LINE BUNDLES ASSOCIATED WITH NORMAL SURFACE SINGULARITIES

ANDRÁS NÉMETHI

1. Introduction

In [15] L. Nicolaescu and the author formulated a conjecture which relates the geometric genus of a complex analytic normal surface singularity (X, 0) (whose link M is a rational homology sphere) with the Seiberg-Witten invariant of M associated with the “canonical” spin^{c} structure of M. (The interested reader is invited to see the articles [15, 16, 17, 13, 19, 20] for the verification of the conjecture in different cases.) Since the Seiberg-Witten theory of the link M provides a rational number for any spin^{c} structure (which are classified by H_{1}(M, ℤ)), it was a natural challenge to search for a complete set of conjecturally valid identities, which involve all the Seiberg-Witten invariants (giving an analytic – i.e. singularity theoretical – interpretation of them).

The formulation of this set of identities is one of the goals of the present article. In fact (similarly as in [15]), we formulate conjecturally valid inequalities which became equalities in special rigid situations. In this way, the Seiberg-Witten invariants determine optimal topological upper bounds for the dimensions of the first sheaf-cohomology of line bundles living on some/any resolution of (X, 0). Moreover, for ℚ-Gorenstein singularities and some “natural” line bundles equality holds.

The first part of the article constructs these “natural” holomorphic line bundles on the resolution X̃ of (X, 0). This construction automatically provides a natural splitting of the exact sequence

0 → P ic^{0}(X̃) → P ic(X̃) →^{c_{1}} H^{2}(X̃, ℤ) → 0.

The line-bundle construction is compatible with abelian covers. This allows us to reformulate the conjecture in its second version which relates the echivariant geometric genus of the universal (unbranched) abelian cover of (X, 0) with the Seiberg-Witten invariants of the link.

In the last section we verify the conjecture for rational singularities.

2. Preliminaries

2.1. Let (X, 0) be a complex analytic normal surface singularity. We fix a good resolution π : X̃ → X of (X, 0) (over a small fixed Stein representative X of (X, 0)) such that the exceptional divisor E := π^{−1}(0) has only normal crossing singularities. We denote the irreducible components of E by {E_{j}}_{j∈J}. The boundary ∂X̃ of X̃ can be identified with the link M of (X, 0), which is an oriented connected 3-manifold. In this article, we will assume that M is an rational homology sphere (i.e. H_{1}(M, ℚ) = 0). This implies that all the curves E_{j} are rational, and H_{1}(X̃, ℤ) = 0.

The exact sequence of ℤ-modules

0 → L →^{i} L′ → H → 0  (1)

will stay for the homological exact sequence

0 → H_{2}(X̃, ℤ) → H_{2}(X̃, M, ℤ) ⟶^{∂} H_{1}(M, ℤ) → 0,  (2)

2000 Mathematics Subject Classification. 14B15, 14E15, 32S25, 57M27, 57R57.

Key words and phrases. normal surface singularities, rational singularities, links of singularities, 3manifolds, ℚ-homology spheres, Seiberg-Witten invariants, Ozsváth-Szabó Floer homology, Reidemeister-Turaev torsion, Casson-Walker invariant.

The author is partially supported by NSF grant DMS-0304759.

2 A. Némethi

or, via Poincaré duality, for

0 → H_{c}^{2}(X̃, ℤ) → H^{2}(X̃, ℤ) → H^{2}(M, ℤ) → 0.  (3)

L, considered as in (2), is freely generated by the homology classes [E_{j}]_{j∈J}. For each j, consider a small transversal disc D_{j} in X̃ with ∂D_{j} ⊂ ∂X̃. Then L′ is freely generated by the classes [D_{j}]_{j∈J}. We set g_{j} := ∂[D_{j}] = [∂D_{j}] ∈ H. In the sequel, for simplicity we write E_{j}, D_{j} for [E_{j}] and [D_{j}].

On L there is a symmetric, non-degenerate, negative definite intersection form (·, ·) (with associated matrix B_{ij} = (E_{j}, E_{j})), cf. [1, 5]. Then the morphism i : L → L′ can be identified with L → Hom(L, ℤ) given by l ↦ (l, ·). The intersection form has a natural extension to L_{ℚ} = L ⊗ ℚ, and it is convenient to identify L′ with a sub-lattice of L_{ℚ}: α ∈ Hom(L, ℤ) corresponds with the unique l_{α} ∈ L_{ℚ} which satisfies α(l) = (l_{α}, l) for any l ∈ L. By this identification, D_{j}, considered in L_{ℚ} (and written in the base {E_{j}}_{j}), is the j^{th} column of B^{−1}, and (D_{j}, E_{i}) = δ_{ji}.

Elements x = ∑_{j} r_{j}E_{j} ∈ L_{ℚ} will be called (rational) cycles. If x_{i} = ∑_{j} r_{j,i}E_{j} for i = 1, 2, then min{x_{1}, x_{2}} := ∑_{j} min{r_{j,1}, r_{j,2}}E_{j}. x^{2} means (x, x); e_{j} := E_{j}^{2}. We define the support |x| of x by ∪E_{j}, where the union runs over {j : r_{j} ≠ 0}.

2.2. One can consider two types of “positivity conditions” for rational cycles. The first one is considered in L: A cycle x = ∑_{j} r_{j}E_{j} ∈ L_{ℚ} is called effective, denoted by x ≥ 0, if r_{j} ≥ 0 for any j. Their set is denoted by L_{ℚ,e}, while L′_{e} := L_{ℚ,e} ∩ L′ and L_{e} := L_{ℚ,e} ∩ L. We write x ≥ y if x − y ≥ 0. x > 0 means x ≥ 0 but x ≠ 0. ≥ provides a partial order of L_{ℚ}.

The second is the numerical effectiveness of the rational cycles, i.e. positivity considered in L′. We define NE_{ℚ} := {x ∈ L_{ℚ} : (x, E_{j}) ≥ 0 for any j}. In fact, NE_{ℚ} is the positive cone in L_{ℚ} generated by {D_{j}}_{j}, i.e. it is {∑_{j} r_{j}D_{j}, r_{j} ≥ 0 for any j}. Since B is negative definite, all the entries of D_{j} are strict negative. In particular, −NE_{ℚ} ⊂ L_{ℚ,e}.

2.3. Characteristic elements. Spin^{c}-structures. The set of characteristic elements are defined by

Char = Char(X̃) := {k ∈ L′ : (k, x) + (x, x) ∈ 2ℤ for any x ∈ L}.

There is a canonical characteristic element K ∈ Char defined by (K, E_{j}) = −(E_{j}, E_{j}) − 2 for any j.

Clearly, Char = K + 2L′. There is a natural action of L on Char by x ∗ k := k + 2x whose orbits are of type k + 2L. Obviously, H acts freely and transitively on the set of orbits by [l′] ∗ (k + 2L) := k + 2l′ + 2L (in particular, they have the same cardinality).

If X̃ is as above, then the first Chern class (of the associated determinant line bundle) realizes an identification between the spin^{c}-structures Spin^{c}(X̃) on X̃ and Char ⊂ L′ = H^{2}(X̃, ℤ) (see e.g. [4], 2.4.16). The restrictions to M defines an identification of the spin^{c}-structures Spin^{c}(M) of M with the set of orbits of Char modulo 2L; and this identification is compatible with the action of H on both sets. In the sequel, we think about Spin^{c}(M) by this identification, hence any spin^{c}-structure of M will be represented by an orbit [k] := k + 2L ⊂ Char.

2.4. Liftings. If H is not trivial, then the exact sequence (1) does not split. Nevertheless, we will consider some “liftings” (set theoretical sections) of the element of H into L′. They correspond to the positive cones in L_{ℚ} considered in 2.2.

More precisely, for any l′ + L = h ∈ H, let l′_{e}(h) ∈ L′ be the unique minimal effective rational cycle in L_{ℚ,e} whose class is h. Clearly, the set {l′_{e}(h)}_{h∈H} is exactly the closed/open unit cube Q := {∑_{j} r_{j}E_{j} ∈ L′; 0 ≤ r_{j} < 1}.

Similarly, for any h = l′ +L, the intersection (l′ +L)∩NE_{ℚ} has a unique maximal element l′_{ne}(h), and the intersection (l′ + L) ∩ (−NE_{ℚ})) has a unique minimal element l̄′_{ne}(h) (cf. [13], 5.4). By their definitions l̄′_{ne}(h) = −l′_{ne}(−h).

The elements l̄′_{ne}(h) were introduced in [13] in order to compute the Ozsváth-Szabó Floer homology of M for some singularity links (their notation in that paper is l′_{[k]}, where [k] = K + 2(l′ + L)). Using these elements, [loc. cit.] defines the distinguished representative k_{r} of [k] by k_{r} := K+2l̄′_{ne}(h) (h = l′ + L).

For some h, l̄′_{ne}(h) might be situated in Q, but, in general, this is not the case (cf. 6.3 and 6.4). In general, the characterization of all the elements l̄′_{ne}(h) is not simple (for the cases when M is a lens space or a Seifert manifold, see [13]).

2.5. RR. In some of our arguments, we use the same notation for l = ∑ n_{j}E_{j} ∈ L and the algebraic cycle ∑ n_{j}E_{j} of X̃ supported by E. E.g., for any l = ∑ n_{j}E_{j} ∈ L one can take the line bundle OX̃(l) := OX̃(∑ n_{j}E_{j}). If l > 0 set O_{l} := OX̃/OX̃(−l), and let χ(l) := χ(O_{l}) = h^{0}(O_{l}) − h^{1}(O_{l}) be its Euler-characteristic. It can be computed combinatorially by Riemann-Roch: χ(l) = −(l, l+K)/2.

By analogy, for any l′ ∈ L_{ℚ}, we define χ(l′) := −(l′, l′ + K)/2.

3. Line bundles on X̃.

3.1. Let π : (X̃, E) → (X, 0) be a fixed good resolution of (X, 0) as in section 2. Since H^{1}(X̃, ℤ) = 0, one has

0 → P ic^{0}(X̃) → P ic(X̃) →^{c_{1}} L′ → 0,  (1)

where P ic^{0}(X̃) = H^{1}(X̃, OX̃), P ic(X̃) = H^{1}(X̃, O∗ ; X̃), and c_{1}(L) = ∑_{j} deg(L|E_{j}) D_{j} is the set of Chern numbers (multidegree) of L. Notice that c_{1}(OX̃(l)) = l, hence c_{1} admits a group-section s_{L} : L → P ic(X̃) above the subgroup L of L′. In the sequel we will construct a (natural) group section s : L′ → P ic(X̃) of (1) which extends s_{L} (and is compatible with abelian coverings).

Clearly, if P ic^{0}(X̃) = 0 (i.e. if (X, 0) is rational, cf. [1]) then there is nothing to construct: c_{1} is an isomorphism, and s := c_{1}^{−1} identifies the lines bundles with their multidegree. Nevertheless, for non-rational singularities, even the existence of any kind of splitting of (1) is not so obvious.

3.2. Notice that X̃ \ E ≈ X \ {0} has the homotopy type of M, hence the abelianization map π_{1}(X̃ \ E) = π_{1}(M) → H defines a regular Galois covering of X̃ \ E. This has a unique extension c : Z → X̃ with Z normal and c finite [6]. The (reduced) branch locus of c is included in E, and the Galois action of H extends to Z as well. Since E is a normal crossing divisor, the only singularities what Z might have are cyclic quotient singularities (situated above Sing(E)). Let r : Z̃ → Z be a resolution of these singular points of Z, such that (c ∘ r)^{−1}(E) is a normal crossing divisor.

Notice that, in fact, Z̃ is a good resolution of the universal (unbranched) abelian cover (X_{a}, 0) of (X, 0). (Here, (X_{a}, 0) is the unique normal singular germ corresponding to the regular covering of X \ {0} associated with π_{1}(X \ {0}) → H.) Hence, one can consider the following commutative diagram:

0 → L → L′ → H → 0
↓ ↓^{p′} ↓^{p_{H}}
0 → L_{a} → L′_{a} → H_{a} → 0

Here, the first, resp. second, horizontal line is the exact sequence 2.1(3) applied for the resolution X̃ → (X, 0), resp. for Z̃ → (X_{a}, 0). The vertical arrows (pull-back of cohomology classes) are induced by p = c ∘ r.

3.3. Lemma. p_{H} = 0. In particular, p′(L′) ⊂ L_{a} (i.e. any element p′(l′), l′ ∈ L′, can be represented by a divisor supported by the exceptional divisor in Z̃).

Proof. Denote by M_{a} the link of (X_{a}, 0). The morphism p_{H} : H^{2}(M, ℤ) → H^{2}(M_{a}, ℤ) is dual to p_{∗} : H_{1}(M_{a}, ℤ) → H_{1}(M, ℤ), which is zero since H_{1}(M_{a}, ℤ) is the abelianization of the commutator subgroup of π_{1}(M). □

This shows that for any l′ ∈ L′, one can take OZ̃(p′(l′)) ∈ P ic(Z̃).

3.4. Theorem. The line bundle OZ̃(p′(l′)) is a pull-back of a unique element of P ic(X̃).

Proof. We brake the proof into several steps. Let f : S → T be one of the maps r : Z̃ → Z, c : Z → X̃ or p : Z̃ → X̃. In each case one has a commutative diagram of type

H^{1}(T, ℤ) → P ic^{0}(T) → P ic(T) → H^{2}(T, ℤ) → 0
↓^{p′′} ↓^{p^{0,∗}} ↓^{p^{∗}} ↓^{p′}
H^{1}(S, ℤ) → P ic^{0}(S) → P ic(S) → H^{2}(S, ℤ) → 0

(I). Assume that f = c. Then:

• c′ is injective since c′ ⊗ ℚ : H^{2}(X̃, ℚ) ≈ H^{2}(Z, ℚ)^{H} ↪ H^{2}(Z, ℚ) is so and H^{2}(X̃, ℤ) is free.

• c^{0,∗} is injective with image P ic^{0}(Z)^{H}. Indeed (cf. also with the proof of 3.7), c_{∗}O_{Z} has a direct sum decomposition ⊕_{χ}L_{χ} into line bundles L_{χ}, where the sum runs over all the characters of H, and for the trivial character L_{1} = OX̃. Therefore, since c is finite, H^{1}(Z, O_{Z}) = H^{1}(X̃, c_{∗}O_{Z}) = ⊕_{χ}H^{1}(X̃, L_{χ}), whose H-invariant part is H^{1}(X̃, OX̃).

• Im(H^{1}(Z, ℤ) → P ic^{0}(Z)) ∩ Imp^{0,∗} = 0. Indeed, since any element of Imp^{0,∗} is H-invariant, the above intersection is in Im(H^{1}(Z, ℤ)^{H} → P ic^{0}(Z)^{H}). But H^{1}(Z, ℤ)^{H} embeds into H^{1}(Z, ℚ)^{H} = H^{1}(X̃, ℚ) = 0, hence it is trivial.

• p^{∗} is injective, a fact which follows from the previous three statements. (II). Assume that f = r. Then r′′ is an isomorphism (since such a resolution does not modify H_{1}(·, ℤ) of the exceptional divisors), r^{0,∗} is an isomorphism (since a quotient singularity has geometric genus zero), and r′ is injective. Hence (by a diagram check) p^{∗} is also injective. (III). Assume that f = p = c ∘ r. Then:

• For any l′ ∈ L′, OZ̃(p′(l′)) is in the image of p^{∗}. Indeed, take a line bundle L ∈ P ic(X̃) with c_{1}L = l′. Then L′ := OZ̃(p′(l′)) ⊗ p^{∗}L^{−1} has trivial multidegree, and it is in P ic^{0}(Z̃)^{H}. But using (I-II) p^{0,∗} is onto on P ic^{0}(Z̃)^{H}, hence L′ is a pull-back. Hence OZ̃(p′(l′)) itself is a pull-back.

• Again, from (I-II), p^{∗} is injective; a fact which ends the proof of 3.4. □

3.5. Notation. Write OX̃(l′) for the unique line bundle L ∈ P ic(X̃) with p^{∗}(L) = OZ̃(p′(l′)).

The proof of 3.4 also implies the following fact.

3.6. Corollary. s : L′ → P ic(X̃) defined by l′ ↦ OX̃(l′) is a group section of the exact sequence (1) which extends s_{L} (cf. 3.1).

The following result will illuminate a different aspect of the line bundles OX̃(l′). In fact, the next proposition could also serve as the starting point of a different construction of these line bundles.

Below we will write Ĥ for the Pontrjagin dual Hom(H, S^{1}) of H. Recall that the natural map

θ : H → Ĥ, induced by [l′] ↦ e^{2πi(l′,·)}

is an isomorphism.

3.7. Theorem. Consider the finite covering c : Z → X̃, and set Q := {∑ r_{j}E_{j} ∈ L′ : 0 ≤ r_{j} < 1} ⊂ L′ (cf. 2.4) as above. Then the H-eigenspace decomposition of c_{∗}O_{Z} has the form:

c_{∗}O_{Z} = ⊕_{χ∈Ĥ}L_{χ},

where L_{θ(h)} = OX̃(−l′_{e}(h)) for any h ∈ H. In particular, c_{∗}O_{Z} = ⊕_{l′∈Q}OX̃(−l′).

Proof. The proof is based on a similar statement of Kollár valid for cyclic coverings, see e.g. [7], §9.

First notice that c_{∗}L_{Z} is free. Indeed, since all the singularities of Z are cyclic quotient singularities, this fact follows from the corresponding statement for cyclic Galois coverings, which was verified in [7]. Moreover, above X̃ \ E the covering is regular (unbranched) corresponding to the regular representation of H. Therefore, rank c_{∗}L_{Z} = |H|, and it has an eigenspace decomposition ⊕_{χ}L_{χ}, where all the characters χ ∈ Ĥ appear, and L_{χ}|X̃ \E is a line bundle. By a similar reduction as above to the cyclic case, one gets that L_{χ} itself is a line bundle. Moreover, L_{1} (corresponding to the trivial character 1) equals OX̃.

Next we identify L_{χ} for any character. Fix χ_{0} ∈ Ĥ. It generates the subgruop ⟨χ_{0}⟩ in Ĥ. Write K̂ := Ĥ/⟨χ_{0}⟩. The morphism π_{1}(M) → H → ⟨χ_{0}⟩ˆ defines a well-defined normal space Y as a cyclic Galois ⟨χ_{0}⟩ˆ -covering of X̃. Clearly, one has the natural maps Z ⟶^{f} Y ⟶^{e} X̃ with e∘f = c. By similar argument as above f_{∗}O_{Z} = ⊕_{ξ∈K̂} L̄_{ξ} for some L̄_{ξ} ∈ P ic(Y) and L̄_{1} = O_{Y}. Since e_{∗}L̄_{ξ} = ⊕_{[χ]=ξ}L_{χ}, and [χ_{0}] = 0 in K̂, one gets that L_{χ_{0}} is one of the summands of e_{∗}L̄_{1} = e_{∗}O_{Y}. In particular, L_{χ_{0}} can be recovered from the cyclic covering e as the χ_{0}-eigenspace of e_{∗}O_{Y}.

Assume that the order of χ_{0} is n, i.e. ⟨χ_{0}⟩ = ℤ_{n}, and we regard χ_{0} as the distinguished generator of ℤ_{n}. Then for any j ∈ J, χ_{0}(g_{j}) has the form e^{2πim_{j}/n} for some (unique) 0 ≤ m_{j} < n (for the definition of g_{j} see 2.1). Using these integers, define the divisor B := ∑_{j} m_{j}E_{j}. Since for any fixed i ∈ J one has

1 = χ_{0}([E_{i}]) = χ_{0}(∑_{j}B_{ij}g_{j}) = e^{2πi(∑_{j} m_{j}E_{j},E_{i})/n},

one gets that B/n ∈ L′. Let h = [B/n] be its class in H. Clearly, θ(h) = χ_{0} since

θ(h)(g_{j}) = e^{2πi(∑ m_{j}E_{j}/n,D_{j})} = e^{2πim_{j}/n} = χ_{0}(g_{j}).

On the other hand, by [7], 9.8 (and from the fact that all the coefficients of B/n are in the interval [0, 1)), the χ_{0}-eigenspace of e_{∗}O_{Y} is some line bundle L^{−1} with the properties (i) L^{⊗n} = OX̃(B) and (ii) e^{∗}L = O_{Y} (e′(B/n)). From (i) follows that c_{1}(L) = B/n, hence (ii), via our definition 3.5, reads as L = OX̃(B/n). Therefore, L^{−1} = OX̃(−B/n). Since B/n is in the unit cube, B/n = l′_{e}(h). □

4. Some cohomological computations

4.1. Let (X, 0) and π : X̃ → X as above. In this section we analyse h^{1}(X̃, L) for any L ∈ P ic(X̃).

The next 4.1.1 is an improvement of the following general (Kodaira, or Grauert-Riemenschneider type) vanishing theorem (cf. [24], page 119, Ex. 15): If c_{1}(L) ∈ K + NE_{ℚ}, then h^{1}(l, L|_{l}) = 0 for any l ∈ L, l > 0, hence h^{1}(X̃, L) = 0.

4.1.1. Assume that (X, 0) is a rational singularity. If c_{1}(L) ∈ NE_{ℚ}, then h^{1}(l, L|_{l}) = 0 for any l > 0, l ∈ L, hence h^{1}(X̃, L) = 0 too.

For the convenience of the reader we sketch a proof. For any l > 0 there exists E_{j} ⊂ |l| such that (E_{j}, l + K) < 0. Indeed, (E_{j}, l + K) ≥ 0 for any j would imply χ(l) = −(l, l + K)/2 ≤ 0, which would contradict the rationality of (X, 0) [1]. Then, from the cohomology exact sequence of

0 → L ⊗ O_{E_{j}}(−l + E_{j}) → L|_{l} → L|_{l−E_{j}} → 0

one gets h^{1}(L|_{l}) = h^{1}(L|_{l−E_{j}}), hence by induction h^{1}(L|_{l}) = 0. We will generalize this proof as follows.

4.2. Proposition. Let X̃ → X be a good resolution of a normal singularity (X, 0) as above. (a) For any l′ ∈ L′ there exists a unique minimal element l_{l′} ∈ L_{e} with e(l′) := l′ − l_{l′} ∈ NE_{ℚ}.

(b) l_{l′} can be found by the following (generalized Laufer’s) algorithm. One constructs a “computation sequence” x_{0}, x_{1}, …, x_{t} ∈ L_{e} with x_{0} = 0 and x_{i+1} = x_{i} + E_{j(i)}, where the index j(i) is determined by the following principle. Assume that x_{i} is already constructed. Then, if l′−x_{i} ∈ NE_{ℚ}, then one stops, and t = i. Otherwise, there exists at least one j with (l′ − x_{i}, E_{j}) < 0. Take for j(i) one of these j’s. Then this algorithm stops after a finitely many steps, and x_{t} = l_{l′}.

(c) For any L ∈ P ic(X̃) with c_{1}(L) = l′ one has:

h^{1}(L) = h^{1}(L ⊗ OX̃(−l_{l′})) − (l′, l_{l′}) − χ(l_{l′}).

In particular (since c_{1}(L⊗OX̃(−l_{l′})) ∈ NE_{ℚ}), the computation of any h^{1}(L) can be reduced (modulo the combinatorics of (L, (·, ·))) to the computation of some h^{1}(L′) with c_{1}(L′) ∈ NE_{ℚ}.

[Although the above decomposition l′ = e(l′) + l_{l′} with l_{l′} ≥ 0 and e(l′) ∈ NE_{ℚ} looks similar to the Zariski decomposition (cf. [25]), in general it is different from it.] Proof. (a) First notice that since B is negative definite, there exists at least one effective cycle l with l′ − l ∈ NE_{ℚ} (take e.g. a large multiple of some x with (x, E_{j}) < 0 for any j). Next, we prove that if l′ − l_{i} ∈ NE_{ℚ} for l_{i} ∈ L_{e}, i = 1, 2, and l := min{l_{1}, l_{2}}, then l′ − l ∈ NE_{ℚ} as well. For this, write x_{i} := l_{i} − l ∈ L_{e}. Then |x_{1}| ∩ |x_{2}| = ∅, hence for any fixed j, E_{j} ⊄ |x_{i}| for at least one of the i’s. Therefore, (l′ − l, E_{j}) = (l′ − l_{i}, E_{j}) + (x_{i}, E_{j}) ≥ 0.

(b) First we prove that x_{i} ≤ l_{l′} for any i. For i = 0 this is clear. Assume that it is true for some i but not for i + 1, i.e. E_{j(i)} ⊄ |l_{l′} − x_{i}|. But this would imply (l′ − x_{i}, E_{j(i)}) = (l′ − l_{l′}, E_{j(i)}) + (l_{l′} − x_{i}, E_{j(i)}) ≥ 0, a contradiction. The fact that x_{i} ≤ l_{l′} for any i implies that the algorithm must stop, and x_{t} ≤ l_{l′}. But then by the minimality of l_{l′} (part a) x_{t} = l_{l′}. (Cf. [8].)

(c) For any 0 ≤ i < t, consider the exact sequence

0 → L ⊗ OX̃(−x_{i+1}) → L ⊗ OX̃(−x_{i}) → L ⊗ O_{E_{j(i)}}(−x_{i}) → 0.

Since deg(L ⊗ O_{E_{j(i)}}(−x_{i})) = (l′ − x_{i}, E_{j(i)}) < 0, one gets h^{0}(L ⊗ O_{E_{j(i)}}(−x_{i})) = 0. Therefore

h^{1}(L ⊗ OX̃(−x_{i})) − h^{1}(L ⊗ OX̃(−x_{i+1})) = −χ(L ⊗ O_{E_{j(i)}}(−x_{i}))

which equals −(l′, x_{i+1} − x_{i}) + χ(x_{i}) − χ(x_{i+1}). Hence the result follows by induction. □

4.3. Examples. If L = OX̃(l′) for some l′ ∈ L′ (cf. 3.5) then 4.2(c) reads as

h^{1}(OX̃(l′))) = h^{1}(OX̃(e(l′))) − (l′, l_{l′}) − χ(l_{l′}).

Additionally, if (X, 0) is rational then h^{1}(OX̃(e(l′))) = 0 by 4.1.1, hence h^{1}(OX̃(l′)) = −(l′, l_{l′}) − χ(l_{l′}). In particular, for (X, 0) rational, h^{1}(L) depends only on topological data and it is independent of the analytic structure of (X, 0).

4.4. Definition. We will distinguish the following set of rational cycles:

𝕃′ := {l′ ∈ L′ : e(l′) = l′_{ne}(h) for some h ∈ H} = ⋃_{h∈H} l′_{ne}(h) + L_{e}.

It is easy to verify the following inclusions: L′_{e} ⊂ ⋃_{l′∈Q} −l′ + L_{e} ⊂ 𝕃′.

5. The main conjecture

5.1. In this section we present a conjecture which provides an optimal upper bound for h^{1}(L) (L ∈ P ic(X̃)) in terms of the topology of (X, 0) and c_{1}(L). The main topological ingredient is provided by the Seiberg-Witten theory of the link M.

For any fixed spin^{c}-structure [k] ∈ Spin^{c}(M), one defines the modified Seiberg-Witten invariant sw_{M}^{0}([k]) as the sum of the number of Seiberg-Witten monopoles and the Kreck-Stolz invariant, see [3, 10, 11] (for this notation, more discussions and references, see [15]). In this article we prefer to change its sign: we will write sw_{M,[k]}^{0} := −sw_{M}^{0}([k]). In general it is very difficult to compute sw_{M,[k]}^{0} using its analytic definition, therefore there is an intense activity to replace this definition with a different one. Presently, there exist a few candidates. In this note we will discuss two of them. One of them is sw_{M,[k]}^{T CW} provided by the sign refined Reidemeister-Turaev torsion (cf. [26]) normalized by the Casson-Walker invariant, cf. 5.1.1; another is sw_{M,[k]}^{OSz} provided by the Ozsváth-Szabó theory [21], see also [22, 13] for possible connections with singularities; cf. with 5.1.2 here.

In particular, we formulate our conjecture for the “symbol” sw_{M,[k]}^{∗}, which can be replaced by any of the above invariants. The conjectured identity sw_{M,[k]}^{0} = sw_{M,[k]}^{T CW} = sw_{M,[k]}^{OSz} makes all the conjectures (for different ∗) equivalent. Nevertheless, different realizations might illuminate essentially different aspects.

5.1.1. sw_{M,[k]}^{T CW}. For every spin^{c}-structure [k], Turaev defines the sign refined Reidemeister-Turaev torsion 𝒯_{M,[k]} = ∑_{h∈H} 𝒯_{M,[k]}(h)h ∈ ℚ[H] (determined by the Euler structure) associated with [k] [26]. Set λ(M) for the Casson-Walker invariant of M (normalized as in [9], 4.7). Then, one defines

sw_{M,[k]}^{T CW} := −𝒯_{M,[k]}(1) + \frac{λ(M)}{|H|},

where 1 denotes the neutral element of the group H (with the multiplicative notation).

5.1.2. sw_{M,[k]}^{OSz}. For any oriented rational homology 3-sphere M and [k] ∈ Spin^{c}(M), the Ozsváth-Szabó ℤ[U]-module HF^{+}(M, [k]) (or, the absolutely graded Floer homology) was introduced in [21] (cf. also with the long list of recent preprints of Ozsváth and Szabó). This has a ℚ-grading compatible with the ℤ[U]-action, where deg(U) = −2.

Additionally, HF^{+}(M, [k]) has an (absolute) ℤ_{2}-grading; HF_{even}^{+}(M, [k]), respectively HF_{odd}^{+}(M, [k]) denote the part of HF^{+}(M, [k]) with the corresponding parity.

For any rational number r ∈ ℚ, we denote by T_{r}^{+} (cf. [22, 13]) the graded ℤ[U]-module defined as the quotient of ℤ[U, U^{−1}] by the submodule U · ℤ[U]. This has a rational grading in such a way that deg(U^{−d}) = 2d + r (d ≥ 0).

By the general theory, one has a graded ℤ[U]-module isomorphism

HF^{+}(M, [k]) = T_{d(M,[k])}^{+} ⊕ HF_{red}^{+}(M, [k]),

(for some well-defined rational number d(M, [k])), where HF_{red}^{+} has a finite ℤ-rank, and an induced (absolute) ℤ_{2}-grading. From this one extracts two numerical invariants: d(M, [k]) and

χ(HF^{+}(M, [k])) := rank_{ℤ} HF_{red,even}^{+}(M, [k]) − rank_{ℤ} HF_{red,odd}^{+}(M, [k]).

Then one defines

sw_{M,[k]}^{OSz} := χ(HF^{+}(M, [k])) − \frac{d(M, [k])}{2}.

5.2. Conjecture (strong version). Let (X, 0) be a normal surface singularity whose link M is a rational homology sphere. Let π : X̃ → X be a fixed good resolution and s := #J the number of irreducible exceptional divisors of π. Consider an arbitrary l′ ∈ 𝕃′ and define the characteristic element k := K − 2l′ ∈ Char. Then

(a) For any line bundle L ∈ P ic(X̃) with c_{1}(L) = l′ one has

h^{1}(L) ≤ −sw_{M,[k]}^{∗} − \frac{k^{2} + s}{8}.  (1)

(b) Additionally, if L = OX̃(l′) and (X, 0) is ℚ-Gorenstein (i.e. some power of the line bundle of holomorphic 2-forms over X \ {0} is analytically trivial), then in (1) one has equality.

(For a generalization see 5.3(d), where the restriction l′ ∈ 𝕃′ is dropped.)

5.3. Remarks.

(a) If L = OX̃ then h^{1}(OX̃) is the geometric genus p_{g}(X, 0) of (X, 0), hence this case corresponds exactly to the conjecture formulated in [15]. For detailed historical remarks and list of cases for which the conjecture (for OX̃) was verified, see [15, 16, 17, 13].

(b) Notice that sw_{M,[k]}^{∗} depends only on the class [k] of k. In particular, the right hand side of (1) consists of the “periodical” term −sw_{M,[k]}^{∗} and the “quadratic” term −(k^{2} + s)/8.

(c) In order to prove the conjecture, it is enough to verify it for line bundles L with c_{1}(L) = l′ of type l′ = l′_{ne}(h) (for some h ∈ H).

Indeed, write l′ in the form l′ = l′_{1} +l where l′_{1} = e(l′) = l′_{ne}(l′ +L) and l ∈ L_{e}. Let RHS(l′), resp. RHS(l′_{1}), be the right hand side of (1) for l′, resp. l′_{1}. Since [K −2l′] = [K −2l′_{1}], the Seiberg-Witten invariants are the same, hence

RHS(l′) − RHS(l′_{1}) = \frac{−(K − 2l′)^{2} + (K − 2l′_{1})^{2}}{8} = −(l, l′) − χ(l).

This combined with 4.2(c) shows that the statements of 5.2 for L and L ⊗ OX̃(−l) are equivalent.

(c’) Consider any set of representatives {l′}_{l′∈R} (R ⊂ 𝕃′) of the classes H, i.e. {l′ + L}_{l′∈R} = H. Then the comparison (c) can be also be done for any L with l′ := c_{1}(L) ∈ R and for L ⊗ OX̃(−l). Therefore, the validity of the conjecture 5.2 follows from the verification of (1) for line bundles L with c_{1}(L) ∈ R.

E.g., one can take R = Q, or R = −Q. The importance of R = −Q is emphasized by 3.7. This fact is exploited in second version of the conjecture.

(d) Similarly, if one verifies the inequality (1) for any l′ ∈ 𝕃′, then one gets automatically the inequality (1) for any l′ ∈ L′, hence for any L ∈ P ic(X̃). This statement follows by induction: if the inequaity (1) is valid for some L, then it is valid for L ⊗ OX̃(−E_{j}) (for any j ∈ J).

Indeed, using the exact sequence 0 → L ⊗ OX̃(−E_{j}) → L → L|_{E_{j}} → 0, one gets that h^{1}(L ⊗ OX̃(−E_{j})) ≤ h^{1}(L) + 1 + (c_{1}(L), E_{j}). The proof ends with similar comparison as in (c).

The point is that if l′ ∉ 𝕃′, then the inequality (1), in general, is not sharp (optimal).

5.4. Example. Almost-rational singularities. [13] First we recall that a singularity is rational if its graph is rational, namely, if for any l > 0 one has χ(l) > 0 (cf. [1, 2]). We say that (X, 0) is almost-rational (in short AR) if one of its good resolution graphs is almost-rational. A graph Γ is almost rational if it is a negative definite connected tree which has a distinguished vertex j_{0} ∈ J such that replacing e_{j_{0}} by some e′_{j_{0}} ≤ e_{j_{0}} we get a rational graph.

The set of AR graphs is rather large (cf. [13] 8.2): all rational, weakly elliptic and star-shaped graphs are AR. Moreover, they contain all the graphs considered by Ozsváth and Szabó in [22]. The class of AR graphs is closed while taking subgraphs and decreasing the Euler numbers e_{j}.

The main point of [13] is that for 3-manifolds with AR plumbing graphs, the Seiberg-Witten invariants can be computed combinatorially using “computational sequences” (objects in general used in singularity theory to evaluate p_{g}). In fact, theorem 9.6 of [13] implies the following.

5.5. Fact. Part (a) of the strong version of the conjecture (hence 5.3(d) as well) is true for any almost rational singularity.

5.6. Discussion. (a) Let (X_{a}, 0) be the universal abelian cover of (X, 0) with its natural H-action. Obviously, if p : Z̃ → X_{a} is a resolution of (X_{a}, 0), then Z̃ inherits a natural H-action. Recall that the geometric genus p_{g}(X_{a}, 0) of (X_{a}, 0) is h^{1}(Z̃, OZ̃). But one can define much finer invariants: consider the eigenspace decomposition ⊕_{χ∈Ĥ}H^{1}(Z̃, OZ̃)_{χ} of H^{1}(Z̃, OZ̃), and take

p_{g}(X_{a}, 0)_{χ} := dim_{ℂ} H^{1}(Z̃, OZ̃)_{χ} (for any χ ∈ Ĥ).

(b) Obviously, we can repeat the above definition for any (unbranched) abelian cover of (X, 0). More precisely, for any epimorphism H → K one can take the composed map π_{1}(M) → H → K which defines a Galois K-covering (X_{K}, 0) → (X, 0) of (X, 0) (with (X_{K}, 0) normal). Similarly as above, one can define p_{g}(X_{K}, 0)_{χ} for any χ ∈ K̂. But these invariants are not essentially new: all of them can be recovered from the corresponding invariants associated with the universal abelian cover. Indeed, consider χ ∈ Ĥ via K̂ ↪ Ĥ. Then p_{g}(X_{K}, 0)_{χ} = p_{g}(X_{a}, 0)_{χ}. In particular,

p_{g}(X_{K}, 0) = ∑_{χ∈K̂} p_{g}(X_{a}, 0)_{χ}.

(c) In the above definition (part (a)), p_{g}(X_{a}, 0)_{χ} is independent of the choice of Z̃, in particular one can take Z̃ considered in the proof of 3.4. Those facts, together with 3.7 show that

p_{g}(X_{a}, 0)_{θ(h)} = h^{1}(OX̃(−l′_{e}(h))) (for any h ∈ H).

Since the set {−l′_{e}(h)}_{h∈H} is a set of representatives (cf. 5.3(c’)), the previous version 5.2 of the conjecture, restricted to the set of line bundles of type OX̃(l′) is completely equivalent with the following.

5.7. Conjecture (weak version). Let (X, 0) be a normal surface singularity whose link M is a rational homology sphere. Let π : X̃ → X be a fixed good resolution and s := #J the number of irreducible exceptional divisors of π. For any h ∈ H = H_{1}(M, ℤ) consider the character χ := θ(h) and the characteristic element k := K + 2l′_{e}(h) ∈ Char. Then for any h ∈ H

p_{g}(X_{a}, 0)_{θ(h)} ≤ −sw_{M,[k]}^{∗} − \frac{k^{2} + s}{8}.  (2)

Additionally, if (X, 0) is ℚ-Gorenstein then in (2) one has equality.

5.8. Assumption. For some sw^{∗}-theory and link M consider the identity

∑_{[k]∈Spin^{c}(M)} sw_{M,[k]}^{∗} = λ(M).  (3)

Notice that for sw^{T CW} (and any M), (3) is true because ∑_{[k]} 𝒯_{M,[k]}(1) = 0. In particular, we expect that (3) is valid in any situation. [Obviously, if for some M one verifies that sw^{OSz} = sw^{T CW}, then (3) will be valid for sw^{OSz} as well. This is the case of Seifert 3-manifolds, see [13].]

5.9. Corollary. (I). (2) and (3) imply

p_{g}(X_{a}, 0) ≤ −λ(M) − ∑_{l′∈Q} \frac{(K + 2l′)^{2} + s}{8}  (4)

for any normal singularity (X, 0), with equality if (X, 0) is ℚ-Gorenstein.

(II). Assume that for some link M the identity (3) works, and also the inequality (2) was verified (see e.g. 5.4). Then the equality (4) implies the Conjecture 5.7 with equalities for all h.

5.10. Example. For example, for singularities with good ℂ^{∗} action the link is a Seifert manifold. In this case the inequality in (2) and the identity (3) was verified in [13] (cf. also 5.4). Therefore, the weak version 5.7 of the conjecture with equalities follows from (4) with equality. We exemplify this by a Brieskorn singularity. The general case will be treated in the forthcoming paper [14].

Assume that (X, 0) = {x^{2} + y^{3} + z^{12t+2} = 0}. Here we assume that t ≥ 1 (if t = 0 then (X, 0) is rational, a case which will be clarified in the next section.) The following invariants of (X, 0) can be computed using [15], section 6.

The minimal good resolution graph of M is star-shaped with three arms and (normalized) Seifer invariants (α, ω) equal to (3, 1), (3, 1), (6t + 1, 2t), the self intersection of the central curve is −1, the orbifold euler number equals −1/(18t + 3), K^{2} + s = 2, λ(M) = −(24t + 1)/12, and H = ℤ_{3}.

Consider the two arms corresponding to Seifert invariants (3, 1). Both of them contain only one vertex with self intersection −3. We denote them by E_{1}, respectively E_{2}. Then Q contains exactly three element. They are 0, l′_{1} := (E_{1} + 2E_{2})/3 and l′_{2} := (2E_{1} + E_{2})/3. One can rewrite

∑_{l′∈Q} \frac{(K + 2l′)^{2} + s}{8} = |H| · \frac{K^{2} + s}{8} − ∑_{l′∈Q} χ(l′).

In our case, by an easy verification χ(l′_{1}) = χ(l′_{2}) = 1/3. Therefore, the right hand side of (4) is (24t + 1)/12 − 3 · 2/8 + 2/3 = 2t.

On the other hand, the universal abelian cover of (X, 0) is isomorphic to a Brieskorn singularity of type (X_{a}, 0) = {u^{3} + v^{3} + w^{6t+1} = 0} (with the action ξ ∗ (u, v, w) = (ξu, ξ̄v, w), ξ^{3} = 1). And one can verify easily that p_{g}(X_{a}, 0) = 2t. Hence, for (X, 0), (2) is valid with equalities.

In fact, in this case p_{g}(X, 0) = 2t as well, hence p_{g}(X_{a}, 0)_{χ} = 0 for any χ ≠ 1.

5.11. It is also instructive to compare the statements of the conjecture applied for both (X, 0) and for some abelian cover (X_{K}, 0) of (X, 0) (cf. 5.6(b)), provided that the link of (X_{K}, 0) is again a rational homology sphere. In the next corollary we exemplify this in a simple situation, the interested reader is invited to work out more complicated cases. Write M_{a}, K_{a} and s_{a} for the corresponding invariants associated with (X_{a}, 0). (In particular, M_{a} is the universal abelian regular cover of M.) Then one has:

5.12. Corollary. Assume that (X, 0) is ℚ-Gorenstein, M is a rational homology sphere and M_{a} is an integral homology sphere. Then expressing −p_{g}(X_{a}, 0) in two different ways, one gets:

λ(M_{a}) + \frac{K_{a}^{2} + s_{a}}{8} = λ(M) + ∑_{l′∈Q} \frac{(K + 2l′)^{2} + s}{8}.

6. The case of rational singularities

6.1. In this section we assume that (X, 0) is rational, and sw^{∗} = sw^{OSz}. Some of the arguments are based on the results of [13] about sw_{M,[k]}^{OSz}. E.g., by [13] 6.3 and 8.3, one has:

−sw_{M,[k]}^{OSz} = \frac{d(M, [k])}{2} = \frac{k_{r}^{2} + s}{8} (cf. also with 2.4).  (1)

Recall also that any rational singularity is automatically ℚ-Gorenstein.

6.2. Theorem. Conjecture 5.2 (hence 5.7 too) is true for any rational singularity if we take sw_{M,[k]}^{∗} = sw_{M,[k]}^{OSz}.

Proof. By 5.3(c), we can assume that l′ = l′_{ne}(h) where h = l′ + L. Using 4.1.1, and the fact that l′_{ne}(h) ∈ NE_{ℚ}, one gets that h^{1}(OX̃(l′)) = 0. On the other hand, by (1), −sw_{M,[k]}^{OSz} = (k_{r}^{2} + s)/8. Here (see also 2.4), k = K −2l′, hence k_{r} = K +2l̄′_{ne}(−l′ +L) with l̄′_{ne}(−l′ +L) = −l′_{ne}(−(−l′ +L)) = −l′_{ne}(l′ + L) = −l′. Hence k_{r} = k and the right hand side of 5.2(1) is also vanishing. □

Theorem 6.2 (and its proof) and the identity 6.1(1) have the following consequence (whose statement is independent of the Seiberg-Witten theory):

6.3. Corollary. Assume that (X, 0) is rational, and fix some h ∈ H. Then

p_{g}(X_{a}, 0)_{θ(h)} = \frac{(K + 2l̄′_{ne}(h))^{2} − (K + 2l′_{e}(h))^{2}}{8} = −χ(l̄′_{ne}(h)) + χ(l′_{e}(h)).

In particular, (X_{a}, 0) is rational if and only if χ(l̄′_{ne}(h)) = χ(l′_{e}(h)) for any h ∈ H.

This emphasizes in an impressive way the differences between the two “liftings” l′_{e}(h) and l̄′_{ne}(h). Recall: for a class h = l′ + L, both l′_{e}(h) and l̄′_{ne}(h) are elements of l′ + L, but the first is minimal in L_{ℚ,e} (i.e. it is the representative in Q), while the second is minimal in −NE_{ℚ}. Since −NE_{ℚ} ⊂ L_{ℚ,e}, one has l′_{e}(h) ≤ l̄′_{ne}(h). In some cases they are not equal.

For example, take the A_{4} singularity, where E has three components E_{1}, E_{2}, E_{3}, all with selfintersection −2, and E_{2} intersecting the others transversely. Then −D_{2} = (1/2, 1, 1/2) = l̄′_{ne}(h) for some h, but it is not in Q: l′_{e}(h) = −D_{2} − E_{2} = (1/2, 0, 1/2). Nevertheless, in this case, their Euler characteristics are the same (corresponding to the fact that – A_{4} being a cyclic quotient singularity – the universal abelian cover is smooth).

6.4. Example. Assume that (X, 0) is a rational singularity with the following dual resolution graph.

[figure: −2; −3; −3; −3]

Let X̃ be its minimal resolution, we write E_{0} for the central exceptional component, the others are denoted by E_{i}, i = 1, 2, 3. Then by a computation l′ := −(D_{1} + D_{2} + D_{3}) = (1, 2/3, 2/3, 2/3). This element is minimal in −NE_{ℚ} but not in L_{ℚ,e}. Its representative in Q is l′ − E_{0}. The character χ = θ([l′]) is χ(g_{0}) = 1 and χ(g_{j}) = e^{4πi/3} for j = 1, 2, 3. By a computation K = −l′, (K+2l′)^{2} = −2 and (K +2(l′ −E_{0}))^{2} = −10. Therefore, p_{g}(X_{a}, 0)_{χ} = (−2+10)/8 = 1. In particular, (X_{a}, 0) cannot be rational.

Indeed, one can verify (using e.g. [18]), that the minimal resolution of (X_{a}, 0) contains exactly one irreducible exceptional curve of genus 1 and self-intersection −3. In particular, (X_{a}, 0) is minimally elliptic with p_{g}(X_{a}, 0) = 1. This also shows that the above eigenspace is the only non-trivial one.

We reverify this last fact for the conjugate χ̄ of χ. In this case h = θ^{−1}(χ̄) is the class of −D_{0} = (1, 1/3, 1/3, 1/3) = l̄′_{ne}(h); and l′_{e}(h) = −D_{0} − E_{0}. By a calculation K + 2l̄′_{ne}(h) = E_{0} and K + 2l′_{e}(h) = −E_{0}, hence their squares are the same. In particular, p_{g}(X_{a}, 0)χ̄ = 0.

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Department of Mathematics, Ohio State University, Columbus, OH 43210

   E-mail address: nemethi@math.ohio-state.edu
   URL: http://www.math.ohio-state.edu/ ~nemethi/