariant Campillo–Delgado–Gusein-Zade type formula about the dimension of relative sections of line bundles, extending former results about rational and minimally elliptic singularities; (3) in particular, we prove that the equivariant, divisorial multi-variable Hilbert–Poincaré series is topological; (4) a combinatorial description of divisors of analytic function-germs;

(5) and an expression for the multiplicity of the singularity from its resolution graph (in particular solving Zariski’s Multiplicity Conjecture for splice-quotient hypersurfaces).

From topological point of view, we get a new combinatorial expression for the Seiberg-Witten invariants of links of splice-quotient singularities.

1. Introduction

Splice-quotient singularities were introduced by Neumann and Wahl and became the subject of an intense mathematical activity [21, 22, 23, 24, 19, 34, 25, 26, 27, 18, 17, 3, 31]. In their definition, one starts with a resolution graph Γ which is a tree, and the genera of all the vertices are zero, and it also satisfies an additional combinatorial restriction. From Γ one constructs an equisingular family of singularities, as analytic realizations supported by the topological type fixed by Γ. Rational and minimally elliptic singularities [26] and weighted-homogeneous singularities [20] are examples of splice-quotients.

In this article we settle some basic results regarding these singularities. The fact that they are constructed from a combinatorial object, gives hope that their basic discrete analytic invariants can be represented from the topology (hence are independent of the analytic moduli). The present article unifies apparently different research directions working on these type of questions for different invariants.

First, we show that the equivariant divisorial multi-variable Hilbert-Poincaré series [6, 14] (see also [5, 4] for the non-equivariant version) is topological. In order to do this, we prove the equivariant Campillo–Delgado–Gusein-Zade type formula for splice-quotients, cf. Theorem 5.1.1(a). This identifies a series P, which codifies the same information as the Hilbert series itself, with a combinatorially defined ‘zeta function’. (Here, for the equivariant divisorial Hilbert series and for the formula we adopt the formalism established in [14].)

Such a formula was proved by Campillo, Delgado and Gusein-Zade for rational singularities [4, 6], by a different method it was reproved for rational and extended for minimal elliptic singularities in [14], but it was unknown (at least by the author) even for weighted-homogeneous singularities. For such a singularity with a ℂ^{∗}- action, its reduction to a one-variable identity (corresponding to the valuation of the ‘central’ divisor) is the celebrated Pinkham-Dolgachev-Neumann formula, which identifies the (ℂ^{∗}-equivariant) Poincaré series with a topologically defined zetafunction [28, (5.7)], [20].

2000 Mathematics Subject Classification. Primary 32S05, 32S25, 57M27; Secondary 32S45, 32S50, 32C35, 57R57.

Key words and phrases. splice-quotient singularities, normal surface singularities, resolutions of singularities, Hilbert series, Poincaré series, divisorial multi-filtration, cohomology of line bundles, multiplicity, plumbed 3-manifolds, low-dimensional topology, Seiberg-Witten invariants.

In fact, for splice-quotients, besides the divisorial multi-filtration, the so called monomial multi-filtration appears naturally too. It is induced by the combinatorics of Γ, and guides most of the geometry of the singularity. It generalizes the weightdecomposition of the monomials in the case of weighted-homogeneous singularities. Here we also show that the monomial and the divisorial multi-filtrations agree, see (6.1.2). This contains Okuma’s result [27, (3.3)] as a special case.

This Hilbert-Poincaré series, surprisingly enough, can be connected with the (equivariant) Seiberg-Witten Invariant Conjecture for ‘natural’ line bundles L (see [16, 11] for the conjecture and [12] for related results). This provides a topological description of the dimensions h^{1}(L) of the cohomology groups (including the geometric genus corresponding to the trivial line bundle) involving the Seiberg-Witten invariant of the link, cf. Theorem 5.1.1(b). Although [3] provides a proof of the conjecture for splice-quotients, that version, as the original conjecture itself, contains a restriction regarding the Chern class of the involved line bundles. Without this restriction, the topological candidate for h^{1}(L), even at conjectural level, was not known. Here we treat the complete general case without any restriction on the Chern class. The ‘correction term’ needed for arbitrary Chern class is provided by a ‘truncated part’ of the combinatorial expression appearing in Campillo–Delgado– Gusein-Zade formula. As a by-product, we get a new combinatorial expression for the Seiberg-Witten invariants of links of splice-quotient singularities, see 6.2.

In the last section we characterize from the graph the local divisors of analytic functions (and natural line bundle sections) for splice-quotient singularities. For general singularities this is a very difficult open task, the corresponding semi-group might depend in a very subtle way on the analytic type of the germ. Nevertheless, for splice-quotients, this semi-group is determined combinatorially (see Theorem 7.1.2), and its saturation is the ‘Lipman semi-group’, cf. Theorem 7.1.4). Moreover, we determine topologically the multiplicity of the singularity too, cf. Theorem 7.3.3. In particular, we answer positively Zariski’s Multiplicity Conjecture for all splicequotients.

2. Notations and motivations

2.1. General surface singularities. Let (X, o) be a complex normal surface singularity whose link is a rational homology sphere. Let π : X̃ → X be a good resolution with dual graph Γ whose vertices are denoted by V. Recall that the link is a rational homology sphere if and only if Γ is a tree and all the irreducible exceptional divisors have genus 0.

Set L := H_{2}(X̃, ℤ). It is freely generated by the classes of the irreducible exceptional curves {E_{v}}_{v∈V}. They will also be identified with the integral cycles supported on E = π^{−1}(o). If L′ denotes H^{2}(X̃, ℤ), then the intersection form ( , ) on L provides an embedding L ↪ L′ with factor the first homology group H of ∂X̃ (or, of the link). Moreover, ( , ) extends to L′. L′ is freely generated by the duals E_{v}^{∗}, where we prefer the convention (E_{v}^{∗}, E_{w}) = −1 for v = w, and = 0 otherwise.

Effective classes l = ∑ r_{v}E_{v} ∈ L′ with all r_{v} ∈ ℚ_{≥0} are denoted by L′_{≥0}, and L_{≥0} := L′_{≥0} ∩ L. Denote by S′ the (Lipman’s) anti-nef cone {l′ ∈ L′ : (l′, E_{v}) ≤ 0 for all v}. It is generated over ℤ_{≥0} by the base-elements E_{v}^{∗}. Since all the entries of E_{v}^{∗} are strict positive, S′ is a sub-cone of L′_{≥0}, and for any fixed a ∈ L′ the set

(2.1.1)  {l′ ∈ S′ : l′ ≱ a} is finite.

Set Q := {∑ l′_{v}E_{v} ∈ L′, 0 ≤ l′_{v} < 1}. For any l′ ∈ L′ write its class in H by [l′], and let r_{[l′]} ∈ L′ be its unique representative in Q with l′ −r_{[l′]} ∈ L. Finally, denote by θ : H → Ĥ the isomorphism [l′] ↦ e^{2πi(l′,·)} of H with its Pontrjagin dual Ĥ.

Next, we list three major motivations of our investigation.

2.2. First motivation: cohomology of line bundles. Most of the analytic geometry of X̃ (hence of (X, o) too) is described by its line bundles and their cohomology groups. One has the following basic goal (below the cohomology groups are considered on X̃):

For any L ∈ Pic(X̃) and effective cycle l ∈ L_{≥0} recover the dimensions

(2.2.1)  (a) dim \frac{H^{0} (L)}{H^{0} (L (−l))} and (b) dim H^{1}(L)

from the combinatorics of Γ (for some families of singularities).

Thanks to a recent intense activity, there is an increasing optimism to understand this problem for ‘special’ line bundles. They are provided by the splitting of the cohomological exponential exact sequence summarized below, cf. [11, §3], [25, (2.2)]:

\xymatrix{ &  &  & L \ar[d] \ar@{-->}[dl] \\ 0 \ar[r] & H^{1}(X̃, OX̃) \ar[r] & Pic(X̃) \ar[r]^{c_{1}} & L′ \ar[r] \ar@{-->}[l]^{O} & 0.}

The first Chern class c_{1} is surjective and it has an obvious section on the subgroup L: it maps every element to its associated line bundle. This section has a unique extension O to L′. We call a line bundle natural if it is in the image of this section.

From the above splitting, one has the next characterization: a line bundle L is natural if and only if there exists n ∈ ℤ_{>0} such that L^{⊗n} is of ‘classical type’ O(l) for some divisor l ∈ L (i.e. supported by E).

Natural line bundles appear in other geometric constructions as well. Let c : (Y, o) → (X, o) be the universal abelian cover of (X, o), π_{Y} : Ỹ → Y the normalized pullback of π by c, and c̃: Ỹ → X̃ the morphism which covers c. Then the action of H on (Y, o) lifts to Ỹ and one has an H-eigenspace decomposition ([11, (3.7)] or [27, (3.5)]):

(2.2.2)  c̃_{∗}OỸ = ⨁_{l′∈Q} O(−l′).

Furthermore, O(−l′) (l′ ∈ Q) in the decomposition (2.2.2), is the θ([l′])-eigenspace of c̃_{∗}OỸ. This is compatible with the eigenspace decomposition of O_{Y,o} too.

Therefore, H^{0}(X̃, O(−l′)) is (and in this article will be regarded as) the θ([l′])- eigenspace of the ℂ-vector space of meromorphic functions 𝔰 on Ỹ with div(𝔰) ≥ c^{∗}(l′) (where, in fact, c^{∗}(l′) is an integral cycle supported on π_{Y}^{−1}(o) by [11, (3.3)]).

Part (2.2.1)(a), even for particular line bundles, is crucial in several geometric problems; next we list two of them.

2.3. Second motivation: equivariant Hilbert series. For those natural line bundles which appear in (2.2.2), the dimensions from (2.2.1)(a) can be organized in a generating function. Indeed, once a resolution π is fixed, O_{Y,o} inherits the divisorial multi-filtration (cf. [14, (4.1.1)]):

(2.3.1)  F(l′) := {f ∈ O_{Y,o} | div(f ∘ π_{Y}) ≥ c^{∗}(l′)}.

Let 𝔥(l′) be the dimension of the θ([l′])-eigenspace of O_{Y,o}/F(l′). Then, one defines the equivariant divisorial Hilbert series by

(2.3.2)  H(t) = ∑ ; l′=∑ l_{v}E_{v}∈L′ 𝔥(l′)t_{1}^{l_{1}} ⋯ t_{s}^{l_{s}} = ∑_{l′∈L′} 𝔥(l′)t^{l′} ∈ ℤ[[L′]].

We are aware that in H(t) many coefficients are repeated several times – cf. e.g. (3.1.1), and also that the ‘sub-series’ ∑_{l′∈L′_{≥0}} 𝔥(l′)t^{l′} contains the same information and additionally it lives in a ring of power series instead of the ℤ[L′]-module ℤ[[L′]] as H(t) does. However, we prefer the definition (2.3.2) because of the concise relation (2.4.1).

Notice that the terms of the sum reflect the H-eigenspace decomposition too: 𝔥(l′)t^{l′} contributes to the θ([l′])-eigenspace. For example, ∑_{l∈L} 𝔥(l)t^{l} corresponds to the H-invariants, hence it is the Hilbert series of O_{X,o} associated with the π^{−1}(o)- divisorial multi-filtration (considered and intensively studied, see e.g. [5] and the citations therein, or [4]). The second central problem is:

Recover (for some families of singularities) H(t) from Γ.

2.4. Third motivation: Campillo–Delgado–Gusein-Zade identity. Comparison of H with topological invariants lead Campillo, Delgado and Gusein-Zade to consider the next series, cf. [4, 6]:

(2.4.1)  P(t) = −H(t) · ∏_{v}(1 − t_{v}^{−1}) ∈ ℤ[[L′]].

Above, ℤ[[L′]] is regarded as a module over ℤ[L′]. Campillo, Delgado and Gusein-Zade for several families of singularities proposed a topological description for P(t) (presented here in 5.1.1). For more positive cases when that description holds, and for some counterexamples too, see [14]. Thus, it is natural to ask:

How general is this phenomenon (that is, the topological characterization of P(t)), and where are its limits?

3. General preliminary results

3.1. As we already mentioned, the above ‘motivating questions’ are not independent. E.g., H can be recovered from (2.2.1)(a) as follows.

Write l′ ∈ L′ as l′ = r_{[l′]} + l for some l ∈ L. Then (see e.g. [14, (4.2.4)])

(3.1.1)  𝔥(l′) = dim \frac{H^{0}(O(−r_{[l′]}))}{H^{0}(O(−r_{[l′]} − max\{0, l\}))}.

Hence in those cases when the dimensions from (2.2.1)(a) — applied for bundles O(−l′), l′ ∈ Q — are topological, H(t) is topologically determined too.

P and H are related as follows. Although the multiplication by ∏_{v}(1 − t_{v}^{−1}) in ℤ[[L′]] is not injective, hence apparently P contains less information then H (cf. (2.4.1)), they, in fact, determine each other (a fact already noticed in [4] for the non-equivariant case): the precise ‘invertion’ follows from (3.2.6) below.

3.2. In fact, the next proposition puts in evidence that in all the invariants listed in the previous section the crucial ingredient is:

(3.2.1)  h_{L} := ∑_{I⊆V}(−1)^{|I|+1} dim \frac{H^{0} (L)}{H^{0} (L (−E_{I}))} ,

where E_{I} = ∑_{v∈I} E_{v} and L ∈ Pic(X̃). If (c_{1}(L), E_{v}) < 0 for some v ∈ V, then for any I ∌ v one has an isomorphism H^{0}(L(−E_{I∪v})) → H^{0}(L(−E_{I})), hence

(3.2.2)  h_{L} = 0 whenever −c_{1}(L) ∉ S′.

Regarding ‘natural line bundles’, the point is that P is also supported on the anti-nef cone S′ (see e.g. [14, (4.2)]), and by (3.1.1) and (2.4.1) (for more details see [14, (4.2.12)]) the generating function for {h_{L} : L natural} is exactly the series P, sitting in the ring ℤ[[S′]]:

(3.2.3)  ∑_{l′∈L′} h_{O(−l′)}t^{l′} = P(t).

The next proposition targets the structure of the invariants (2.2.1)(a)-(b).

Proposition 3.2.4. Let (X, o) be a normal surface singularity whose link is a rational homology sphere. Fix one of its good resolutions X̃ → X. Denote by K ∈ L′ the canonical class satisfying (K + E_{v}, E_{v}) = −2 for all v ∈ V. Then for any line bundle L ∈ Pic(X̃) the following facts hold:

(1) For every effective cycle l ∈ L_{≥0}

(3.2.5)  dim \frac{H^{0} (L)}{H^{0} (L (−l))} = ∑_{a∈L_{≥0},a≱l} h_{L(−a)}.

This (applied for L = O(−r_{[l′]})) and (3.1.1), for any l′ ∈ L′ imply

(3.2.6)  𝔥(l′) = ∑_{a∈L, a≱0} h_{O(−l′−a)}.

(2) There exists a constant const_{[L]}, depending only on the class of [L] ∈ P ic(X̃)/L, such that

(3.2.7)  − h^{1} (L) = ∑_{a∈L, a≰0} h_{L(a)} + const_{[L]} + \frac{(K − 2c_{1} (L))^{2} + |V|}{8} .

(Notice that above the sums are finite because of (3.2.2) and (2.1.1).)

The proof is given in 3.3. Some remarks are in order.

Remark 3.2.8. (1) The above proposition shows that the set {h_{L} : L natural} determines all the ranks dim H^{0}(O(l′))/H^{0}(O(l′ − l)). For h^{1}(O(l′)) additionally one also needs the constants {const_{h}}_{h∈H}, where const_{[l′]} = const_{[O(l′)]}.

(2) The identity (3.2.7) identifies in a natural way (in the world of algebraic geometry) the constants {const_{h}}_{h∈H}. There are two distinguished regions of special interest for the Chern classes l′. First, if one takes in (3.2.7) L = O(−r_{h} + l) with l ∈ L_{≥0}, then, by (3.2.2),

(3.2.9)  − h^{1} (O(−r_{h} + l)) = const_{−h} + \frac{(K + 2r_{h} − 2l)^{2} + |V|}{8} .

Hence, const_{−h} + ((K + 2r_{h})^{2} + |V|)/8 appears as −h^{1} (O(−r_{h})), or as the constant term of the ‘quadratic function’ l ↦ −h^{1} (O(−r_{h} + l)). Note that the geometric genus of Y is h^{1}(OỸ), hence, by (2.2.2), {h^{1}(O(−r_{h}))}_{h} are the equivariant geometric genera of the universal abelian cover.

For the second case, consider L = O(−l′) with l′ ∈ −K + S′. Then by the vanishing of h^{1}(L) [8, (3.2)], one has

(3.2.10)  const_{−[l′]} = − ∑_{a∈L, a≱0} h_{O(−l′−a)} − \frac{(K + 2l′)^{2} + |V|}{8}.

Hence, const_{−[l′]} is determined from the asymptotic behaviour of the cohomological sum on the right hand side of this identity.

On the other hand, and rather surprisingly, const_{−[l′]} appears in a completely different context as well. Indeed, identity (3.2.9) can be compared with the identity predicted by the ‘Seiberg-Witten Invariant Conjecture’ [16, 14, 3]. The only difference between the two identities is that in the second one const_{−[l′]} is replaced by the Seiberg-Witten invariant of the link. (For the general version of this identity see here (5.1.1)(b).) Let us remind that the Seiberg-Witten invariants of the link M are indexed by the spin^{c}-structures of M. Spin^{c}(M) is an H-torsor (with action (h, σ) ↦ h ∗ σ), and with a distinguished element, the canonical spin^{c}-structure σ_{can}. This is induced by the complex structure of X, but it can be recovered from the combinatorics of Γ too (see [16]).

Hence, the Seiberg-Witten Invariant Conjecture has the next reinterpretation:

Conjecture 3.2.11. For ‘nice’ normal surface singularities with rational homology sphere link, the constants {const_{h}}_{h} are the Seiberg-Witten invariants of the link:

(3.2.12)  const_{h} = sw_{h∗σ_{can}} .

3.3. Proof of Proposition (3.2.4). We will use the following rewriting of (3.2.1). For any fixed u ∈ V, by grouping the subsets of V in pairs {I, I ∪ u}_{I∌u}, we get

(3.3.1)  h_{L} = ∑_{I∌u}(−1)^{|I|} \frac{H^{0}(L(−E_{I}))}{H^{0}(L(−E_{I} − E_{u}))}.

Notice that part (2) implies part (1) by Riemann-Roch formula (use (3.3.4)). Nevertheless, in our proof of (2) we will use part (1) too. Hence first we prove (1).

By comparing (3.2.5) for l and l + E_{v} (for some v ∈ V) and using induction, it is enough to prove (3.2.5) only for l = E_{v} (and arbitrary L).

We simplify the right-hand side of (3.2.5) in several steps. First, via (3.2.2), it is ∑_{a} h_{L(−a)}, where a runs over a = ∑_{u≠v} a_{u}E_{u}, with 0 ≤ a_{u} < k for some k ≫ 0. Then, by (3.3.1) applied for u = v, the right hand side of (3.2.5) for l = E_{v} is

(3.3.2)  ∑_{a} ∑_{I∌v}(−1)^{|I|} dim \frac{H^{0}(L(−a − E_{I}))}{H^{0}(L(−a − E_{I} − E_{v}))}.

Since all the involved spaces of global sections are finite codimensional subspaces in H^{0}(L), by combinatorial cancellation, (3.3.2) equals

(3.3.3)  ∑_{I∌v}(−1)^{|I|} dim \frac{H^{0}(L(−kE_{I}))}{H^{0}(L(−kE_{I} − E_{v}))}.

But, the inclusion H^{0}(L(−kE_{I} − E_{v})) ↪ H^{0}(L(−kE_{I})) is an isomorphism for I ≠ ∅, I ∌ v, and k ≫ 0. In order to prove this, embed again both groups in H^{0}(L). Then notice that H^{0}(L(−l)) \ {0} = {𝔰 ∈ H^{0}(L) \ {0} : div_{L}(𝔰) ≥ l} for any l ∈ L_{≥0}. On the other hand, since (div_{L}(𝔰) − c_{1}(L), E_{u}) = 0 for all u ∈ V and 𝔰 ≠ 0, the compactly supported part D(𝔰) of div_{L}(𝔰) satisfies D(𝔰) ∈ c_{1}(L) + S′. On the other hand, since S′ is generated by {E_{u}^{∗}}_{u} whose coefficients are all strict positive, S′ has the following property: for all c_{1} ∈ L′ there is a k ≫ 0 such that for all D ∈ c_{1} + S′ with D ≥ kE_{I} one has D ≥ kE_{I} + E_{v} too. Hence, the only contribution in (3.3.3) comes for I = ∅, and (3.2.5) for l = E_{v} follows.

For (3.2.4)(2), one compares the sheaves L and L(−l) for l ∈ L_{≥0}. The cohomology long exact sequence of L(−l) ↪ L ↠ L ⊗ O_{l} provides

(3.3.4)  dim \frac{H^{0}(L)}{H^{0}(L(−l))} = χ(L ⊗ O_{l}) − h^{1}(L(−l)) + h^{1}(L).

Then (3.2.5) and the Riemann-Roch formula shows that the expression

h^{1}(L) + ∑_{a≰0} h_{L(a)} + \frac{(K − 2c_{1}(L))^{2} + |V|}{8}

is the same for L and L(−l).

4. A short review of splice-quotients singularities

4.1. Splice-quotient singularities were introduced by Neumann and Wahl in [23]. From any fixed graph Γ (which has some special arithmetical properties) one constructs an equisingular family of singularities whose analytic type is strongly linked with special properties of one of its resolutions whose dual graph is exactly Γ.

Splice-quotients very fast became the subject of an intense research activity developing their structural properties. In particular, in present, there are three different approaches how one can introduce and study them, each based on a different geometric property. The first is the original construction of Neumann and Wahl, which imposes the semigroup and congruence condition for the graph, and it emphasizes more the arithmetical/combinatorial properties of the graph and its universal abelian cover [23]. In our short review in this section we will follow Okuma’s approach from [27], which slightly differs from [23, (7.1)]. According to this, Γ has to satisfy a combinatorial restriction: the monomial condition (see 4.2.1) (a property which is needed in the proofs of § 7). The monomial condition is equivalent to the semigroup and congruence conditions of Neumann-Wahl (see [23, §13]). The third approach targets the end curve condition (i.e. the existence of ‘end curve functions’), whose definition will be reviewed and it will be developed further in § 5. The fact that splice-quotient singularities (satisfying the ‘original’ definition) satisfy the ‘end curve condition’ follows immediately from the definitions (the converse is harder: it is the subject of the End Curve Theorem of Neumann and Wahl, and it is not needed in our proofs).

4.2. Fix a resolution graph Γ. Let δ_{v} denote the valency of the vertex v and E the set of ends, i.e. the set of those vertices v with δ_{v} = 1. ℂ{z} denotes the ring of local convergent power series in variables {z_{i}}_{i∈E}. H acts on ℂ{z} by h ∗ z_{i} = θ([E_{i}^{∗}])(h)z_{i}. A vertex v is called node if δ_{v} ≥ 3. For each node v, the connected components of Γ \ v are called branches of v. The set of ends contained in a branch C are denoted by E_{C}.

Definition 4.2.1. [27] (a) A monomial cycle has the form D(α) = ∑_{i∈E} α_{i}E_{i}^{∗} ∈ L′, where α_{i} ∈ ℤ_{≥0}.

(b) We say that Γ satisfies the monomial condition if for any node v and any branch C of v, there exists a monomial cycle D(α) such that D(α) − E_{v}^{∗} is an effective integral cycle supported on C. (In particular, α_{i} = 0 for i ∉ E_{C}.) In this case, z^{α} ∈ ℂ{z} is called an admissible monomial belonging to C.

Fix a vertex v ∈ V. For any monomial z^{α} ∈ ℂ{z} one defines its v-degree by v-deg(z^{α}) = −(D(α), E_{v}^{∗}). The v-degree of any power series is the minimum of the v-degrees of its monomials having non-zero coefficient. Clearly, any admissible monomial belonging to any branch of v has v-degree −(E_{v}^{∗}, E_{v}^{∗}) and is a θ([E_{v}^{∗}])- eigenvector of the H-action. (Defined in this way, v-deg has values in ℚ, which slightly differs from the literature where the present degrees are multiplied to make them integers.)

Definition 4.2.2 (Splice diagram equations [23, 27]). Assume that Γ satisfies the monomial condition. We make some arbitrary choices. We select an admissible monomial M_{v,C} for each node v and branch C of v. We select complex numbers a_{v,C,j} (where v is a node and 1 ≤ j ≤ δ_{v} − 2) such that for every fixed v all the maximal minors of the matrix (a_{v,C,j})_{C,j} have full rank. Finally, we choose convergent power series g_{v,j} ∈ ℂ{z} (1 ≤ j ≤ δ_{v} −2) which are θ([E_{v}^{∗}])-eigenvectors of the H-action and have v-degree greater than −(E_{v}^{∗}, E_{v}^{∗}). The set of splice diagram equations is the collection for all nodes v of equations of the form

∑_{C} a_{v,C,j}M_{v,C} + g_{v,j} = 0 (1 ≤ j ≤ δ_{v} − 2),

where C runs over the branches of v.

Theorem 4.2.3. [23, (7.2)] The splice diagram equations of Γ define an isolated complete intersection singularity in (ℂ^{|E|}, 0) on which H acts freely off the origin. It is the universal abelian cover of its factor by H. The factor has a good resolution with dual graph Γ and which admits ‘end curve functions’ (for definition see 5.2.1).

A singularity defined as in (4.2.3) is called splice-quotient. Its equisingularity type is independent of the different choices (cf. [23, (10.1)], [26]).

Splice-quotients include the rational singularities (with arbitrary, not necessarily minimal resolution graph Γ) [26], minimally elliptic singularities (with resolutions where the support of the minimal elliptic cycle is not proper) [26], and weightedhomogeneous singularities (with their minimal good resolutions) [20]. For more details see also [23] and [27].

See Example 7.3.4 for an explicit computation of a possible system of splice diagram equations for a graph.

5. The Campillo–Delgado–Gusein-Zade identity and the Seiberg-Witten Invariant Conjecture for splice-quotients

5.1. Let (X, o) be a splice-quotient associated with the graph Γ. Denote by (Y, o) its universal abelian cover (which is a complete intersection in ℂ^{|E|}, cf. § 4).

Theorem 5.1.1. (a) Let (X, o) be a splice-quotient singularity associated with the fixed graph Γ. Consider its resolution π with graph Γ and the analytic invariants H(t) and P(t) associated with π. Then they can be recovered from Γ. More precisely:

(5.1.2)  P(t) = ∏_{v∈V} (1 − t^{E_{v}^{∗}})^{δ_{v}−2}.

(The expression of H(t) follows from (5.1.2) and (3.2.6).)

(b) Conjecture 3.2.11 is true for splice-quotient singularities. Moreover, h^{1}(O(l′)) is topological for any l′ ∈ L′. More precisely, define the coefficients c_{l′} by the Taylor expansion at the origin

(5.1.3)  ∏_{v∈V} (1 − t^{E_{v}^{∗}})^{δ_{v}−2} = ∑_{l′∈S′} c_{l′}t^{l′} ∈ ℤ[[S′]]

(hence c_{l′} = 0 whenever l′ ∉ S′). Then

(5.1.4)  − h^{1} (O(−l′)) = ∑_{a∈L,a≱0} c_{l′+a} + sw_{[−l′]∗σ_{can}} + \frac{(K + 2l′)^{2} + |V|}{8} .

The identity (5.1.2) is the generalization of the ‘Campillo–Delgado–Gusein-Zade identity’ (stated in the terminology of [14]). This, for rational singularities was proved in [6], for minimally elliptic ones in [14]. This is a common generalization of them, which includes the weighted homogeneous singularities as well.

Part (b) is a direct consequence of (3.2.7), Part (a) and [3]. Indeed, for the first sentence combine [3, (2.2.4)] (which proves the Seiberg-Witten Invariant Conjecture for splice-quotients singularities and for line bundles O(−r_{h}+l) with l ∈ L_{≥0} getting the analogue of the identity (3.2.9) with sw_{−h∗σ_{can}} instead of const_{−h}) and (3.2.9). This identifies the constants {const_{h}}_{h}. For the identity (5.1.4) use (3.2.7) and (5.1.2) too.

In order to prove Theorem 5.1.1(a), and to be able to formulate some further consequences, we need some preparations: they constitute subsections 5.2 and 5.3. The proof is given in 5.4 and runs over induction (on the number of vertices of the graph) based on the ‘end curve condition’. The presentation is self-consistent (and independent of the End Curve Theorem of Neumann-Wahl).

The author studied splice-quotient singularities in a joint project with G. Braun; we even prepared a manuscript about some new results. Nevertheless, the manuscript was split in two independent parts: Braun’s article [2] contains a new proof of the End Curve Theorem of Neumann and Wahl, the remaining part targeting the Campillo–Delgado–Gusein-Zade type identity is incorporated in the present section. Inevitably, some (technical) overlaps remained (e.g. Lemma 5.3.1); nevertheless they emphasize different aspects and target different applications.

5.2. Preparations. Singularities satisfying the ‘end curve condition’. We fix a singularity (X, o) and one of its good resolutions π : X̃ → X with graph Γ. As above, V (resp. E) denotes the set of vertices (resp. ends) of Γ.

Consider an irreducible curve on X̃ transversal to E intersecting exactly one E_{v} (v ∈ V) at one point. A cut function for this curve is a function f ∈ O_{X,o} such that the divisor of f ∘ π is supported on the union of E with this curve. The curve is a cut of E_{v} if it has a cut function.

Definition 5.2.1. [23] The end curve condition for π requires a cut H_{i} for each end component E_{i}, i ∈ E. We call such an H_{i} end curve and the corresponding function end curve function.

Recall that any splice-quotient singularity satisfies the end curve condition by their construction ([23, (7.2)(6)]): the |H|-powers of the coordinate functions of ℂ^{|E|} serve as end curve functions.

5.2.2. End curve sections. [23] For a fixed v ∈ V, let e_{v} be the order of [E_{v}^{∗}] in H. Since Pic(X̃) has no torsion, if H_{v} is a cut of E_{v}, one can take a cut function f_{v} such that div(f_{v} ∘ π) = e_{v}(H_{v} + E_{v}^{∗}). In particular, O(−E_{v}^{∗}) = O(H_{v}) in Pic(X̃).

Consider now the universal abelian cover of (X, o), maps c and c̃ as in (2.2), and a function f_{v} for some v ∈ V as in the previous paragraph. Then f ∘ π ∘ c̃ is an e_{v}-power of some z_{v} ∈ H^{0}(Ỹ, OỸ) (≃ O_{Y,o}). Since z_{v} is in the θ([E_{v}^{∗}])-eigenspace, by (2.2.2) and divisor verification one also has that z_{v} ∈ H^{0}(X̃, O(−E_{v}^{∗})), and the divisor of z_{v}, as global section of O(−E_{v}^{∗}), is H_{v}. (In this construction of z_{v} from f_{v} there is an ambiguity with multiplication by an e_{v}-root of unity what we will disregard. Also, in the above discussion one can replace e_{v} by any of its multiples.)

If π : X̃ → X satisfies the end curve condition, the sections z_{i} for i ∈ E constructed from (a fixed set of) end curve functions will be called end curve sections.

Usually, for each i ∈ E, one fixes one cut. Nevertheless, in order to run the theory (and inductive steps) properly, for a graph which has only one vertex, one fixes two disjoint cuts, hence one gets two end curve sections. (For this special case, the reader is invited to replace some of the indices used below accordingly.)

5.2.3. Monomials in H^{0}(X̃, O(−l′)). Fix a subset W ⊂ V, and assume that π admits cut functions and cuts H_{w} for any w ∈ W. Fix l′ ∈ L′. For any collection α_{w} ∈ ℤ_{≥0} (w ∈ W), with [∑_{w∈W} α_{w}E_{w}^{∗}] = [l′] in H, consider the monomial z^{α} = ∏_{w∈W} z_{w}^{α_{w}} in {z_{w}}_{w}, considered as a function on Ỹ. It is in the θ([l′])-eigenspace, hence it is a meromorphic section of O(−l′). Since its divisor div_{O(−l′)}(z^{α}), as section of O(−l′), is ∑_{w} α_{w}(H_{w} + E_{w}^{∗}) − l′, z^{α} is a global holomorphic section of O(−l′) if and only if ∑_{w} α_{w}E_{w}^{∗} − l′ ∈ L_{≥0}. We write D(α) := ∑_{w∈W} α_{w}E_{w}^{∗}.

Definition 5.2.4. Assume that π satisfies the end curve condition. Consider the end curve sections {z_{i}}_{i∈E} associated with a fixed set of end curve functions. An element of H^{0}(X̃, O(−l′)) of the form z^{α} with W = E is called monomial section.

A global sections z^{α} of O(−l′), where {α_{v}}_{v∈W} is indexed by a subset W ⫌ E (provided that each {z_{w}}_{w∈W} exists) is called an extended monomial section.

Example 5.2.5. For any l′ ∈ L′ and l ∈ L_{≥0}, the classes of non-zero monomial sections in H^{0}(X̃, O(−l′))/H^{0}(X̃, O(−l′ − l)) are indexed by

{α | α_{i} ≥ 0 for all i ∈ E; a ≥ 0, a ≱ l, where a = D(α) − l′ ∈ L}.

E.g., if l = E_{v} for some v ∈ V with (l′, E_{v}) > 0, then this index set is empty: the existence of such an α would imply 0 > (D(α) − l′, E_{v}) ≥ 0. This chimes in with the fact that H^{0}(O(−l′))/H^{0}(O(−l′ − E_{v})) embeds into H^{0}(E_{v}, O(−l′)) = 0.

5.2.6. Restrictions. Next we introduce the notations of the inductive step used in the proof. Let π be a fixed resolution with dual graph Γ and |V| > 1. We fix a vertex v ∈ E; let E_{w} be the unique irreducible exceptional curve which intersects E_{v}. Set Ē = ∪_{u∈V\v}E_{u}. Let X̃(Ē) be a sufficiently small neighbourhood of Ē in X̃, and Γ̄ the dual graph of Ē. We denote by Ē_{u} (= E_{u}), u ∈ V \ v, the base elements of the new lattice L̄, and by Ē_{u}^{∗} their Γ̄-duals in L̄′. Notice that Ē analytically can be contracted (denote this by π̄) giving rise to a singularity (X̄, o). Set Ē for the ends of Γ̄ and E′ = E \ v.

If (X, o) admits the end curves {H_{i}}_{i∈E} in X̃ cut out by the end curve functions {f_{i}}_{i∈E}, then X̃(Ē) inherits some compatible cuts and end curves. Indeed:

Lemma 5.2.7. [27, (2.15)]. The curves {H_{i}}_{i∈E′} and H̄_{w} = E_{v} ∩ X̃(Ē) are cuts of Ē in X̃(Ē). In particular, the resolution π̄ of (X̄, o) satisfies the end curve condition with distinguished end curve functions inherited from {f_{i}}_{i∈E}.

Proof. Let m_{i} be the vanishing order of f_{i} along E_{v}. Then for π̄, f_{v}|X̃(Ē) is a cut function for H̄_{w}, and f̄_{i} = f_{i}^{m_{v}}/f_{v}^{m_{i}}|X̃(Ē) is a cut function for H_{i}, i ∈ E′. □

Let i : L̄ ↪ L be the natural lattice embedding Ē_{u} ↦ E_{u}, u ∈ V \ v. Its dual R : L′ → L̄′ is defined by (R(l′), l̄) = (l′, i(l̄)) (or by R(E_{v}^{∗}) = 0 and R(E_{u}^{∗}) = Ē_{u}^{∗} for u ∈ V \ v), and one also has

(5.2.8)  R(E_{v}) = −Ē_{w}^{∗} and R(E_{u}) = Ē_{u} for u ∈ V \ v.

Lemma 5.2.9. The restriction of any natural line bundle to X̃(Ē) is natural. In fact, the restriction of OX̃(−l′) is isomorphic to O_{X̃(Ē)}(−R(l′)).

Proof. We need to show that some power of the restriction has the form O_{X̃(Ē)}(l̄) for some l̄∈ L̄. Hence, by taking |H|-power of OX̃(−l′), the proof reduces to the cases OX̃(E_{u})|X̃(Ē). But this, for u ≠ v is O_{X̃(Ē)}(Ē_{u}), and for u = v is O_{X̃(Ē)}(H̄_{w}), which equals O_{X̃(Ē)}(−Ē_{w}^{∗}) by 5.2.7 and 5.2.2; all of them are natural. □

Here, a word of warning is necessary. Consider the restriction map

R^{0} : H^{0}(X̃, OX̃(−l′)) ⟶ H^{0}(X̃(Ē), OX̃(−l′)|_{X̃(Ē)}).

Although OX̃(−l′)|_{X̃(Ē)} and O_{X̃(Ē)}(−R(l′)) are isomorphic, the isomorphism connecting them might not send the restriction of a monomial section of OX̃(−l′) to a monomial section of O_{X̃(Ē)}(−R(l′)) (hence some caution in the inductive procedure is necessary).

5.3. This subsection constitute of the following technical lemma:

Lemma 5.3.1. [2] Let π : X̃ → X be the resolution which satisfies the end curve condition. Let L = OX̃(−l′) (l′ ∈ L′) be a natural line bundle on X̃. Then:

(1) For some l ∈ L_{≥0} consider the vector space V := H^{0}(X̃, L)/H^{0}(X̃, L(−l)). Assume that a subset of H^{0}(X̃, L) has the following property: for any nonzero class in V of a monomial section of L the set contains an element with the same divisor. Then the classes of the elements of the set generate V as a vector space.

(2) Assume that |V| > 1, v ∈ E and (c_{1}(L), E_{v}) ≥ 0. Set E_{w} and X̃(Ē) as in 5.2.6. Then the restriction map induces an isomorphism

(5.3.2)  R^{♭} : \frac{H^{0}(X̃, L)}{H^{0}(X̃, L(−E_{w}))} ⟶ \frac{H^{0}(X̃(Ē), L|_{X̃(Ē)})}{H^{0}(X̃(Ē), L|_{X̃(Ē)}(−E_{w}))}.

Proof. We prove both statements by a simultaneous induction on the number of vertices of the resolution graph. For this we will use the notations 5.2.6 and 5.2.7. The cut functions {f̄}_{i∈E′} and f_{v}|X̃(Ē) induce sections in H^{0}(X̃(Ē), O_{X̃(Ē)}(−Ē_{i}^{∗})),

resp. in H^{0}(X̃(Ē), O_{X̃(Ē)}(−Ē_{w}^{∗})), denoted by z̄_{i} for i ∈ E′, resp. by y_{w} (cf. 5.2.2 and 5.2.7). Clearly, if δ_{w} > 2, then w ∉ Ē, hence Ē = E′. Hence the new monomial sections have the form ∏_{i∈E′} z̄_{i}^{α_{i}} (α_{i} ≥ 0); a product of type ∏_{i∈E′} z̄_{i}^{α_{i}} · y_{w}^{β} (β ≥ 0) is an extended monomial section. If δ_{w} = 2, then Ē = E′ ∪ w, hence all {H̄_{i}}_{i} and H̄_{w} are end curves of X̃(Ē), and the new monomials sections have the form ∏_{i∈E′} z̄_{i}^{α_{i}} · y_{w}^{β} (α_{i} ≥ 0, β ≥ 0). If δ_{w} = 1 then for X̃(Ē) we preserve both (old and new) cuts H_{w} and H̄_{w}, and the new monomials sections have the form z̄_{w}^{α}y_{w}^{β}.

The induction starts with a one-vertex graph with two end curves H_{1} and H_{2}, in which case we have to check only (1). In this case the picture is very explicit (and can be verified easily by the reader):

Set p = −E^{2}. Then E^{∗} = (1/p)E and H = ℤ_{p}. The abelian cover Y is ℂ^{2}, O_{Y,o} = ℂ{z_{1}, z_{2}}, where z_{1} and z_{2} are the two end curve sections (see e.g. [1, III.5]). Moreover, Ỹ is the blow up of Y at o with irreducible exceptional curve Ẽ, and c^{∗}(E) = pẼ. Set L = OX̃(−kE^{∗}) for some k ∈ ℤ. Then H^{0}(X̃, L) = H^{0}(Ỹ, O(−kẼ))_{θ([l′])} which equals the subspace of ℂ{z_{1}, z_{2}} generated by monomials of degree deg ≥ k and deg ≡ k (mod p). Therefore, if l = nE, the monomial sections z^{α} of L with α_{1}+α_{2} = k+ip (0 ≤ i < n) are linearly independent and the space spanned by them projects bijectively onto V. Since div_{L}(z^{α}) = α_{1}H_{1} + α_{2}H_{2} + iE, the vanishing orders of all these divisors at the intersection point H_{1} ∩ E are all distinct. Hence, any set of section of L with the same cardinality and divisors projects into a linearly independent set in V, which necessarily form a basis of V.

Now, we consider a resolution π : X̃ → X whose graph has |V| > 1 vertices. We fix a vertex v ∈ E, and we will use the notations of 5.2.6. By induction, we assume that the statements of the lemma are true for π̄.

First we prove statement (2) for π. Since R^{♭} is injective, we have to show its surjectivity. For this consider all the monomial sections of L|_{X̃(Ē)} with non-zero class in the target of R^{♭}. They have the form M̄ = ∏_{i∈E′} z̄_{i}^{α_{i}} · y_{w}^{β} (where α_{i} ≥ 0 and β ≥ 0, and β = 0 if w ∉ Ē) such that

∑_{i∈E′} α_{i}Ē_{i}^{∗} + βĒ_{w}^{∗} + R(c_{1}(L)) = l̄

for some l̄∈ L̄_{≥0} and l̄≱ Ē_{w}. In particular, l̄is supported on the closure of Ē \Ē_{w}, hence (i(l̄), E_{v}) = 0. Since R(∑_{i∈E′} α_{i}E_{i}^{∗} − βE_{v} + c_{1}(L) − i(l̄)) = 0, one has

(5.3.3)  ∑_{i∈E′} α_{i}E_{i}^{∗} − βE_{v} + c_{1}(L) − i(l̄) = −α_{v}E_{v}^{∗}

for some α_{v} ∈ ℤ. Since βE_{v} + i(l̄) ∈ L_{≥0}, and α_{v} = −βE_{v}^{2} + (c_{1}(L), E_{v}) ≥ 0, we get that z^{α} = ∏_{i∈E} z_{i}^{α_{i}} is a monomial section of L, such that the divisors of R^{0}(z^{α}) and M̄ coincide. By the inductive step the collection {R^{0}(z^{α})}_{α} is a generator set of the target of R^{♭}. In particular, R^{♭} is onto proving part (2).

Moreover, if we replace above each section z^{α} by another section 𝔰_{α} of L with the same divisor, then by the same inductive argument as above the set {R^{0}(𝔰_{α})}_{α} form a basis of the target of R^{♭}. But, since R^{♭} is an isomorphism, the classes of s_{α} necessarily form a basis too. This proves part (1) for l = E_{w} and any L with (c_{1}(L), E_{v}) ≥ 0.

Now we turn to statement (1) for general L and l ∈ L_{≥0}. First we consider the following infinite ‘computation sequence’ {x_{n}}_{n≥0}, x_{n} ∈ L_{≥0}, such that x_{0} = 0, x_{n+1} = x_{n} + E_{u(n)}, where u(n) ∈ V is provided by the following principle:

(a) If (−c_{1}(L) + x_{n}, E_{u}) ≤ 0 for all u ∈ V then u(n) = w.

(b) If (−c_{1}(L) + x_{n}, E_{u}) > 0 for some u, then take one of them for u(n). One can show (see e.g. [11, (4.2)]) that both steps occur infinitely many times. Moreover, at the beginning of step (a), −c_{1}(L) + x_{n} ∈ S′. Since S′ ∩ {l′ ∈ L′ : l′ ≱ l − c_{1}(L)} is finite, cf. (2.1.1), we get that x_{k} ≥ l for some sufficiently large k.

Next we analyze the steps of {x_{n}}_{n=0}^{k} and we show by induction on n the validity of (1) for V_{n} := H^{0}(L)/H^{0}(L(−x_{n})). In case (b), the natural projection V_{n+1} → V_{n} is an isomorphism and the divisors of the monomials can also be identified (cf. 5.2.5). In case (a), one has the exact sequence

0 ⟶ \frac{H^{0}(L(−x_{n}))}{H^{0}(L(−x_{n} − E_{w}))} ⟶ V_{n+1} ⟶ V_{n} ⟶ 0.

(1) is valid for V_{n} by the inductive hypothesis, and for the left hand side too by the particular case already proved (l = E_{w}); hence it works for the middle term as well. This ends the induction showing (1) for V_{k}. Finally, consider the projection V_{k} → V = H^{0}(L)/H^{0}(L(−l)). Any set of H^{0}(L) which satisfy the assumption of (1) for V can be completed (by adding monomial sections whose classes in V are zero) to a set which satisfies the assumption of (1) for V_{k}. Hence, since V_{k} satisfies (1), so does V. □

5.4. Proof of 5.1.1(a). By the ‘change of variables’ {x_{u} = t^{E_{u}^{∗}}}_{u∈V}, and using (3.2.3) and the support restriction (3.2.2), the identity (5.1.2) is equivalent to

(5.4.1)  ∑_{k_{u}≥0} h_{O(− ∑_{u} k_{u}E_{u}^{∗})} ∏_{u∈V} x_{u}^{k_{u}} = ∏_{u∈V}(1 − x_{u})^{δ_{u}−2}.

Next we prove (5.4.1) for all singularities which satisfies the end curve condition. Since splice-quotients satisfies this condition, the result follows.

The proof is by induction on the number of vertices (where we will also use that the restriction preserves the class of singularities satisfying the end curve condition, and also their natural line bundles, cf. 5.2.7 and 5.2.9). The statement is clear for graphs having only one vertex u, since it is just a reformulation of dim H^{0} (O (−kE_{u}^{∗}))/H^{0} (O (−kE_{u}^{∗} − E_{u})) = h^{0}(O_{ℙ^{1}}(k)) = k + 1 for all k ≥ 0.

For a graph Γ with |V| > 1, let v, w, Γ̄ be as in 5.2.6, and denote the restriction of B = O(− ∑_{u} k_{u}E_{u}^{∗}) ∈ Pic(X̃) to X̃(Ē) by B̄. Then consider (3.3.1) for u = w:

(5.4.2)  h_{B} = ∑_{I∌w}(−1)^{|I|} \frac{H^{0}(B(−E_{I}))}{H^{0}(B(−E_{I} − E_{w}))}.

Since (c_{1}(B), E_{v}) = k_{v} ≥ 0 and −(E_{I}, E_{v}) ≥ 0, 5.3.1(2) can be applied for each L = B(−E_{I}). Separating the cases when I contains v or not (and using (5.2.8)), and then by reorganizing it in expressions as (5.4.2) for hB̄ and h_{B̄(Ē_{w}^{∗})}, we get

h_{B} = hB̄ − h\overline{B(−E_{v})} = hB̄ − h_{B̄(Ē_{w}^{∗})}.

Notice that, by (3.2.2), h_{B̄(Ē_{w}^{∗})} = 0 whenever k_{w} = 0. Therefore, if P_{X,o} denotes the left hand side of (5.4.1), we have

P_{X,o} = P_{X̄,o} · ∑_{k_{v}≥0} x_{v}^{k_{v}} · (1 − x_{w}) = P_{X̄,o} · \frac{1 − x_{w}}{1 − x_{v}} .

Since the right hand side of (5.4.1) satisfies the same inductive formula, this ends the proof.

6. First consequences and remarks

6.1. Comparison of the divisorial and monomial filtrations. For splicequotients, besides the divisorial multi-filtration {F(l′)}_{l′∈L′} (cf. (2.3.1)), there exists another filtration on O_{Y,o}, indexed by the same index set L′. One starts with the monomial filtration on ℂ{z} defined by the degrees of monomials:

(6.1.1)  G(l′) := { ∑_{α} a_{α}z^{α} ∈ ℂ{z} : ∑_{i∈E} α_{i}E_{i}^{∗} ≥ l′ whenever a_{α} ≠ 0}.

Let ϕ : ℂ{z} → O_{Y,o} send the variable z_{i} to the corresponding end curve section (denoted by z_{i} too, cf. 5.2.2). By definition, the monomial filtration of O_{Y,o} is the image of G by ϕ. By construction and definitions ϕ(G(l′)) ⊂ F(l′).

Corollary 6.1.2. ϕ(G(l′)) = F(l′) for all l′ ∈ L′.

Proof. We provide two proofs, each emphasizing different aspects of the identity.

The first proof points out that the rational expression (5.1.3) is the Poincaré series of a weighted complete intersection.

Let H_{ϕ(G)}, respectively P_{ϕ(G)}, be the series associated with ϕ(G) similarly as H and P are associated with the filtration F, cf. (2.3.2) and (2.4.1). One shows similarly as in 3.3 that H_{ϕ(G)} can be recovered from P_{ϕ(G)} by a similar ‘inversion’ combinatorial formula as H from P (cf. (3.2.6)). The point is the following Fact:

(6.1.3)  P_{ϕ(G)}(t) = ∏_{v∈V} (1 − t^{E_{v}^{∗}})^{δ_{v}−2}.

Then by 5.1.1(a) one has P = P_{ϕ(G)}, hence by the two ‘inversion’ formulae H = H_{ϕ(G)} too. This together with ϕ(G(l′)) ⊂ F(l′) implies ϕ(G) = F.

Hence, it remains only too verify (6.1.3), which also exposes the right hand side of (5.1.2) in a new light. Recall (cf. § 4) that the abelian cover (Y, o) is a complete intersection in (ℂ^{|E|}, o) with δ_{v} − 2 equations for each node v. G is that filtration of ℂ{z}, which has weights in L′ so that each variable z_{i} (i ∈ E) has weight E_{i}^{∗}; i.e. the weight w(z^{α}) of the monomial z^{α} is D(α), and as usual, if f = ∑ a_{α}z^{α}, then w(f) := inf_{a_{α}≠0}D(α). Therefore, the Poincaré series P_{G}(t) (of ℂ{z}) is exactly ∏_{i∈E}(1 − t^{E_{i}^{∗}})^{−1}. Consider now a (splice diagram) equation f_{v,j} = ∑_{C} a_{v,C,j}M_{v,C} + g_{v,j} (j = 1, …, δ_{v} − 2) associated with the node v, cf. 4.2.2. Since M_{v,C} is admissible monomial, there exists an integral effective cycle Z_{C} supported on the branch C such that D(M_{v,C}) = E_{v}^{∗} + Z_{C}. In particular,

w ( ∑ a_{v,C,j}M_{v,C}) = inf_{C}{E_{v}^{∗} + Z_{C}} = E_{v}^{∗}.

Next, regarding g_{v,j}, by the construction 4.2.2, its v-degree satisfies

(6.1.4)  − (w(g_{v,j}), E_{v}^{∗}) > −(E_{v}^{∗}, E_{v}^{∗}).

We claim that w(g_{v,j}) > E_{v}^{∗} holds too. Indeed, take D := w(g_{v,j}) − E_{v}^{∗} and write it as D = D′ + a_{v}E_{v}, where the support of D′ does not contain E_{v}. Then (6.1.4) just says that a_{v} > 0. Since w(g_{v,j}) is a sum of monomial cycles, (D, E_{u}) ≤ 0 for all u ≠ v. Hence (D′, E_{u}) ≤ 0 too for all E_{u} in the support of D′ implying D′ ≥ 0.

Therefore, w(f_{v,j}) = E_{v}^{∗}, and the usual Koszul-complex argument for the complete intersection ends the proof of Fact, and the first proof of Corollary too.

The second possible proof uses the m-adic completion (of the maximal ideal) and a classical theorem of Housel [7]. The argument runs as follows. In the ring ℂ{z} the m-adic filtration and the monomial multi-filtration determine the same completion, since every ideal of either filtration contains an ideal from the other one. This remains true for their projection on O_{Y,o} too. Also, the m-adic filtration and the divisorial filtration of O_{Y,o} have the same property as well. Indeed, every ideal of the divisorial filtration clearly contains a large power of the maximal ideal; the other direction (namely that F(ka) ⊂ m_{Y,o}^{k} for some integral cycle a = ∑ a_{v}E_{v} with all a_{v} > 0 and any k ≥ 2) follows e.g. from [9, (3.2)].

Next, recall that by [26, (3.2)], the θ(h)-eigenspace of c̃_{∗}(OỸ (−c^{∗}(l′))) is O(−r_{h}+ [[r_{h} − l′]]) (where [[l′]] denotes the greatest integral cycle smaller than or equal to l′ for any l′ ∈ L′). Hence applying 5.3.1(1) to the eigenspaces, we obtain

ϕ(G(l′)) + F(l′ + l) = F(l′) for any l ∈ L_{≥0}.

Therefore, for any fixed l′, the m-adic completion ϕ̂: { \atop G(l′)} → { \atop F(l′)} is onto. Hence, by [7], ϕ : G(l′) → F(l′) is onto too. □

This generalizes a result of Okuma (for splice-quotients) [27, (3.3)], where the filtrations are associated with only one node. For a similar discussion see [2, (8.2.1)] too. We also note that a similar identity (of the divisorial and a combinatorial Poincaré series), valid for affine toric varieties, was verified by A. Lemahieu in [10].

Remark 6.1.5. (Semigroups associated with divisorial valuations) For any fixed vertex u of Γ consider the divisorial filtration F_{u} associated with the irreducible component E_{u}, and let P_{u} be the Poincaré series of Gr_{F_{u}}O_{Y,o}. Then, by [14], Gr_{F_{u}}O_{Y,o} is obtained from P(t) by substitutions t_{w} = 1 for all w ∈ V \ u. Hence, by taking H-invariants (for details see e.g. [14]), Theorem 5.1.1 implies that the Poincaré series of Gr_{F_{u}}O_{X,o} is

(6.1.6)  ∑_{i≥0} c_{i}t^{i} = \frac{1}{|H|} · ∑_{ρ∈Ĥ} ∏_{v∈V} ( 1 − ρ([E_{v}^{∗}]) t_{u}^{−(E_{v}^{∗},E_{u}^{∗})} )^{δ_{v}−2}.

This provides topologically the semigroup S_{u} := {i : c_{i} ≠ 0} of the E_{u}-valuation too (this agrees with the set of integers which appear as multiplicities of functions along E_{u}). In general, it is extremely hard to determine these semigroups, even to characterize their qualitative properties (for more details regarding this subject see e.g. the recent articles of Cutkosky and Teissier and the references therein).

6.2. A new Seiberg-Witten invariant formula. Theorem 5.1.1 and the identity (3.2.10) imply the following topological consequence too:

Corollary 6.2.1. Let Γ be the graph of any splice-quotient singularity. For any class [l′] ∈ H, take any representative l′ in l′ ∈ −K +S′. Then, with the notation of (5.1.3), the Seiberg-Witten invariants of the associated plumbed 3-manifold satisfy:

(6.2.2)  sw_{[−l′]∗σ_{can}} + \frac{(K + 2l′)^{2} + |V|}{8} = − ∑_{a∈L,a≱0} c_{l′+a}.

In other words, the expression from the right hand side (i.e. special truncations of the series (5.1.3)) admits a quadratic generalized Hilbert polynomial, whose free coefficient is the (renormalized) Seiberg-Witten invariant.

The point is that the SW Invariant Conjecture is proved in [3] for line bundles of the form O(−r_{h} + l) with l ∈ L_{≥0}, then via the general identities (3.2.4) this can be reinterpreted in the ‘special vanishing region’ l′ ∈ −K + S′, cf. (3.2.10). This provides the above new combinatorial formula for the Sieberg-Witten invariant for links of splice-quotients emphasizing the role of the ‘zeta function’ (5.1.3). Its generalization for arbitrary negative definite graphs is treated in [15].

7. Principal cycles and the multiplicity

7.1. Principal ℚ-cycles. Let (X, o) be a normal surface singularity and π : X̃ → X one of its fixed resolutions. We use the notations of 2.1 and also define

S := S′ ∩ L = {l ∈ L : (l, E_{v}) ≤ 0 for all v ∈ V}.

Let supp(l) ⊂ E denote the support of a cycle l ∈ L. For any set {l′_{j}}_{j∈J} with l′_{j} = ∑_{v} l′_{j,v}E_{v} ∈ L′ set inf_{j} l′_{j} := ∑_{v} l′_{v}E_{v} ∈ L ⊗ ℚ, where l′_{v} = min_{j} l′_{j,v}.

Definition 7.1.1. A rational cycle l′ ∈ L′ is called a principal ℚ-cycle if O(−l′) has a global holomorphic section 𝔰 which is not zero on any of the exceptional components. Their set will be denoted by Pr′. The set Pr of principal cycles is defined as Pr′ ∩ L; it consists precisely of the restrictions to E of the divisors of π-pullbacks of analytic functions from O_{X,o}.

Clearly, Pr′ (respectively Pr) is a sub-semigroup of S′ (resp. of S).

In general, for an arbitrary singularity, it is very difficult (unsolved) task to decide if an element of S′ is principal or not. E.g., if (X, o) is rational (and π is arbitrary), or minimally elliptic (and π is minimal and good) then Pr = S; but, in general, Pr ≠ S. The point is that, in general, the semigroup Pr cannot be characterized topologically, it strongly depends on the analytic structure supported by the topological type fixed by Γ.

The next result characterizes Pr′ combinatorially in L′(Γ) for a splice-quotient singularity associated with Γ. Recall that E denotes the set of end-vertices of Γ and any {α_{i}}_{i∈E} (α_{i} ∈ ℤ_{≥0}) provides a monomial cycle D(α) = ∑_{i} α_{i}E_{i}^{∗} ∈ L′.

Theorem 7.1.2. [Characterization of principal ℚ-cycles] Let (X, o) be a splice-quotient singularity associated with the graph Γ, and consider its resolution π with dual graph Γ. Then for any l′ ∈ S′ the following facts are equivalent:

(1) l′ ∈ Pr′;

(2) for each v ∈ V there exists an effective cycle l_{v} ∈ L_{≥0} such that: (a) E_{v} ⊄ supp(l_{v}),

(b) (l_{v}, E_{u}) = −(l′, E_{u}) for any u ∉ E,

(c) (l_{v}, E_{u}) ≤ −(l′, E_{u}) for any u ∈ E;

(3) there exists finitely many monomial cycles {D(α_{(k)})}_{k} ∈ l′ + L so that l′ = inf_{k} D(α_{(k)}).

Proof. (1)⇒(2). If l′ ∈ Pr′ then the quotient H^{0}(O(−l′))/H^{0}(O(−l′ − E_{v})) ≠ 0 for each v ∈ V. Then, by Lemma 5.3.1(1) (see also 5.2.5), there exists a monomial z^{α_{(v)}} so that

(7.1.3)  ∑_{i∈E} α_{(v),i} E_{i}^{∗} − l′ = l_{v},

where l_{v} ∈ L_{≥0} and E_{v} ⊄ supp(l_{v}). Hence (2) follows. In order to prove (2)⇒(3), define the non-negative integers α_{(v),i} := −(l′ + l_{v}, E_{i}) for all i ∈ E. This means that (7.1.3) is satisfied. Since inf_{v} D(α_{(v)}) = inf_{v}(l′ + l_{v}) = l′, (3) follows too. For (3)⇒(1) notice that if l′ = inf_{k} D(α_{(k)}), then a linear combination of the monomials z^{α_{(k)}} works for 𝔰. □

The next result shows that the ‘saturation’ of the sub-semigroup Pr in S (resp. S′) is S (resp. S′) itself.

Theorem 7.1.4. Let (X, o) be as in Theorem 7.1.2. Then m · S′ ⊂ Pr for some m ∈ ℤ_{>0}. In fact, if δ_{v} ≠ 2 then E_{v}^{∗} ∈ Pr′, hence e_{v}E_{v}^{∗} ∈ Pr, where e_{v} is the order of [E_{v}^{∗}] in H.

Proof. For v ∈ E, E_{v}^{∗} ∈ Pr′ by the definition of splice-quotients (see e.g. 5.2.2). Next, fix a node v. Let {C_{k}}_{1≤k≤δ_{v}} be its branches. By the monomial condition there exists a monomial cycle D(α_{k}) so that D(α_{k}) − E_{v}^{∗} is effective, integral and supported on C_{k}. Hence E_{v}^{∗} = inf_{k} D(α_{k}). Therefore, e_{v}E_{v}^{∗} ∈ Pr′ ∩ L = Pr.

More generally, take a vertex v with δ_{v} ≥ 2 adjacent vertices {w_{k}}_{k} sitting in the branches {C_{k}}_{k} of v. Regard C_{k} as a subgraph, let Ē_{u}^{∗} be the C_{k}-dual of E_{u} in this subgraph (u ∈ V(C_{k})). Next, consider in C_{k} an arbitrary integral monomial cycle l̄_{k} of type ∑_{u∈E_{C_{k}}} α_{k,u}Ē_{u}^{∗}, where E_{C_{k}} are the ends of Γ in C_{k}. Since α_{k,u} are positive l̄_{k} is effective. Let m_{k} be the multiplicity of E_{w_{k}} in l̄_{k}. Then, one verifies (by intersecting both parts with all E_{w}, w ∈ V) that ∑_{u∈E_{C_{k}}} α_{k,u}E_{u}^{∗} − m_{k}E_{v}^{∗} = i(l̄_{k}), hence it is effective, integral and supported on C_{k}. (Here i : L(C_{k}) ↪ L is the inclusion of lattices.) Hence m_{v} = lcm_{k}{m_{k}} works because m_{v}E_{v}^{∗} = inf_{k} D(α_{k}m_{v}/m_{k}). □

Remark 7.1.5. (a) It can happen that e_{v}E_{v}^{∗} ∉ Pr (hence E_{v}^{∗} ∉ Pr′ neither); e.g. in the case of the −13-vertex of the right graph from 7.2.3 e_{v} = 1 and E_{v}^{∗} ∉ Pr.

(b) One of the motivations for the characterization of Pr is the Nash Conjecture. One way to separate the components of the arc space is to use the ratios l_{i}/l_{j} of the coefficients of the principal cycles ∑_{i} l_{i}E_{i} ∈ Pr, see e.g. [29]. By 7.1.4, for splice-quotients, the set of these ratios remain the same if we replace Pr by S.

7.2. The minimal and maximal cycle. Let us recall the following definitions:

Definition 7.2.1. Let Z_{min} ∈ S be the minimal (or fundamental) cycle of Γ, i.e. the unique minimal non-zero element of S. Let OX̃(−Z_{max}) be the divisorial part of the pullback π^{∗}(m_{X,o}) of the maximal ideal. Equivalently, Z_{max} is the unique minimal non-zero element of Pr. It is called the maximal (ideal) cycle of (X, o).

Clearly Z_{min} ≤ Z_{max}. Z_{min} can be determined by a combinatorial algorithm of Laufer [8, (4.1)], but, in general, for arbitrary singularities Z_{max} cannot be read from Γ, it might depend on the analytic structure supported by the fixed topological type.

We denote the (finitely generated) semigroup of integral monomial cycles by

L_{E} := { ∑_{i∈E} α_{i}E_{i}^{∗} : α_{i} ∈ ℤ_{≥0}} ∩ L.

Theorem 7.1.2 implies the following:

Corollary 7.2.2. Let (X, o) be as in 7.1.2. Then Z_{max} = inf(L_{E} \ {0}).

Example 7.2.3. Consider the following two graphs, both with L′ = L.

[figure: −2; −7; E_{1}; E_{0}; −1; −2; −3; −2; −1; −13; E_{1}; −1; −2; −3; −3]

For the left graph Z_{min} ∈ Pr, nevertheless Pr ≠ S. In fact, Pr \ S = {E_{1}^{∗}}, where E_{1} is the −7-curve. To show that E_{1}^{∗} ∉ Pr in 7.1.2(2) for E_{v} choose E_{0}.

The right graph is an example with Z_{min} ∉ Pr and with infinite Pr \ S. Indeed, let E_{1} denote the (−13)-curve and let E_{i} be any exceptional curve with valency 1. Then E_{1}^{∗} + kE_{i}^{∗} ∉ Pr for every k ≥ 0 (take in 7.1.2(2) for v the node adjacent to the end vertex i). For this graph one has Z_{max} = 2Z_{min}.

7.3. Base points and multiplicity. In general, π^{∗}m_{X,o} has the form OX̃(−Z_{max})⊗ ⨂_{P ∈B} I_{P}, where B denotes the finite set of base-points and I_{P} is an m_{P} -primary ideal for P ∈ B. Next we provide a combinatorial description of the ideals I_{P}.

Assume that π satisfies the end curve condition, we fix some end curves H_{i} for i ∈ E as in 5.2. Consider D := E ∪ (∪_{i∈E}H_{i}), and let Sing(D) denote the singular (double) points of D. Fix such a point P ∈ Sing(D) and order the two components of D containing it. Associate with each α ∈ L_{E} the multiplicity orders (a_{α}, b_{α}) in ∑_{i∈E} α_{i}(H_{i} + E_{i}^{∗}) − Z_{max} of the two components of D containing P.

The results 7.1.2 and 7.2.2 have the following consequence too:

Corollary 7.3.1. (1) B ⊂ Sing(D). In particular, if H_{i} ∩ E ∈ B for some i ∈ E, then the intersection point H_{i} ∩ E is independent of the choice of H_{i} (i.e. it is imposed by the analytic structure).

(2) Fix P ∈ Sing(D). In some local coordinates (t, s) of P (so that ts = 0 is the local equation of D), I_{P} is the monomial ideal generated by {t^{a_{α}}s^{b_{α}}}_{α∈L_{E}}.

Example 7.3.2. On the left graph of 7.2.3, B consists of only one point, the intersection of a cut with E_{0}, the unique curve E_{i} with (Z_{min}, E_{i}) < 0. In the case of the right graph B = ∅. In the next example 7.3.4, B consists of one point, which is the intersection of two irreducible exceptional components.

For every P ∈ B, the pairs (a_{α}, b_{α})_{α} determine a (convenient) Newton diagram:

N_{P}^{−} = ℝ_{≥0}^{2} \ convex closure{ ⋃_{α∈L_{E}} ((a_{α}, b_{α}) + ℝ_{≥0}^{2}) }.

Theorem 7.3.3. The multiplicity mult(X, o) of (X, o) is topological:

mult(X, o) = −Z_{max}^{2} + 2 · ∑_{P ∈B} area(N_{P}^{−}).

Proof. For each P let ϕ_{P} and ψ_{P} be two plane curve singularities with Newton diagram N_{P}^{−} and generic coefficients. Then (see e.g. the proof of [33, (2.7)])

mult(X, o) = −Z_{max}^{2} + ∑_{P ∈B}(ϕ_{P} , ψ_{P})_{P} ,

where (−, −)_{P} denotes the intersection multiplicity at P. One the other hand it is well-known that (ϕ_{P} , ψ_{P})_{P} = 2 · area(N_{P}^{−}), see e.g. [32], or [30, page 276]. □

Example 7.3.4. Consider the following graph.

[figure: −2; −1; −8; −8; −1; −3; −3; −2; z_{1}; z_{2}; z_{3}; z_{4}]

In this case Z_{max} = Z_{min} with Z_{max}^{2} = −2. Let P be the intersection point of the two (−8)-curves. Then π^{∗}(m_{X,o}) = OX̃(−Z_{max}) ⊗ m_{P}, hence mult(X, o) = 3.

In fact, in this case we can even write the equations of (X, o). Indeed, if {z_{i}}_{1≤i≤4} are the end curve sections (as indicated in the picture), then a possible choice for the splice diagram equations is z_{1}^{2} +z_{2}^{3} +z_{3}^{5} = z_{3}^{2} +z_{4}^{3} +z_{1}^{5} = 0. H ≃ ℤ_{3} is generated by [E_{2}^{∗}], and acts by [E_{2}^{∗}] ∗ (z_{1}, z_{2}, z_{3}, z_{4}) = (z_{1}, ψ̄z_{2}, z_{3}, ψz_{4}), where ψ = e^{2πi/3}. The generators of the invariants are z_{1}, z_{3}, a = z_{2}^{3}, b = z_{4}^{3} and c = z_{2}z_{4}, hence the equations of (X, o) are z_{1}^{2}+a+z_{3}^{5} = z_{3}^{2}+b+z_{1}^{5} = ab−c^{3} = 0, or c^{3} = (z_{1}^{2}+z_{3}^{5})(z_{3}^{2}+z_{1}^{5}).

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Rényi Institute of Mathematics, Budapest, Reáltanoda u. 13–15, 1053, Hungary

   E-mail address: nemethi@renyi.hu