JACOBI FORMS AND THE STRUCTURE OF DONALDSON INVARIANTS FOR 4-MANIFOLDS WITH b_{+} = 1

LOTHAR GÖTTSCHE AND DON ZAGIER

Contents

1. Introduction 2 2. Preliminaries 5 2.1. Walls and chambers 6 2.2. Some elementary computations with modular forms 8 2.3. Wall-crossing formula 10 3. Theta functions for indefinite lattices 11 3.1. The function F 12 3.2. Definition of the theta functions 12 3.3. Jacobi forms 15 3.4. Properties of Θ_{L}^{f,g} for f, g ∈ S_{L} 16 3.5. The structure theorem for the theta functions 19 4. Application to Donaldson invariants 22 4.1. Extension of the Donaldson invariants 22 4.2. Blowup formulas 24 4.3. The structure theorem for the differences 26 4.4. Speculations about the relation to Seiberg-Witten theory 29 5. The case of rational algebraic surfaces 30 5.1. Donaldson invariants of ℙ_{1} × ℙ_{1} and ℙ_{2}#ℙ̄_{2} 30 5.2. The structure theorem for rational surfaces 33 5.3. Examples 34 References 36

Key words and phrases. Blowup-formulas, Donaldson invariants, Jacobi forms, modular forms.

1. Introduction

In the last few years the Donaldson invariants of smooth 4-manifolds have been very powerful tools. They are defined using a Riemannian metric g on the 4-manifold X, but in the case that b_{+}(X) > 1 they are independent of g. Recently very much progress has been made in the understanding of these invariants. In particular Kronheimer and Mrowka [Kr-M1], [Kr-M2] and Fintushel and Stern [F-S2] have shown some important structure theorems.

For C ∈ H^{2}(X, ℤ) and c_{2} ∈ ℤ we denote by Φ_{C,d}^{X} the Donaldson invariant of X with respect to a complex rank 2 bundle E with Chern classes C and c_{2} with 4c_{2} − C^{2} − \frac{3}{2}(1 + b_{+}(X)) = d. Φ_{C,d}^{X} is a linear map from the set A_{d}(X) of polynomials of degree d in H_{2}(X, ℚ) ⊕ H_{0}(X, ℚ) to ℚ, where the classes in H_{2}(X, ℚ) are given degree 1 and the class p ∈ H_{0}(X, ℤ) corresponding to a point is given degree 2. We put Φ_{C,d}^{X} := 0 if b_{+}(X) is even or if d is not congruent to −C^{2} − \frac{3}{2}(1 + b_{+}(X)) modulo 4, and set Φ_{C}^{X} := ∑_{d≥0} Φ_{C,d}^{X}, thought of as a linear map from A_{∗}(X) = ∑_{d≥0} A_{d}(X) to ℚ. The polynomials Φ_{C,d}^{X} depend only on C modulo 4H^{2}(X, ℤ) and up to sign only on C modulo 2H^{2}(X, ℤ).

Kronheimer and Mrowka introduce the notion of simple type: The manifold X is of simple type if Φ_{C}^{X}(α(p^{2}−4)) = 0 for all α ∈ A_{∗}(X) and all C ∈ H^{2}(X, ℤ). In [Kr-M1], [Kr-M2], [F-S2] the structure of the Donaldson invariants of 4-manifolds of simple type is then described in a very compact way: There exist finitely many so-called basic classes K_{i} ∈ H^{2}(X, ℤ) (independent of C) and rational numbers a_{i} (depending on C) such that for all x ∈ H_{2}(X, ℤ) there is an identity

Φ_{C}^{X}((1 + p/2)e^{xz}) = e^{Q(x)z^{2}/2} ∑_{i} a_{i}e^{K_{i}xz},

of formal power series in z, where Q is the quadratic form on H_{2}(X, ℤ). Note that this formula determines Φ_{C}^{X} completely since Φ_{C,d}^{X}(p^{i}x^{n}) is 0 unless 2i+n ≡ −C^{2} − \frac{3}{2}(1+b_{+}(X)) modulo 4. Very many 4-manifolds with b_{+} > 1 have been shown to be of simple type, and the class of 4-manifolds of simple type is closed under several natural operations on 4-manifolds, like connected sum with ℙ̄_{2} and rational blowdown [F-S3].

Fintushel and Stern and Kronheimer and Mrowka have also introduced a generalization of the concept of simple type, the k-th order simple type, where one instead requires that Φ_{C}^{X}((p^{2} −4)^{k}α) = 0. They showed that all manifolds with b_{+} > 1 are of k-th order simple type for some k, but it is not known whether k > 1 is ever needed.

A new light has been shed on these structure theorems by the Seiberg-Witten invariants [S-W],[W1]. A class K in H^{2}(X, ℤ) is called SW -basic if the corresponding Seiberg-Witten invariant does not vanish, and X is of SW -simple type if for all SW -basic classes the corresponding moduli space is 0-dimensional. It is conjectured that the SW -basic classes are the same as the basic classes in Donaldson theory, and that the condition of simple type in Donaldson theory and in Seiberg-Witten theory are equivalent. From the viewpoint of theoretical physics the precise relation between the Donaldson and the Seiberg-Witten invariants should be given in terms of the modular curve ℌ/Γ(2).

There is a project towards giving a mathematical proof of this relationship [P-T1], [P-T2]. All symplectic 4-manifolds with b_{+} > 1 are known to be of SW -simple type [T2], and no 4-manifold with b_{+} > 1 is known not to be of SW -simple type.

In this paper we study the case b_{+} = 1, where the invariants have been introduced in [K1]. In this case the invariants are no longer independent of the metric, but in [K-M] they were shown to depend only on the so-called period point of the metric in the positive cone H^{2}(X, ℝ)^{+}. We will therefore denote them by Φ_{C,d}^{X,L}, where L is the period point. In fact there is a collection of cohomology classes ξ ∈ H^{2}(X, ℤ) defining “walls” (hyperplanes), and Φ_{C,d}^{X,L} depends only on the chamber (= connected component of the complement of the walls) to which L belongs. The change of the Donaldson invariants when passing through a wall is given by wall-crossing terms δ_{ξ,d}^{X}. Kotschick and Morgan made a conjecture about the structure of the δ_{ξ,d}^{X}. In particular they should only depend on ξ^{2}, d and the homotopy type of X. Morgan and Ozsváth have announced a proof of this conjecture [M-O]. In this whole paper we will assume the conjecture.

In [K-L] the wall-crossing formulas were used in combination with the blowup formulas to compute Donaldson invariants of ℙ_{2} and ℙ_{1} × ℙ_{1} and to show in particular that ℙ_{2} is not of simple type and ℙ_{1} × ℙ_{1} is not of simple type for any chamber. In [E-G1],[F-Q],[E-G2] the δ_{ξ,d}^{S} were studied in the case of algebraic surfaces S with p_{g} = 0. In [E-G1],[F-Q] they were expressed in terms of Hilbert schemes of points on S, and the leading terms were computed. In [E-G2] the Bott residue formula is applied to computing the δ_{ξ,d}^{S}, and the Donaldson invariants, for rational surfaces with help of the computer. In [G], assuming the conjecture of Kotschick and Morgan and using the blowup formulas, the δ_{ξ,d}^{X} were determined completely (for arbitrary X) in terms of modular forms. On a rational algebraic surface one can, after possibly blowing up, always find a chamber where the Donaldson invariants vanish. Therefore the Donaldson invariants of rational surfaces can always be expressed in terms of modular forms. In particular this gives the Donaldson invariants of ℙ_{2}. The results made it look very unlikely that ℙ_{2} or ℙ_{1} × ℙ_{1} can be of k-th order simple type for any k. By just straightforwardly applying the results from [G] one will in general get a very complicated expression for the Donaldson invariants, from which it is very hard to read off structural results.

On the other hand Morgan and Szabó showed [M-Sz] that, for some rational surfaces admitting an elliptic fibration with CF odd (F ∈ H^{2}(X, ℤ) dual to the class of a fibre), the limit of the Donaldson invariants Φ_{C}^{X,L}, for the period point L tending to F, fulfills the simple type condition Φ_{C}^{X,L}(α(p^{2} −4)) = 0, and the corresponding Donaldson power series is given by formulas analoguous to those of [Kr-M2] and [F-S2]. In fact this is a special case of their more general results.

In the current paper we want to apply the results of [G] to understand the structure of the Donaldson invariants of 4-manifolds with b_{+} = 1. Looking at the previously known results (e.g. [K-L],[E-G2]), it seems very unlikely that results analogous to [Kr-M2] and [F-S2] can hold for period points in the inside of the positive cone, while the results of [M-Sz] and the structure of the formulas in [G] suggest that one should restrict the attention to the boundary. One first has to find the correct definition of the Donaldson invariants for a period point F there. If, for a primitive representative F ∈ H^{2}(X, ℤ), the number CF is odd, then F lies in the closure of a unique chamber and on no wall, and one just takes the values of the invariants in this chamber. In case CF is even, F will in general lie in the closure of infinitely many chambers, and one has to take a certain kind of “renormalized average” over all of them.

With this correction the Donaldson invariants are well-defined, and we show that for F, G ∈ H^{2}(X, ℤ) on the boundary of the positive cone, the difference Φ_{C}^{X,F} − Φ_{C}^{X,G} always fullfils the k-th order simple type condition for k = (W^{2} − σ(X))/8, where σ(X) is the signature of X and W is a characteristic element with W^{2} maximal in a certain sector of H^{2}(X, ℤ) determined by F and G. We also give a universal formula for the precise structure of this difference in terms of modular forms and an explicit set of basic cohomology classes, which are again characteristic elements in the above sector of H^{2}(X, ℤ). With k as above the leading terms of this formula give an expression for (Φ_{C}^{X,F} − Φ_{C}^{X,G})(e^{xz}(1 + p/2)(1 − p^{2}/4)^{k−1}) analogous to that of [Kr-M2] and [F-S2]. There is however one difference: While in the case b_{+} > 1 there is only a finite number of basic classes, one can interpret the formulas in our case as saying that there are infinitely many (all but finitely many orthogonal to F or G), and that the Donaldson invariants are obtained as a “renormalized” (by analytic continuation) sum of their contributions.

We note that the basic classes not orthogonal to F or G are precisely the characteristic cohomology classes whose corresponding Seiberg-Witten invariants (with respect to the unperturbed equations) for metrics with period points near F and near G differ. This leads to a conjectural formula for the relationship between Seiberg-Witten and Donaldson invariants in the case b_{+} = 1 on the boundary of the positive cone.

If X is a rational algebraic surface, it is easy to show that there always exists a point on the boundary of the postive cone such that all Donaldson invariants vanish. Therefore for these surfaces the formulas above give the Donaldson invariants and not just their differences. In particular the above-mentioned conjecture about the relation of Donaldson and Seiberg-Witten invariants holds for rational surfaces. In particular this shows that for any k very many (X, F) are of strictly k-th order simple type, related to the fact that in case b_{+} = 1, the chamber structure in Seiberg-Witten theory will very often make SW -basic classes of arbitrary expected dimension appear.

In the case of ℙ_{2}#ℙ̄_{2} or ℙ_{1} × ℙ_{1}, the limit of the Donaldson invariants at the boundary of the positive cone is also defined without the correction, but the results of [K-L] show that they are not of simple type, whereas with the correction they vanish. The limit can therefore be computed as the negative of the correction we introduce, proving a conjecture from [E-G2]. We conclude by giving a number of examples for our formulas.

To prove our results we observe that by the results of [G] the formula for the difference of the Donaldson invariants at two boundary points is very closely related to a new kind of theta functions, associated to a lattice L of type (r − 1, 1) and a pair (f, g) of elements in L with self-intersection 0 (in fact the lattice is H_{2}(X, ℤ) with the negative of the intersection form). We show that these theta functions are Jacobi forms for the theta group Γ_{θ}. Using this fact one can show that the difference of the Donaldson invariants has a development in terms of modular functions for a subgroup Γ_{u} ⊂ Γ_{θ} which are holomorphic except at the cusps −1, 1, ∞ of Γ_{u}. It follows that they can be expressed as a rational function in a certain modular function u, and that the relevant information about the Donaldson invariants can be read off from the poles at the cusp −1. We note that Γ_{u} is conjugate to Γ(2) by an element of GL(2, ℤ), which maps the cusps −1, 1, ∞ to ∞, 0, 1, thus giving a connection to the description from theoretical physics. In particular all the computations could be rephrased in terms of Γ(2).

We also mention the connection with the recent work of Borcherds on automorphic forms on orthogonal groups ([Bo], in particular the results of §10 and the examples and problems concerning Donaldson polynomials in §15 and §16).

The first named author would like to thank Barbara Fantechi, with whom he had useful discussions on several aspects of this paper, Ronald Stern, who told him about the notion of higher order simple type, Victor Pidstrigach and Andrei Tyurin who explained their program for proving the relationship of Donaldson and Seiberg-Witten invariants to him, and John Morgan and Zoltan Szabó, who informed him about their results about the structure of some Donaldson invariants e.g. on ℙ_{2}#9\overline{ℙ}_{2}; in fact this paper was motivated in part by trying to understand that result. With Zoltan Szabó he also had some further discussions, which helped to clarify our ideas. This work was started while the first named author was at the Max-Planck-Institut für Mathematik Bonn, and carried out during his stay at Pisa, with a grant of MAP.

2. Preliminaries

In this paper let X be a simply connected smooth 4-manifold with b_{+}(X) = 1.

Notation 2.1. We will usually denote by upper case letters the classes in H^{2}(X, ℂ), unless these appear as walls (see below), when we denote them by Greek letters. Elements of H_{2}(X, ℂ) are denoted by lower case letters. We usually denote the Poincaré dual class of A ∈ H^{2}(X, ℂ) by the corresponding lower case letter a. For A, B ∈ H^{2}(X, ℂ) and the corresponding dual classes a, b ∈ H_{2}(X, ℂ), the canonical pairing of H^{2}(X, ℂ) and H_{2}(X, ℂ) and the intersection product on H^{2}(X, ℂ) are just denoted by Ab and AB respectively. We denote Q(a) the quadratic form (given by A^{2}) on H_{2}(X, ℂ) and by σ(X) the signature of X. We denote X̂ := X#ℙ̄_{2}, and E the canonical generator of H^{2}(ℙ̄_{2}, ℤ). We will always identify H^{2}(X, ℚ) with the orthogonal subspace E^{⊥} ⊂ H^{2}(X̂, ℚ). We trust that there will be no confusion between the exponential function and the Poincaré dual of E.

Let E a complex rank 2 bundle with first Chern class C and second Chern class c_{2}. We put d := 4c_{2}−C^{2}−3, and denote by Φ_{C,d}^{X,g} the Donaldson invariant corresponding to E and the (generic) metric g (cf. [Do]). They depend up to sign only on C modulo 2, and the sign is determined by C modulo 4 and by a so-called homology orientation. We use the conventions of e.g. [F-S1]. Let p ∈ H_{0}(X, ℤ) be the class of a point. Let A_{d}(X) be the set of polynomials of weight d in H_{2}(X, ℂ) ⊕ H_{0}(X, ℂ), where a ∈ H_{2}(X, ℂ) has weight 1 and p has weight 2, and put A_{∗}(X) := ⨁_{d≥0} A_{d}(X). Then Φ_{C,d}^{X,g} is a linear map A_{d}(X) ⟶ ℚ. The letters z and t will denote indeterminates; z will also be sometimes a complex variable, and x usually denotes an element of H_{2}(X, ℂ). We will very often write 1^{a} instead of e^{2aπi} for a rational number a.

2.1. Walls and chambers. In the case b_{+} = 1 the Donaldson invariants are no longer independent of the metric.

Definition 2.2. Firstly we recall that a Riemannian metric g determines a ray in

H^{2}(X, ℝ)^{+} = {H ∈ H^{2}(X, ℝ) | H^{2} > 0};

namely the set of self-dual harmonic forms. This ray, or any representative in H^{2}(X, ℝ)^{+}, is called the period point of g and denoted ω(g). The quotient H^{2}(X, ℝ)^{+}/ℝ^{+} has two connected components. The choice of a homology orientation amounts to the choice of one of them, which we call ℍ_{X}. We will in future always assume that we have chosen a homology orientation, and view the period points as lying in the corresponding component ℍ_{X}.

The space ℍ_{X} is a model of hyperbolic r-space, where r = b_{2}(X) − 1. In particular we can complete it to \overline{ℍ}_{X} := ℍ_{X} ∪ 𝕊_{X}, where

𝕊_{X} := ({H ∈ H^{2}(X, ℚ) |H^{2} = 0} \ {0})/ℚ^{∗}

is the set of cusps. We will usually not distinguish between an element H ∈ ℍ_{X} and a representative in H^{2}(X, ℝ)^{+}, and similar for 𝕊_{X} (with the representative on the boundary of the chosen component).

Definition 2.3. (see e.g. [K1], [K-M]) By a wall in ℍ_{X} we mean the intersection of ℍ_{X} with a set

W^{ξ} := {L ∈ H^{2}(X, ℝ)^{+} | ξL = 0}/ℝ^{+}.

where ξ ∈ H^{2}(X, ℚ) with ξ^{2} < 0. If C ∈ H^{2}(X, ℤ), and d ∈ ℤ_{≥0}, we will say that an element ξ ∈ H^{2}(X, ℚ) is of type (C, d) if C/2 − ξ ∈ H^{2}(X, ℤ) and (d + 3)/4 + ξ^{2} ∈ ℤ_{≥0}. A chamber of type (C, d) is a connected component of the complement of the walls in ℍ_{X} defined by elements of type (C, d).

Theorem 2.4. [K-M] 1. Φ_{C,d}^{X,g} depends only on the chamber of the period point ω(g).

2. For all ξ ∈ \frac{1}{2}H^{2}(X, ℤ) of type (C, d) there exists a linear map δ_{ξ,d}^{X} : A_{d}(X) ⟶ ℂ such that for all generic metrics g_{1} and g_{2}

Φ_{C,d}^{X,g_{1}} − Φ_{C,d}^{X,g_{2}} = 1^{C^{2}/8} ∑_{ξ}(−1)^{(ξ−C/2)C} δ_{ξ,d}^{X},

where the sum runs through all ξ of type (C, d) with ξω(g_{2}) < 0 < ξω(g_{1}).

Note that the conventions are different from [G],[K-M],[K-L]. We have changed the sign conventions and replaced ξ by ξ/2 to get below a more direct relation to theta functions.

Actually in [K-M] the result is only proven for the restriction of Φ_{C,d}^{X,g} to Sym^{d}(H_{2}(X, ℚ)); see [G] for the extension to A_{d}(X). We mention that this extension also works outside of the so-called stable range, i.e. when C ≡ 0 modulo 2 and for classes α ∈ A_{d}(X) containing monomials x^{k}p^{r} with x ∈ H_{2}(X, ℚ) and k < 2(r + 1).

Kotschick and Morgan make a conjecture about the structure of the wallcrossing terms.

Conjecture 2.5. [K-M] δ_{ξ,d}^{X}|_{Sym^{d}(H_{2}(X,ℚ))} is a polynomial in ξ and the quadratic form Q whose coefficients depend only on ξ^{2}, d, and the homotopy type of X.

In this statement, polynomials in ξ and Q are considered as maps on Sym^{d}(H_{2}(X), ℚ) by Q^{k}ξ^{ℓ}(x^{d}) = Q(x)^{k} (ξx)^{ℓ} if 2k + ℓ = d (and 0 otherwise) for x ∈ H_{2}(X, ℚ), and then by multilinearity for arbitrary elements of Sym^{d}(H_{2}(X), ℚ).

A proof of Conjecture 2.5 has been announced [M-O]. We will assume 2.5 for the whole of the paper.

By Theorem 2.4 the Donaldson invariant Φ_{C,d}^{X,g} depends only on the chamber of the period point ω(g) ∈ ℍ_{X}. For H ∈ ℍ_{X} not on a wall defined by a class of type (C, d) we will therefore put Φ_{C,d}^{X,H} = Φ_{C,d}^{X,g}, where g is a generic metric whose period point lies in the same chamber of type (C, d) as H. Apparently it is not known whether every element of ℍ_{X} appears as period point of a metric. If there is no period point in the chamber of H, then we define Φ_{C,d}^{X,H} by requiring that

Φ_{C,d}^{X,H} − Φ_{C,d}^{X,g} = 1^{C^{2}/8} ∑_{ξ}(−1)^{(ξ−C/2)C} δ_{ξ,d}^{X},  (2.5.1)

the sum running through all ξ of type (C, d) with ξω(g) < 0 < ξH, where g is any generic Riemannian metric on X. Theorem 2.4 implies that in this way Φ_{C,d}^{X,H} is well-defined.

Finally, we mention that if C ≡ 0 modulo 2 then Φ_{0,d}^{X,H}(α) was originally defined only for α in the stable range but can be defined for arbitrary α by

Φ_{C,d}^{X,H}(α) := Φ_{C+E,d+1}^{X̂,H+ϵE}(eα)  (ϵ > 0 sufficently small)  (2.5.2)

(see [G] definition 2.7). Note that now by the blowup formulas (e.g. [F-S1],[T1]) the formula (2.5.2) holds for all d and all α ∈ A_{d}(X).

We put Φ_{C,d}^{X,H} = 0 if d is not congruent to −C^{2} − 3 modulo 4 and, if Hξ ≠ 0 for all ξ ∈ H^{2}(X, ℤ) + C/2 (i.e. H does not lie on a wall defined by a class of type (C, d) for any d), we put Φ_{C}^{X,H} := ∑_{d≥0} Φ_{C,d}^{X,H}. We also put δ_{ξ}^{X} := ∑_{d≥0} δ_{ξ,d}^{X}. For an indeterminate z and a_{n} ∈ A_{∗}(X) for all n ≥ 0 we put

Φ_{C}^{X,H}( ∑_{n≥0} a_{n}z^{n}) := ∑_{n≥0} Φ_{C}^{X,H}(a_{n})z^{n}, δ_{ξ}^{X}( ∑_{n≥0} a_{n}z^{n}) := ∑_{n≥0} δ_{ξ}^{X}(a_{n})z^{n}

With this convention we define, for x ∈ H_{2}(X, ℤ) and P (p) a polynomial in p (the class of a point) the following two formal power series in variables z, t:

Ψ_{C}^{X,H}(x · z, P (p)) := Φ_{C}^{X,H}(e^{xz}P (p)), \overline{Ψ}_{C}^{X,H}(x · z, t) := ∑_{r=0}^{∞} Ψ_{C}^{X,H}(x · z, p^{r}) t^{r+1}. (2.5.3)

(The change of argument from e^{xz} in Φ to x · z in Ψ will become important in Section 4, where the invariant Ψ —but not Φ —is defined for H belonging to the boundary of the positive cone.)

Following [Kr-M1],[Kr-M2] we say that the pair (X, H) is of k-th order simple type if Φ_{C}^{X,H}(α(p^{2} − 4)^{k}) = 0 for all α ∈ A_{∗}(X) and all C ∈ H^{2}(X, ℤ). It is of strictly k-th order simple type if it is of k-th order simple type but not of (k − 1)-th order simple type.

2.2. Some elementary computations with modular forms. Let ℌ := {τ ∈ ℂ | Im(τ) > 0} be the complex upper half-plane. For τ ∈ ℌ let q := e^{2πiτ} and q^{1/n} := e^{2πiτ/n}. Elements A =

(\begin{matrix}a & b \\ c & d \end{matrix}) ∈ SL(2, ℤ) act on ℌ by Aτ := \frac{aτ + b}{cτ + d} and, given k ∈ ℤ, on functions g : ℌ ⟶ ℂ by

(g|_{k}A)(τ) := (cτ + d)^{−k}g (Aτ) .

For k ∈ \frac{1}{2}ℤ and (c, d) ≠ (0, −1) we define (g|_{k}A) by the same formula, where (cτ + d)^{−k} stands for \sqrt{cτ + d}^{−2k}, where we always use the principal branch of the square root (whose real part is positive on complex numbers with argument strickly between −π and π). Note that this in general does not define an action anymore. If g is a modular form of weight k or k is otherwise understood we just write g|A. Let

T := (\begin{matrix}1 & 1 \\ 0 & 1 \end{matrix}) , V := T^{2} = (\begin{matrix}1 & 2 \\ 0 & 1 \end{matrix}) , S := (\begin{matrix}0 & −1 \\ 1 & 0 \end{matrix}) , W := T^{−1}S = (\begin{matrix}−1 & −1 \\ 1 & 0 \end{matrix}) . (2.5.4)

Let Γ_{θ} = ⟨V, S⟩ ⊂ SL(2, ℤ) be the theta group. The quotient ℌ/Γ_{θ} has two cusps: −1 and ∞. Let η(τ) := q^{1/24} ∏_{n>0}(1 − q^{n}) be the Dirichlet eta function and Δ := η^{24} the discriminant. We denote σ_{k}(n) := ∑_{d|n} d^{k} and by σ_{1}^{odd}(n) the sum of the odd divisors of n. For even k ≥ 2 let

G_{k}(τ) := −\frac{B_{k}}{2k} + ∑_{n>0} σ_{k−1}(n)q^{n}

be the classical Eisenstein series, where B_{k} is the k-th Bernoulli number. For odd k we put G_{k}(τ) := 0. Note that G_{k} is a modular form of weight k for SL(2, ℤ) for k ≥ 4, but is only “quasi-modular” for k = 2, i.e., it transforms by equation 3.11.1.

Recall the classical theta functions

θ_{μν}(τ, z) := ∑_{n∈ℤ}(−1)^{nν}q^{(n+μ/2)^{2}/2}e^{2πi(n+μ/2)z} (μ, ν ∈ {0, 1})  (2.5.5)

and their “Nullwerte”

θ(τ) := θ_{00}(τ, 0) = \frac{η(τ)^{5}}{η(τ/2)^{2}η(2τ)^{2}} , θ_{01}^{0}(τ) = θ_{01}(τ, 0) = \frac{η(τ/2)^{2}}{η(τ)}
θ_{10}^{0}(τ) = θ_{10}(τ, 0) = 2\frac{η(2τ)^{2}}{η(τ)} , θ_{11}^{0}(τ) = 0.

We set

f(τ) := 1^{−1/8} \frac{η(τ)^{3}}{θ(τ)} = 1^{−1/8} \frac{η(τ/2)^{2}η(2τ)^{2}}{η(τ)^{2}} = \frac{1^{−1/8}}{2} θ_{01}^{0}(τ)θ_{10}^{0}(τ).  (2.5.6)

The transformation laws

η(τ + 1) = 1^{1/24}η(τ), η(−1/τ) = \sqrt{\frac{τ}{i}} η(τ)

(where we again use the principal branch of the square root) imply the following.

1. θ|V = θ, θ|S = 1^{−1/8}θ, θ|T = θ_{01}^{0} and θ|W (τ) = 1^{−1/8}θ_{10}^{0}(τ) = 1^{−1/8} \frac{2η(2τ)^{2}}{η(τ)} .

2. f|V = if, f|S = −if, f|T^{−1} = 1^{−1/4} \frac{η(τ)^{3}}{θ_{01}^{0}} and f|W (τ) = −\frac{η(τ)^{4}}{2η(2τ)^{2}} . In particular f^{4} is a modular form of weight 4 for Γ_{θ}.

3. The function

R(τ) := \frac{Δ(τ)^{2}}{Δ(τ/2)Δ(2τ)}

is a modular function for Γ_{θ} and R|W (τ) = −2^{12}Δ(2τ)/Δ(τ).

Finally let e_{1}, e_{2}, e_{3} be the 2-division values of the Weierstraß ℘-function at 1/2, τ/2 and (1 + τ)/2 respectively, i.e.

e_{1}(τ) = −\frac{1}{6} − 4 ∑_{n>0} σ_{1}^{odd}(n)q^{n},
e_{2}(τ) = \frac{1}{12} + 2 ∑_{n>0} σ_{1}^{odd}(n)q^{n/2},
e_{3}(τ) = \frac{1}{12} + 2 ∑_{n>0}(−1)^{n}σ_{1}^{odd}(n)q^{n/2}.

We write

U(τ) := −\frac{3e_{3}(τ)}{f(τ)^{2}}, u(τ) := \frac{1}{U(τ)}, G(τ) := G_{2}(τ) + e_{3}(τ)/2.  (2.5.7)

The first two are modular functions, the third a quasi-modular form.

Lemma 2.6. 1. e_{3} is a modular form of weight 2 for Γ_{θ} and e_{3}|T = e_{2}, e_{3}|W = e_{1}. 2. We have the identities

U(τ)^{2} − 4 = −\frac{1}{16}R(τ),  (2.6.1)
q \frac{d}{dq} U(τ) = −\frac{1}{16}R(τ)f(τ)^{2}.  (2.6.2)

Proof. 1. is standard and can be shown e.g. by observing that e_{3}(τ) = −2G_{2}(τ/2)+8G_{2}(τ)−8G_{2}(2τ) and using the transformation behaviour of the quasi-modular form G_{2}. For 2., we note that the four functions f^{4}, e_{3}^{2} = U^{2}f^{4}/9, Rf^{4} and f^{2}q \frac{d}{dq} U are modular forms of weight 4 on Γ_{θ}. Therefore they are linear combinations of the two Eisenstein series G_{4}(τ) and G_{4}((τ + 1)/2), which generate the space of modular forms of weight 4 for Γ_{θ}. The identities (2.6.1) and (2.6.2) are then obtained by comparing the first few Fourier coefficients.

Remark 2.7. Let Γ_{u} = ±⟨V^{2}, V S, SV ⟩; this is a subgroup of index 2 of Γ_{θ}. The quotient ℌ/Γ_{u} has 3 cusps 1, −1 and ∞. From the above it is clear that U(τ) is a modular function for Γ_{u} and that f(τ)^{2} is a modular form of weight 2 for Γ_{u}. We note that U(τ) defines an isomorphism of from ℌ/Γ_{u} ∪ {1} ∪ {−1} ∪ {∞} to ℙ_{1}, as the smallest power of T contained in Γ_{u} is T^{4} and the lowest power of q in the Fourier development of U(τ) is −1/4. We see that U(∞) = ∞, U(−1) = 2, U(1) = −2. In fact W maps ∞ to −1 and V W maps ∞ to 1, and

Ũ(τ) := U|W (τ) = −\frac{12e_{1}(τ)η(2τ)^{4}}{η(τ)^{8}} = 2 + 64q + 512q^{2} + 2816q^{3} + …, U|V W (τ) = −Ũ(τ).

We denote ũ(τ) = u|W (τ) = 1/Ũ(τ). In particular every modular function for Γ_{u} will be a rational function in U(τ), and a polynomial in U(τ), (U(τ) − 2)^{−1} and (U(τ) + 2)^{−1} if it is holomorphic on ℌ. We note that Γ_{u} is conjugate to Γ(2) via the matrix (\begin{matrix}1 & −1 \\ 1 & 1 \end{matrix}) ∈ GL(2, ℚ), which sends the

cusps ∞, 1, −1 to 1, 0, ∞ respectively.

We finish this section by giving a list of the leading terms of the Fourier developments of some of the modular forms and functions we introduced, which will play a role in our later computations. This should help the reader to check and apply our computations and results.

θ(τ) = 1 + 2q^{\frac{1}{2}} + 2q^{2} + 2q^{\frac{9}{2}} + 2q^{8} + …
f(τ) = 1^{− \frac{1}{8}} q^{\frac{1}{8}} (1 − 2q^{\frac{1}{2}} + q − 2q^{\frac{3}{2}} + 2q^{2} + 3q^{3} − 2q^{\frac{7}{2}} − 2q^{\frac{9}{2}} + 2q^{5} − 2q^{\frac{11}{2}} + q^{6} − 2q^{\frac{13}{2}} + …)
R(τ) = q^{− \frac{1}{2}} + 24 + 276q^{\frac{1}{2}} + 2048q + 11202q^{\frac{3}{2}} + 49152q^{2} + 184024q^{\frac{5}{2}} + 614400q^{3}

+ 1881471q^{\frac{7}{2}} + 5373952q^{4} + 14478180q^{\frac{9}{2}} + …

e_{1}(τ) = −\frac{1}{6} − 4q − 4q^{2} − 16q^{3} − 4q^{4} − 24q^{5} − 16q^{6} − 32q^{7} − 4q^{8} − …
e_{3}(τ) = \frac{1}{12} − 2q^{\frac{1}{2}} + 2q − 8q^{\frac{3}{2}} + 2q^{2} − 12q^{\frac{5}{2}} + 8q^{3} − 16q^{\frac{7}{2}} + 2q^{4} − 26q^{\frac{9}{2}} + 12q^{5} − 24q^{\frac{11}{2}} + …
U(τ) = 1^{\frac{1}{4}} q^{− \frac{1}{4}} (−\frac{1}{4} + 5q^{\frac{1}{2}} + \frac{31}{2}q + 54q^{\frac{3}{2}} + \frac{641}{4}q^{2} + 409q^{\frac{5}{2}} + \frac{1889}{2}q^{3} + 2062q^{\frac{7}{2}}

+ \frac{17277}{4}q^{4} + 8666q^{\frac{9}{2}} + \frac{33439}{2}q^{5} + 31328q^{\frac{11}{2}} + 57313q^{6} + …)

u(τ) = 1^{\frac{1}{4}} q^{\frac{1}{4}} (4 + 80q^{\frac{1}{2}} + 1848q + 42784q^{\frac{3}{2}} + 990100q^{2} + 22911600q^{\frac{5}{2}} + 530190104q^{3}

+ 12268965984q^{\frac{7}{2}} + 283912371144q^{4} + …)

G(τ) = −q^{\frac{1}{2}} + 2q − 4q^{\frac{3}{2}} + 4q^{2} − 6q^{\frac{5}{2}} + 8q^{3} − 8q^{\frac{7}{2}} + 8q^{4} − 13q^{\frac{9}{2}} + 12q^{5} − 12q^{\frac{11}{2}} + 16q^{6} + …
Ũ(τ) = 2 + 64q + 512q^{2} + 2816q^{3} + 12288q^{4} + 45952q^{5} + 153600q^{6} + 470528q^{7}

+ 1343488q^{8} + 3619136q^{9} + 9280512q^{10} + …

θ_{10}^{0}(τ) = q^{\frac{1}{8}} (2 + 2q + 2q^{3} + 2q^{6} + 2q^{10} + …)
\frac{η(τ)^{4}}{η(2τ)^{2}} = 1 − 4q + 4q^{2} + 4q^{4} − 8q^{5} + 4q^{8} − 4q^{9} + 8q^{10} + …

2.3. Wall-crossing formula. In [G] definition 2.7 we extended the definition of the wall-crossing terms δ_{ξ,d}^{X} to all classes ξ ∈ H^{2}(X, ℤ) by use of the blowup formulas [F-S1],[T1]. If ξ^{2} < 0 we have in particular δ_{ξ,d}^{X} = 0, if ξ is not of of type (2ξ, d). The main theorem of [G] is:

Theorem 2.8. [G] Let X be a simply connected 4-manifold with b_{+} = 1 and signature σ(X). For x ∈ H_{2}(X, ℤ) put

Δ_{ξ}^{X}(τ, x · z) := −\frac{4θ(τ)^{σ(X)}}{f(τ)} · q^{−ξ^{2}/2}e^{−ξxz/f(τ)} · e^{−Q(x)G(τ)z^{2}/f(τ)^{2}}.

Then

δ_{ξ}^{X}(e^{xz}p^{r}) = Coeff_{u(τ)}^{r+1}[Δ_{ξ}^{X}(τ, x · z)] .  (2.8.1)

Here the symbol “Coeff” means that we expand the following expression (or more precisely, the coefficient of z^{n} in it for each fixed power n) as a fractional Laurent series in q, then rewrite it as as a fractional Laurent series in u(τ) = 4iq^{\frac{1}{4}} + ⋯, and then take the coefficient of the indicated power of u(τ).

Theorem 2.8 was stated in [G] (up to some differences in conventions) in the form

δ_{ξ}^{X}(e^{xz}p^{r}) := Res_{q=0}[\frac{1}{4}f(τ)^{2}U(τ)^{r}R(τ)Δ_{ξ}^{X}(τ, x · z)\frac{dq}{q} ].  (2.8.2)

Equation (2.8.1) follows from this by

Proposition 2.9. Let F be a meromorphic function on ℌ having a Laurent development in powers of q^{1/n} for some n ∈ ℤ_{>0}. Then

Res_{q=0}[\frac{1}{4}f(τ)^{2}U(τ)^{r}R(τ)F (τ)\frac{dq}{q} ] = Coeff_{u(τ)}^{r+1}[F (τ)].

Proof. We use the formula (2.6.2) to obtain

\frac{1}{4}Res_{q=0}[f(τ)^{2} \frac{R(τ)}{u(τ)^{r}} F (τ)\frac{dq}{q} ] = 4Res_{q=0}[F (τ)(\frac{d}{dq} u(τ))/u(τ)^{r+2} dq]

= Res_{u(τ)=0}[F (τ)\frac{d u(τ)}{u(τ)^{r+2}} ] = Coeff_{u(τ)^{r+1}}[F (τ)].

Remark 2.10. 1. The term q^{−ξ^{2}/2}e^{−ξxz} occuring in Δ_{ξ}^{X}(τ, x · z) looks like the summand corresponding to a vector ξ ∈ L in the theta series for the lattice L = H_{2}(X, ℤ) with the negative of the intersection form as quadratic form. Therefore the difference \overline{Ψ}_{C}^{X,H_{1}} − \overline{Ψ}_{C}^{X,H_{2}} for two points H_{1}, H_{2} ∈ \overline{ℍ}_{X} should be given in terms of a theta function Θ_{L,c,c}^{h_{1},h_{2}} (where as usual the small letter denotes the Poincaré dual class in L. In the next section we will define such theta functions and show that, at least for f and g rational points with Q(f) = Q(g) = 0, Θ_{L,c,c}^{f,g} has the properties of usual theta functions, enabling us to prove our main structural results.

2. Conjecture 5.1 from [G] can be interpreted as saying that Theorems 2.4 and 2.8 hold more generally in the case H^{1}(X, ℚ) = 0, if by H^{2}(X, ℤ) + C/2 we mean the set of all expressions (2ξ + C)/2 (counted with repetitions), with ξ running through H^{2}(X, ℤ).

3. Theta functions for indefinite lattices

The classical theta series associated to a positive definite lattice L with quadratic form Q and associated bilinear form x · y (see notation 3.1 below) is the sum

Θ_{L}(τ, x) := ∑_{ξ∈L} q^{Q(ξ)}e^{2πiξ·x} , (τ ∈ ℌ, x ∈ L_{ℂ} = L ⊗ ℂ).

These theta series have well-known transformation properties. In particular the “Nullwert” Θ_{L}(τ, 0) is a modular form of weight r/2, where r is the rank of L. In this chapter we give a generalization to the case when L is allowed to be indefinite. In particular we will consider the case when the type of Q is (r − 1, 1). The theta series that we define depends not only on L, but also on two vectors f, g ∈ L_{ℚ} with Q(f) = Q(g) = 0, f · g < 0. It is defined in a certain open subset of ℌ × L_{ℂ} by the formula

Θ_{L}^{f,g}(τ, x) := ( ∑_{ξ·f≥0>ξ·g} − ∑_{ξ·g≥0>ξ·f} )q^{Q(ξ)}e^{2πiξ·x}  (3.0.1)

(ξ running through L). We will also sketch a generalization to the case of type (n − s, s) with n − s ≥ s > 1, but this is not needed for our applications to the Donaldson invariants.

The main properties of Θ_{L}^{f,g} are proved by using an alternative definition. The idea of the construction is simple. When L is the standard hyperbolic lattice H generated by vectors f, g with Q(f) = 0, Q(g) = 0, f · g = −1 then Θ_{L}^{f,g} will be the function F (τ; u, v) studied in [Z] (but in fact going back to Kronecker, see [We]). If L is the direct sum H ⊕ L_{0}, with L_{0} positive definite, then Θ_{L}^{f,g} is just the product of this function with the usual theta series of L_{0}. The general case is reduced to this by considering the sublattice L′ = ⟨f, g⟩ ⊕ ⟨f, g⟩^{⊥} of L and averaging over cosets of L′ in L.

3.1. The function F. The main building block for the construction of theta functions will be the function F (τ; u, v) : ℌ × ℂ^{2} ⟶ ℂ, studied in [Z], which is defined for 0 < −ℜ(u)/ℑ(τ) < 2π, 0 < ℜ(v)/ℑ(τ) < 2π by the formula

F (τ; u, v) := ∑_{n≥0,m>0} q^{nm}e^{−nu−mv} − ∑_{n>0,m≥0} q^{nm}e^{nu+mv}.  (3.0.2)

(This formula is not given explicitly in [Z], but is easily proved to be equivalent to 3. below.) This funcion has the following properties (see [Z]).

1. F (τ; u, v) has a meromorphic continuation to ℌ×ℂ^{2} with simple poles for u or v in 2πi(ℤτ +ℤ) and no other poles.

2. F (τ; u, v) = F (τ; v, u) = −F (τ; −u, −v).

3. For |ℜ(u)/ℑ(τ)| < 2π, |ℜ(v)/ℑ(τ)| < 2π it has the Fourier expansion

F (τ; u, v) := −\frac{1}{1 − e^{u}} + \frac{1}{1 − e^{−v}} − 2 ∑_{n>0,m>0} sinh(nu + mv)q^{nm}.  (3.0.3)

4. F (\frac{aτ + b}{cτ + d}; \frac{u}{cτ + d}, \frac{v}{cτ + d}) = (cτ + d)e^{\frac{cuv/2πi}{cτ+d}} F (τ; u, v) for all (\begin{matrix}a & b \\ c & d \end{matrix}) ∈ SL(2, ℤ).

5. F (τ; u + 2πi(nτ + s), v + 2πi(mτ + t)) = q^{−nm}e^{−nu−mv}F (τ; u, v) for all n, m, s, t ∈ ℤ.

6. F (τ; u, v) = \frac{η(τ)^{3}θ_{11}(τ, u + v)}{θ_{11}(τ, u)θ_{11}(τ, v)}.
7. F (τ; u, v) = \frac{u + v}{uv} exp ( ∑_{k>0} \frac{2}{k!}[(u^{k} + v^{k} − (u + v)^{k}]G_{k}(τ)).

3.2. Definition of the theta functions.

Notation 3.1. For us a lattice is a free ℤ module L together with a quadratic form Q : L → \frac{1}{2}ℤ, such that the associated bilinear form x·y := Q(x+y)−Q(x)−Q(y) is nondegenerate and ℤ-valued. The extensions of the quadratic and bilinear form to L_{ℂ} := L ⊗ ℂ, or more generally L_{R} := L ⊗_{ℤ} R for any ℤ-module R, are denoted in the same way. Let L^{∨} := {μ ∈ L_{ℚ} | μ · L ⊆ ℤ}. L is unimodular if and only if L = L^{∨}. The type of L is the pair (r − s, s), where r is the rank of L and s the largest rank of a sublattice of L on which Q is negative definite, and the signature σ(L) is the number r−2s.

From now on we assume that s = 1, i.e., that L has type (r −1, 1). Then the set of vectors f ∈ L_{ℝ} with Q(f) < 0 has two components, the scalar product of any two such vectors being negative if they belong to the same component and positive if they belong to opposite components. We fix a vector f_{0} ∈ L_{ℝ} with Q(f_{0}) < 0 and let

C_{L} := {f ∈ L_{ℝ} | Q(f) < 0, f · f_{0} < 0}

(“positive light-cone”) be the component containing f_{0}; we further set

S_{L} := {f ∈ L | f primitive, Q(f) = 0, f · f_{0} < 0}.

(The (r−1)-dimensional hyperbolic space C_{L}/ℝ_{+} is the natural domain of definition of automorphic forms with respect to O(L), and S_{L} is a set of representatives for the corresponding set of cusps {f ∈ L_{ℚ} | Q(f) < 0, f · f_{0} < 0}/ℚ_{+}.) For f ∈ S_{L} put

D(f) := {(τ, x) ∈ ℌ × L_{ℂ} | 0 < ℑ(f · x) < ℑ(τ) },

and for f ∈ C_{L} put D(f) := ℌ × L_{ℂ}.

Notation 3.2. For t ∈ ℝ we put μ(t) := \begin{cases}1, & t ≥ 0, \\ 0, & t < 0. \end{cases}

Definition 3.3. Let f, g ∈ C_{L} ∪S_{L}. For (τ, x) ∈ D(f)∩D(g), we define the theta function of L with respect to (f, g) by the formula (3.0.1). More generally we put for c, b ∈ L and (τ, x) ∈ D(f) ∩ D(g)

Θ_{L,c,b}^{f,g}(τ, x) := ∑_{ξ∈L+c/2}(μ(ξ · f) − μ(ξ · g)) q^{Q(ξ)} e^{2πiξ·(x+b/2)},  (3.3.1)

so that Θ_{L,0,0}^{f,g} = Θ_{L}^{f,g}. It is clear that Θ_{L,c,b}^{f,g} depends up to sign only on the class of c and b in L/2L. We will later want to show that in case f, g ∈ S_{L} the function Θ_{L,c,b}^{f,g}(τ, x) has nice analytical properties. To see that Θ_{L,c,b}^{f,g}(τ, x) is well-defined we have to see that the sum (3.3.1) converges absolutely and locally uniformly on D(f) ∩ D(g).

Case 1: f, g ∈ S_{L}. We check the convergence only for (3.0.1) (for (3.3.1) this is analogous). Let N := −f · g. It is enough to show the absolute convergence of (3.0.1) with ξ running in \frac{1}{N} (⟨f, g⟩ ⊕ ⟨f, g⟩^{⊥}) instead of in L. So it is enough to check the absolute convergence of ∑_{η∈ \frac{1}{N} ⟨f,g⟩^{⊥}} q^{Q(η)}e^{2πiη·x}

(which holds even on ℌ × L_{ℂ} as ⟨f, g⟩^{⊥} is negative definite), and that of

∑_{a=1}^{∞} ∑_{b=0}^{∞} q^{ab/N}e^{2πi(af−bg)·x/N},  ∑_{b=1}^{∞} ∑_{a=0}^{∞} q^{ab/N}e^{2πi(−af+bg)·x/N},

which converge absolutely on D(f)∩D(g). We shall see in 3.4 that Θ_{L}^{f,g} and Θ_{L,c,b}^{f,g} have meromorphic continuations to ℌ × L_{ℂ}.

Case 2: f, g ∈ C_{L}. Then Θ_{L,c,b}^{f,g}(τ, x) depends only on the classes of f and g modulo ℝ^{+}. To check the absolute and locally uniform convergence of (3.3.1) on ℌ × L_{ℂ}, we write ξ ∈ L_{ℝ} as af + bg + ξ_{⊥} with a, b ∈ ℝ and ξ_{⊥} ∈ ⟨f, g⟩_{ℝ}^{⊥}. Since then Q(ξ) = Q(af + bg) + Q(ξ_{⊥}), it is enough to show that there is an C > 0 such that for ξ = af +bg ∈ ⟨f, g⟩_{ℝ} with (ξ ·f)(ξ ·g) ≤ 0 we have Q(ξ) > C(a^{2} +b^{2}). On ⟨f, g⟩ the quadratic form has type (1, 1), therefore we have (f · g)^{2} > 4Q(f)Q(g). The inequality (af · g + 2bQ(g))(2aQ(f) + bf · g) ≤ 0 can be rewritten \frac{(f·g)^{2}+4Q(f)Q(g)}{f·g} ab ≥ −2Q(f) a^{2} − 2Q(g) b^{2}. Therefore we get

Q(ξ) = Q(f) a^{2} + (f · g) ab + Q(g) b^{2} ≥ \frac{(f · g)^{2} − 4Q(f)Q(g)}{(f · g)^{2} + 4Q(f)Q(g)} (−Q(f) a^{2} − Q(g) b^{2}),

and, using that Q(f) < 0, Q(g) < 0, the result follows. Note that the argument also shows that in case f, g ∈ C_{L} the sum (3.3.1) makes sense as a formal power series: for each integer k it contains only a finite number of summands q^{Q(ξ)}e^{2πiξ·(x+b/2)} with Q(ξ) ≤ k.

Case 3: f ∈ S_{L}, g ∈ C_{L}. To see that (3.3.1) converges, we note that Θ_{L,c,b}^{g,f}(τ, x) = Θ_{L,c,b}^{h,f}(τ, x) + Θ_{L,c,b}^{g,h}(τ, x) if h ∈ C_{L} and any two of the three theta series converge. By the absolute convergence of (3.3.1) for f, g ∈ C_{L} we can therefore assume that g ∈ L. We split the sum (3.3.1) into two parts, the first consisting of the summands with ξ · f ≠ 0 and the second consisting of those with ξ · f = 0. The first sum converges for |ℑ(f · x)/ℑ(τ)| < 1. The second sum can be rewritten for (τ, x) ∈ D(f) as

∑_{n≥0}e^{2πinf·(x+b/2)} ∑_{\begin{matrix}ξ·f=0 \\ f·g≤ξ·g<0 \end{matrix}} q^{Q(ξ)}e^{2πiξ·(x+b/2)}

= \frac{1}{1 − e^{2πif·(x+b/2)}} ∑_{t∈P_{0}} ∑_{ξ∈⟨f,g⟩^{⊥}} q^{Q(ξ+t)}e^{2πi(ξ+t)·(x+b/2)},

for a suitable finite set P_{0}. Thus the result follows as ⟨f, g⟩^{⊥} is positive definite. This also shows that Θ_{L,c,b}^{f,g}(τ, x) for f ∈ S_{L} and g ∈ C_{L} has a meromorphic extension to |ℑ(f · x)/ℑ(τ)| < 1, and is given there by the Fourier expansion

Θ_{L,c,b}^{f,g}(τ, x) := ∑_{ξ·f≠0} (μ(ξ · f) − μ(ξ · g))q^{Q(ξ)}e^{2πiξ·(x+b/2)}  (3.3.2)

+ \frac{1}{1 − e^{2πif·(x+b/2)}} ∑_{\begin{matrix}ξ·f=0 \\ f·g≤ξ·g<0 \end{matrix}} q^{Q(ξ)}e^{2πiξ·(x+b/2)},

the sums running through ξ ∈ L + c/2. The argument also shows that (3.3.2) makes sense as a formal power series, i.e. for every k ∈ ℤ_{>0} there are only finitely many ξ in the sum with Q(ξ) < k.

Remark 3.4. For f, g, h ∈ C_{L} ∪ S_{L} and (τ, x) ∈ D(f) ∩ D(g) ∩ D(h) we have the cocycle condition Θ_{L,c,b}^{f,g}(τ, x) + Θ_{L,c,b}^{g,h}(τ, x) = Θ_{L,c,b}^{f,h}(τ, x).

Proof. This is immediate from the definitions.

Note that the set D(f) ∩ D(g) ∩ D(h) is always nonempty. Therefore the cocycle condition continues to hold after meromorphic extension of the Θ_{L,c,b}^{f,g}(τ, x).

3.3. Jacobi forms. We briefly recall the notion of a Jacobi form in the form in which we need it. (For more details in the one variable case see [E-Z]. In the one-variable case Q(x) is just mx^{2}, where m is the index.)

Definition 3.5. Let L be a lattice of rank r, and denote the quadratic form on L by Q : L ⟶ \frac{1}{2}ℤ as usual. Denote by M_{L} the set of meromorphic maps f : ℌ × L_{ℂ} → ℂ. For v = (λ, μ) ∈ L_{ℝ^{2}} and

for A = (\begin{matrix}a & b \\ c & d \end{matrix}) ∈ Γ, and k ∈ ℤ we define maps |v : M_{L} ⟶ M_{L} and |_{k}A : M_{L} ⟶ M_{L} by putting

f|v(τ, x) := q^{Q(λ)} exp(2πi(λ · (x + μ/2))f(τ, x + λτ + μ)  (3.5.1)
f|_{k}A(τ, x) := (cτ + d)^{−k} exp (−2πi\frac{cQ(x)}{cτ + d}) f (\frac{aτ + b}{cτ + d}, \frac{x}{cτ + d}) .  (3.5.2)

We view elements v = (v_{1}, v_{2}), w = (w_{1}, w_{2}) ∈ L_{ℝ^{2}} as row vectors with entries in L_{ℝ} and denote by ⟨v, w⟩ := v_{1}·w_{2}−v_{2}·w_{1} the corresponding “determinant” and by vA the application of A ∈ SL(2, ℝ) to v. Then |v and |_{k}A have the following compatibility properties

Remark 3.6. 1. |_{k}A defines an action of the group SL(2, ℝ) on M_{L}.

2. (f|v)|w = e^{πi⟨v,w⟩}f|(v + w) = e^{2πi⟨v,w⟩}(f|w)|v,

3. (f|v)|_{k}A = (f|_{k}A)|vA.

Proof. The proof is elementary and is completely analogous to that of Theorem 1.4 in [E-Z].

In the situation of Definition 3.5, we call a function f ∈ M_{L} a holomorphic Jacobi form of weight k with respect to (Λ, Γ) if

1. f|(λ, μ) = (−1)^{λ·μ}f for all (λ, μ) ∈ L^{2}.

2. f|_{k}A = f for all A ∈ Γ.

3. f is holomorphic in ℌ × L_{ℂ} and holomorphic in the cusps (in a suitable sense). A function f ∈ M_{L} satisfying only 1. and 2. will be called a meromorphic Jacobi form of weight k. In this paper we use only meromorphic Jacobi forms and hence do not explain the holomorphy condition at the cusps in 3.

Remark 3.7. It is evident from the definitions that for a Jacobi form f of weight k for a lattice L and Γ ⊂ SL(2, ℤ) the function (τ, z) ↦ f(τ, yz) on ℌ × ℂ will be for every y ∈ L a Jacobi form of weight k and index Q(y) for Γ in the sense of [E-Z].

Example 3.8. Let L be a positive definite lattice of rank r, and let T, V, S be as in (2.5.4). The theta function Θ_{L} : ℌ × L_{ℂ} → ℂ of L is given by

Θ_{L}(τ, x) := ∑_{ξ∈L} q^{Q(ξ)}e^{2πiξ·x} = ∑_{ξ∈L} 1|(ξ, 0).

We obviously have Θ_{L}|(λ, μ) = (−1)^{λ·μ}Θ_{L} for all (λ, μ) ∈ L × L, and θ^{−r}Θ_{L}|V = Θ_{L}, θ^{−r}Θ_{L}|T = (θ_{01}^{0})^{−r}Θ_{L}|(0, w/2) for a characteristic vector w of L. Note that

Θ_{L} = ∑_{l∈P} Θ_{L′}|(l, 0)

for any sublattice L′ ⊂ L and P a system of representatives of L/L′.

Let N be the index of L in L^{∨}, and let P be a system of representatives for L^{∨} modulo L. Then it is a standard fact that

(θ^{−r}Θ_{L})|S = \frac{1}{\sqrt{N}} ∑_{t∈P}(θ^{−r}Θ_{L})|(t, 0).

In particular, if L is unimodular, then θ^{−r}Θ_{L} is a (meromorphic) Jacobi form of weight 0 for Γ_{θ}, and if L is in addition even, it is a Jacobi form of weight 0 for SL(2, ℤ).

3.4. Properties of Θ_{L}^{f,g} for f, g ∈ S_{L}.

Theorem 3.9. Let L be a unimodular lattice of type (r − 1, 1); let f, g ∈ S_{L} and c, b ∈ L. Then 1. Θ_{L}^{f,g} and Θ_{L,c,b}^{f,g} have meromorphic extensions to ℌ × L_{ℂ}.

2. Θ_{L,c,b}^{f,g}(τ, x) = −(−1)^{c·b}Θ_{L,c,b}^{f,g}(τ, −x).

3. For |ℑ(f · x)/ℑ(τ)| < 1, |ℑ(g · x)/ℑ(τ)| < 1 they have the Fourier developments

Θ_{L}^{f,g}(τ, x) = \frac{1}{1 − e^{2πif·x}} ∑_{{ξ·f=0 \atop f·g≤ξ·g<0}} q^{Q(ξ)}e^{2πiξ·x} − \frac{1}{1 − e^{2πig·x}} ∑_{{ξ·g=0 \atop f·g≤ξ·f<0}} q^{Q(ξ)}e^{2πiξ·x}
+ 2 ∑_{ξ·f>0>ξ·g} q^{Q(ξ)} sinh(2πiξ · x),  (3.9.1)
Θ_{L,c,b}^{f,g}(τ, x) = \frac{1}{1 − e^{2πif·(x+b/2)}} ∑_{{ξ·f=0 \atop f·g≤ξ·g<0}} q^{Q(ξ)}e^{2πiξ·(x+b/2)}
− \frac{1}{1 − e^{2πig·(x+b/2)}} ∑_{{ξ·g=0 \atop f·g≤ξ·f<0}} q^{Q(ξ)}e^{2πiξ·(x+b/2)}  (3.9.2)

+ 2 ∑_{ξ·f>0>ξ·g} q^{Q(ξ)} sinh(2πiξ · (x + b/2)),

where in (3.9.1) and (3.9.2) the ξ run through L and L + c/2 respectively.

4. Θ_{L}^{f,g}/θ^{σ(L)} is a meromorphic Jacobi form of weight 1 for (L, Γ_{θ}).

5. For all λ, μ in L and for any characteristic vector w of L we have

Θ_{L,c,b}^{f,g}|(λ, μ) = (−1)^{cμ−bλ+λμ}Θ_{L,c,b}^{f,g},
(Θ_{L,c,b}^{f,g}/θ^{σ(L)})|_{1}S = 1^{−b·c/4}Θ_{L,b,c}^{f,g}/θ^{σ(L)},
(Θ_{L,c,b}^{f,g}/θ^{σ(L)})|_{1}T = 1^{3Q(c)/4−c·w/4}Θ_{L,c,b−c+w}^{f,g}/(θ_{01}^{0})^{σ(L)},
(Θ_{L,c,b}^{f,g}/θ^{σ(L)})|_{1}V = 1^{Q(c)/2}Θ_{L,c,b}^{f,g}/θ^{σ(L)},
(Θ_{L,c,b}^{f,g}/θ^{σ(L)})|_{1}W = 1^{−Q(c)/4−c·b/4}Θ_{L,w−c+b,c}^{f,g}/(θ_{10}^{0})^{σ(L)}.

Proof. The main idea in the proof is to give an alternative definition of the functions Θ_{L}^{f,g} and Θ_{L,c,b}^{f,g}, which allows us to relate them to the function F (τ; u, v) from Section 3.1 and theta functions of positive definite lattices.

In this proof |A with A ∈ SL(2, ℤ) always stands for |_{1}A. If H is the hyperbolic lattice of type (1, 1), and f, g are generators of H with Q(f) = Q(g) = 0, f · g = −1, then we get immediately from the formulas (3.0.1) and (3.0.2) that

Θ_{H}^{f,g}(τ, x) = F (τ; −2πif · x, 2πig · x)

for all (τ, x) ∈ ℌ × H_{ℂ} with 0 < ℑ(f · x) < ℑ(τ), 0 < ℑ(g · x) < ℑ(τ), and by the results cited in Section 3.1 this shows properties 1., 2., 3. and 4. for Θ_{H}^{f,g}(τ, x).

Similarly let now f and g be vectors generating a lattice L with quadratic form given by Q(f) = Q(g) = 0, f · g = −N ∈ ℤ_{<0}. Then we also see from the results of Section 3.1 that

Θ_{f,g}(τ, x) := F (Nτ; −2πif · x, 2πig · x)

fulfills Θ_{f,g}|(λ, μ) = (−1)^{λ·μ}Θ_{f,g} for all (λ, μ) ∈ L × L^{∨}. For a system R of representatives of L modulo L^{∨} we also get immediately that

Θ_{f,g}|S = \frac{1}{N} ∑_{t∈R} Θ_{f,g}|(t, 0),

and Θ_{f,g}|T = Θ_{f,g}

Now let L be arbitrary of type (r − 1, 1) and f, g ∈ S_{L} with f · g = −N ∈ ℤ_{<0}. Let L_{0} := ⟨f, g⟩ ⊕ ⟨f, g⟩^{⊥}. Let P be a system of representatives for L modulo L_{0}. For each x ∈ L_{ℂ} we denote by x_{f,g} and x_{⊥} the orthogonal projections to ⟨f, g⟩_{ℂ} and ⟨f, g⟩_{ℂ}^{⊥}. We write Θ_{⊥} for Θ_{⟨f,g⟩^{⊥}}. For τ ∈ ℌ, x ∈ L_{ℂ}, t ∈ L_{ℚ} we denote

(Θ_{f,g} ⊠ Θ_{⊥})(τ, x) := Θ_{f,g}(τ, x_{f,g}) · Θ_{⊥}(τ, x_{⊥}),
((Θ_{f,g} ⊠ Θ_{⊥})|(t, 0)) (τ, x) := (Θ_{f,g}|(t_{f,g}, 0)) (τ, x_{f,g}) · (Θ_{⊥}|(t_{⊥}, 0)) (τ, x_{⊥}).

Note that for t ∈ L the function (Θ_{f,g} ⊠ Θ_{⊥})|(t, 0) depends only class of t in L/L_{0}. We define the function Θ̃_{L}^{f,g} on ℌ × L_{ℂ} by

Θ̃_{L}^{f,g} := ∑_{t∈P}(Θ_{f,g} ⊠ Θ_{⊥})|(t, 0),  (3.9.3)

and for c, b ∈ L we write Θ̃_{L,c,b}^{f,g} := Θ̃_{L}^{f,g}|(c/2, 0)|(0, b/2).

Claim. For 0 < ℑ(f · x) < ℑ(τ), 0 < ℑ(g · x) < ℑ(τ) we have Θ_{L,c,b}^{f,g}(τ, x) = Θ̃_{L,c,b}^{f,g}(τ, x) (in particular we get the meromorphic continuation of Θ_{L,c,b}^{f,g} to ℌ × L_{ℂ}).

To prove the claim we can choose a system of representatives P of L/L_{0} such that all t ∈ P satisfy −f · g > (t + c/2) · f ≥ 0, −f · g > (t + c/2) · g ≥ 0. Then we get for every ξ ∈ L_{0} that ξ · f ≥ 0 if and only if (ξ + t + c/2) · f ≥ 0 and ξ · g ≥ 0 if and only if (ξ + t + c/2) · g ≥ 0. Using this the formula (3.5.1) gives for t ∈ P and (τ, x) ∈ D(f) ∩ D(g) that

((Θ_{f,g} ⊠ Θ_{⊥})|(t, 0)|(c/2, 0)|(0, b/2))(τ, x) = ∑_{ξ∈L_{0}+c/2+t}(μ(ξ · f) − μ(ξ · g))q^{Q(ξ)}e^{2πiξ·(x+b/2)},

and the claim follows by summing over t ∈ P. We will in future also write Θ_{L}^{f,g} and Θ_{L,c,b}^{f,g} for their meromorphic extensions Θ̃_{L}^{f,g} and Θ̃_{L,c,b}^{f,g}

2. By Section 3.1 we have Θ_{f,g}(τ, −x_{f,g}) = −Θ_{f,g}(τ, x_{f,g}) and by Example 3.8 Θ_{⊥}(τ, −x_{⊥}) = Θ_{⊥}(τ, x_{⊥}). Thus the definition of Θ̃_{L}^{f,g} gives that Θ_{L}^{f,g}(τ, −x) = −Θ_{L}^{f,g}(τ, x). Therefore

Θ_{L,c,b}^{f,g}(τ, −x) = (Θ_{L}^{f,g}|(c/2, 0)|(0, b/2))(τ, −x)
= −(Θ_{L}^{f,g}|(−c/2, 0)|(0, −b/2))(τ, x) = −(−1)^{b·c}Θ_{L,c,b}^{f,g}(τ, x).

3. We can assume b = 0. By definition we have on D(f) ∩ D(g):

Θ_{L,c,0}^{f,g}(τ, x) = ( ∑_{{ξ·f=0 \atop ξ·g<0}} − ∑_{{ξ·g=0 \atop ξ·f<0}} )q^{Q(ξ)}e^{2πiξ·x} + 2 ∑_{ξ·f>0>ξ·g} q^{Q(ξ)} sinh(2πiξ · x),

the sums running over ξ ∈ L + c/2. The last sum converges obviously for |ℑ(f · x)/ℑ(τ)| < 1, |ℑ(g · x)/ℑ(τ)| < 1, and on D(f) ∩ D(g) we have

∑_{{ξ·f=0 \atop ξ·g<0}} q^{Q(ξ)}e^{2πiξ·x} = \frac{1}{1 − e^{2πif·x}} ∑_{{ξ·f=0 \atop f·g≤ξ·g<0}} q^{Q(ξ)}e^{2πiξ·x}.

The sum on the right hand side converges on ℌ × L_{ℂ}, as it can be rewritten as

∑_{{t∈P +c/2 \atop t·f=0}} ( ∑_{ξ∈⟨f,g⟩^{⊥}} q^{Q(ξ)}e^{2πiξ·x})|(t, 0)

for a suitable system P of representatives of L/L_{0}, and the inner sum converges absolutely because ⟨f, g⟩_{⊥} is positive definite.

To prove the theorem it is now enough to show 5. Let (λ, μ) ∈ L^{2}. Then

Θ_{L,c,b}^{f,g}|(λ, μ) = ∑_{t∈P}(Θ_{f,g} ⊠ Θ_{⊥})|(t, 0)|(c/2, 0)|(0, b/2)|(λ, μ)

= ∑_{t∈P}(−1)^{c·μ−b·λ+λ·μ}(Θ_{f,g} ⊠ Θ_{⊥})|(0, μ)|(λ + t, 0)|(c/2, 0)|(0, b/2)

= ∑_{t′∈P′}(−1)^{c·μ−b·λ+λ·μ}(Θ_{f,g} ⊠ Θ_{⊥})|(t′, 0)|(c/2, 0)|(0, b/2)

= (−1)^{c·μ−b·λ+λ·μ}Θ_{L,c,b}^{f,g},

where P′ is another system of representatives. Let w be a characteristic vector of L. Then

(Θ_{L,c,b}^{f,g}/θ^{σ(L)})|T = ∑_{t∈P}((Θ_{f,g} ⊠ Θ_{⊥})/θ^{σ(L)})|(t, 0)|(c/2, 0)|(0, b/2)|T

= ∑_{t∈P}((Θ_{f,g} ⊠ Θ_{⊥})/θ^{σ(L)})|T |(t, t)|(c/2, c/2)|(0, b/2)

= (θ_{01}^{0})^{−σ(L)} ∑_{t∈P}(Θ_{f,g} ⊠ Θ_{⊥})|(0, w/2)|(t, t)|(c/2, c/2)|(0, b/2)

= (θ_{01}^{0})^{−σ(L)} ∑_{t∈P}(−1)^{t·(t−w)}(Θ_{f,g} ⊠ Θ_{⊥})|(t, 0)|(0, w/2)|(c/2, c/2)|(0, b/2)

= 1^{3Q(c)/4−c·w/4}(θ_{01}^{0})^{−σ(L)}Θ_{L,c,b−c+w}^{f,g}.

Applying |T twice we get

(Θ_{L,c,b}^{f,g}/θ^{σ(L)})|V = 1^{Q(c)/2}(Θ_{L,c,b}^{f,g}/θ^{σ(L)}).

Let R be a system of representatives for L_{0}^{∨}/L_{0} and let N be the index of L_{0} in L. Then

(Θ_{L,c,b}^{f,g}/θ^{σ(L)})|S = ∑_{t∈P}(Θ_{f,g}|S ⊠ (Θ_{⊥}/θ^{σ(L)})|S)|(0, −t)|(0, −c/2)|(b/2, 0)

= ∑_{t∈P} ∑_{r∈R} \frac{1}{N} θ^{−σ(L)}(Θ_{f,g} ⊠ Θ_{⊥})|(r, 0)|(0, −t)|(0, −c/2)|(b/2, 0)

= 1^{−b·c/4} ∑_{r∈R} (∑_{t∈P} \frac{1}{N} 1^{−ir·t}θ^{−σ(L)}(Θ_{f,g} ⊠ Θ_{⊥})|(r, 0)) |(b/2, 0)|(0, c/2).

The inner sum is zero unless r ∈ L^{∨} = L. Thus we get (Θ_{L,c,b}^{f,g}/θ^{σ(L)})|S = 1^{−b·c/4}Θ_{L,b,c}^{f,g}/θ^{σ(L)}. Finally we get

(Θ_{L,c,b}^{f,g}/θ^{σ(L)})|W = (Θ_{L,c,b}^{f,g}/θ^{σ(L)})|V^{−1}|T |S
= 1^{Q(c)/4−c·w/4} ((θ_{01}^{0})^{−σ(L)}Θ_{L,c,b−c+w}^{f,g}) |S
= 1^{−Q(c)/4−b·c/4}(θ_{10}^{0})^{−σ(L)}Θ_{L,w−c+b,c}^{f,g}.

Remark 3.10. We mention a (very) partial generalization of the theta functions to unimodular lattices L of type (r − s, s) with r − s ≥ s. Let F := (f_{1}, …, f_{s}), G := (g_{1}, …, g_{s}), where the f_{j}, g_{j} are primitive elements of L with Q(f_{j}) = Q(g_{j}) = 0, f_{j} · g_{j} < 0 and f_{j} · f_{l} = f_{j} · g_{l} = g_{j} · g_{l} = 0 if j ≠ l. On the set of (τ, x) with 0 < ℑ(f_{j} · x) < ℑ(τ), 0 < ℑ(g_{j} · x) < ℑ(τ) for all j we define Θ_{L}^{F,G} and Θ_{L,c,b}^{F,G} for c, b ∈ L by

Θ_{L}^{F,G}(τ, x) = ∑_{ξ∈L} ( ∏_{i=1}^{s}(μ(f_{i} · ξ) − μ(g_{i} · ξ)))q^{Q(ξ)}e^{2πiξ·x},  (3.10.1)
Θ_{L,c,b}^{F,G}(τ, x) = ∑_{ξ∈L+c/2} ( ∏_{i=1}^{s}(μ(f_{i} · ξ) − μ(g_{i} · ξ)))q^{Q(ξ)}e^{2πiξ·(x+b/2)}.  (3.10.2)

Now let f := f_{s}, g := g_{s}, F_{s} := (f_{1}, …, f_{s−1}), G_{s} := (g_{1}, …, g_{s−1}), and let L_{0} = ⟨f, g⟩ ⊕ ⟨f, g⟩^{⊥} and Θ_{⊥} := Θ_{⟨f,g⟩^{⊥}}^{F_{s},G_{s}}. Then the same arguments as above show that Θ_{L}^{F,G} is defined inductively by the right hand side of (3.9.3) and that Θ_{L,c,b}^{F,G} = Θ_{L}^{F,G}|(c/2, 0)|(0, b/2). Furthermore 1., 2., 3. and 5. of Theorem 3.9 still hold, if we replace everywhere Θ_{L}^{f,g} by Θ_{L}^{F,G} and Θ_{L,c,b}^{f,g} by Θ_{L,c,b}^{F,G} and interpret 3. as an inductive definition of a Fourier development of Θ_{L}^{F,G}, Θ_{L,c,b}^{F,G} on the set of (τ, x) with |ℑ(f_{j} · x)/ℑ(τ)| < 1, |ℑ(g_{j} · x)/ℑ(τ)| < 1 for all j, using the notations just introduced. We also have to replace |_{1} by |_{s}, and instead of 4. we obtain that Θ_{L}^{F,G}/θ^{σ(L)} is a meromorphic Jacobi form of weight s for L and Γ_{θ}. The proofs are staightforward generalizations of the proof of Theorem 3.9.

3.5. The structure theorem for the theta functions.

Notation 3.11. During this section let L be a unimodular lattice of type (r − 1, 1), let c ∈ L, and let f, g ∈ S_{L}. Let x ∈ L, and assume f · x ≠ 0 if c · f is even and g · x ≠ 0 if c · g is even. Then we put for (τ, z) ∈ ℌ × ℂ

φ_{L,c}^{f,g}(τ, x · z) := 1^{−3/8c·c} \frac{2θ(τ)^{−σ(L)}}{f(τ)} Θ_{L,c,c}^{f,g}(τ, \frac{xz}{2πif(τ)})e^{2Q(x)G(τ)z^{2}/f(τ)^{2}}.

When in future we write φ_{L,c}^{f,g}(τ, x·z) we will assume implicitly that the above conditions on f, g, c, x are fulfilled.

It is well known that for a (one-variable) Jacobi form ϕ(τ, z) of weight k and index m for Γ ⊂ SL(2, ℤ) the coefficient w_{n}(τ) of z^{n} in the Taylor development

ϕ(τ, z)e^{−8π^{2}mz^{2}G_{2}(τ)} := ∑_{n≥0} w_{n}(τ)z^{n}

is a modular form of weight k + n for Γ. This follows readily from the transformation behaviour

G_{2} (\frac{aτ + b}{cτ + d}) = (cτ + d)^{2}G_{2}(τ) − \frac{c(cτ + d)}{4πi},  (\begin{matrix}a & b \\ c & d \end{matrix}) ∈ SL(2, ℤ)  (3.11.1)

of the quasi-modular form G_{2}. We show a similar result for φ_{L,c}^{f,g}(τ, x · z), giving however a much more precise description of the coefficients.

Definition 3.12. Let m = m(L, f, g) := min{w · w | w ∈ L characteristic , w · f ≥ 0 ≥ w · g}. Note that (σ(L) − m)/8 is an integer. Let

B(L, f, g) := {w ∈ L characteristic | w · w < σ(L), (w · f)(w · g) ≤ 0},
B_{i}(L, f, g) := {w ∈ B(L, f, g) | w · f < 0 < w · g},
B_{f}(L, f, g) := {w ∈ B(L, f, g) | 0 = w · f > w · g ≥ 2f · g},
B_{g}(L, f, g) := {w ∈ B(L, f, g) | 0 = w · g > w · f ≥ 2f · g}.

The elements w ∈ B(L, f, g) are called basic classes for (L, f, g). Usually we will drop (L, f, g) in the notation. Note that the sets B_{i}, B_{f} and B_{g} are finite. For (τ, x) ∈ ℌ ⊗ L_{ℂ} we put

Ω_{L,c}^{f,g}(τ, x) := − ∑_{w∈B_{f}} \frac{(−1)^{c·(w+c)/2}q^{Q(w)/4}e^{−w·x}}{1 − (−1)^{c·f}e^{−2f·x}} + ∑_{w∈B_{g}} \frac{(−1)^{c·(w+c)/2}q^{Q(w)/4}e^{−w·x}}{1 − (−1)^{c·f}e^{−2g·x}}
+ ∑_{w∈B_{i}} q^{Q(w)/4}((−1)^{c·(w+c)/2}e^{−w·x} − (−1)^{c·(−w+c)/2}e^{w·x}).  (3.12.1)

For x ∈ L_{ℂ} we also write

O_{L,c}^{f,g}(x) := − ∑_{{w∈B_{f} \atop w·w=m}} \frac{(−1)^{c·(w+c)/2}e^{−w·x}}{1 − (−1)^{c·f}e^{−2f·x}} + ∑_{{w∈Bg \atop w·w=m}} \frac{(−1)^{c·(w+c)/2}e^{w·x}}{1 − (−1)^{c·g}e^{−2g·x}}
+ ∑_{{w∈B_{i} \atop w·w=m}}((−1)^{c·(w+c)/2}e^{−w·x} − (−1)^{c·(−w+c)/2}e^{w·x}),  (3.12.2)

so that O_{L,c}^{f,g}(x) is the leading term of Ω_{L,c}^{f,g}(τ, x) (coefficient of q^{m/2}) as q → 0.

Theorem 3.13. 1. φ_{L,c}^{f,g}(τ, x · z) has a Laurent development

φ_{L,c}^{f,g}(τ, x · z) = ∑_{n≥−1} w_{n}(τ)z^{n},

where each w_{n}(τ) is a modular function for Γ_{u}. More precisely:

2.

w_{n}(τ) = P_{n}(\frac{1}{U(τ) − 2}) − 1^{(c·c−1−n)/4}P_{n}(\frac{1}{−U(τ) − 2}) + R_{n}(U(τ)),

where P_{n}(t) is a polynomial of degree (σ(L) − m)/8 in t without constant term and R_{n}(t) is a polynomial of degree ≤ (n + 1)/2.

3. The polynomials P_{n}(t) are determined by the fact that ∑_{n=−1}^{∞} P_{n}(\frac{1}{Ũ(τ) − 2})z^{n} is the principal

part in (Ũ(τ) − 2) of the development of

\frac{4η(2τ)^{2}}{θ_{10}^{0}(τ)^{σ(L)}η(τ)^{4}} Ω_{L,c}^{f,g}(τ, xz \frac{η(2τ)^{2}}{η(τ)^{4}} ) exp (Q(x)(8G_{2}(τ) + 4e_{1}(τ))\frac{η(2τ)^{4}}{η(τ)^{8}} z^{2})

as a Laurent series ∑_{n=−1}^{∞} ( ∑_{k=−(σ(L)−m)/8}^{∞} a_{n,k}(Ũ(τ) − 2)^{m})z^{n} in z and Ũ(τ) − 2 .

4. The leading coefficients a_{n} of t^{(σ(L)−m)/8} in P_{n}(t) are given by

∑_{n=−1}^{∞} a_{n} z^{n} = 2^{2−(σ(L)+3m)/4} O_{L,c}^{f,g}(xz) e^{−Q(x)z^{2}}.

Proof. 1. The properties of f from 2.2 and Theorem 3.9 show that for l ∈ ℤ congruent to 2Q(c) modulo 2 the function θ(τ)^{−σ(L)}f(τ)^{l}Θ_{L,c,c}^{f,g}(τ, x) is a (meromorphic) Jacobi form of weight l + 1 for Γ_{u} and L. Now let x satisfy the conditions of notation 3.11. The Fourier development of Θ_{L,c,c}^{f,g}(τ, x) from part 2. of Theorem 3.9, the fact that e_{3} is a modular form for Γ_{θ} and (3.11.1) and (3.5.2) show that we get a Laurent development

θ(τ)^{−σ(L)} f(τ)^{l} Θ_{L,c,c}^{f,g}(τ, xz) e^{−8π^{2}Q(x)G(τ)z^{2}} = ∑_{n≥−1} v_{n}(τ)z^{n},

where each v_{n}(τ) has the transformation behaviour of a modular form of weight l + n + 1 for Γ_{u}, which has only poles at the zeroes of θ(τ)^{σ(L)}f(τ)^{−l}. We see that w_{n}(τ) = 1^{− \frac{3}{8} c.c} \frac{2v_{n}(τ)}{(2πi)^{n}f(τ)^{1+l+n}}, and the result follows as f(τ)^{2} is a modular form of weight 2 for Γ_{u} (v_{n} is zero unless 1 + l + n ∈ 2ℤ by part 3. of Theorem 3.9).

2. As U(τ) defines an isomorphism ℌ/Γ_{u} ∪{−1}∪{1}∪{∞} → ℙ_{1} we see that w_{n}(τ) is a rational function in U(τ). The functions θ(τ) and f(τ) are holomophic and nonzero on ℌ, therefore w_{n}(τ) can only have poles in the cusps. As U(τ) sends the cusps ∞, 1, −1 to ∞, −2, 2 respectively, this shows that w_{n}(τ) = P_{n}(\frac{1}{U(τ)−2}) + Q_{n}(\frac{1}{U(τ)+2}) + R_{n}(U(τ)) for suitable polynomials P_{n}, Q_{n} and R_{n}. As θ(τ) is holomorphic and nonzero at q = 0, and the q-developments of f(τ) and U(τ) start in degree 1/8 and −1/4, the degree of R_{n} is at most (n + 1)/2. To determine P_{n} and its degree we apply W ∈ SL(2, ℤ), which sends ∞ to −1. As U|W = Ũ, and the q-development of Ũ(τ) − 2 starts in degree 1, the degree of P_{n} is the order of pole of the q-development of w_{n}|W (τ). By definition the w_{n}|W (τ) are the coefficients in the Laurent development in z of φ_{L,c}^{f,g}(W τ, x · z), and by Theorem 3.9, and the results of Section 2.2 we get that

φ_{L,c}^{f,g}(W τ, x · z) := −(1^{c·c/4} \frac{4η(2τ)^{2}}{θ_{10}^{0}(τ)^{σ(L)}η(τ)^{4}} Θ_{L,w,c}^{f,g}(τ, −xz \frac{η(2τ)^{2}}{πiη(τ)^{4}} ) ·  (3.13.1)

· exp (Q(x)(8G_{2}(τ) + 4e_{1}(τ))\frac{η(2τ)^{4}}{η(τ)^{8}} )).

By 3.9 the lowest power of q occuring in the Fourier development of Θ_{L,w,c}^{f,g}(τ, x) is q^{m/8}. On the other hand the Fourier development of (θ_{10}^{0})^{−σ(L)} starts with q^{−σ(L)/8}, and G_{2}(τ), e_{1}(τ), η(2τ)^{2}/η(τ)^{4} are holomorphic and nonzero at q = 0. Therefore the degree of P_{n} is (σ(L) − m)/8. The formula Q_{n}(t) = 1^{(c·c−n−1)/4}P_{n}(−t) follows from the fact that V W transports ∞ to 1 and f|V (τ) = if(τ), U|V (τ) = −U(τ), Θ_{L,c,c}^{f,g}|V = 1^{c·c/4}Θ_{L,c,c}^{f,g}, and therefore φ_{L,c}^{f,g}(V W τ, x · z) = 1^{(c·c−1)/4}φ_{L,c}^{f,g}(W τ, −ix · z), and finally U|V (τ) − 2 = −(U(τ) + 2).

The formula (3.13.1) shows that 3. holds if we replace Ω_{L,c}^{f,g}(τ, x) by −(1^{c·c/4}Θ_{L,w,c}^{f,g}(τ, −\frac{x}{πi})). But by definition their q-developments are congruent modulo the ideal generated by q^{σ(L)/8}. Therefore the result follows.

To show 4. we just have to compute the leading terms of the expressions occuring in 3. So we use the congruences

η(τ)^{4}/η(2τ)^{2} ≡ 1, G_{2}(τ) + e_{1}(τ)/2 ≡ −1/8, q^{σ(L)/8}(θ_{10}^{0})^{−σ(L)}≡2^{−σ(L)}

modulo the ideal generated by q and Ũ(τ) − 2 ≡ 64q modulo the ideal generated by q^{2}, and the result follows.

4. Application to Donaldson invariants

We want to apply the results about theta functions from Section 3 to get structural results for the Donaldson invariants of a simply connected 4-manifold with b_{+} = 1. For this our lattice L will be the lattice H_{2}(X, ℤ) with the negative of the intersection form, i.e. Q = −2Q, σ(X) = −σ(L) and for a, b in H_{2}(X, ℤ) we have a · b = −AB where AB is the intersection product of the Poincaré duals.

4.1. Extension of the Donaldson invariants. The Donaldson invariant Φ_{C}^{X,H} is defined for H ∈ ℍ_{X} with Hξ ≠ 0 for all ξ ∈ H^{2}(X, ℤ) + C/2. In (2.5.3) we used it to define new formal power series Ψ_{C}^{X,H} and \overline{Ψ}_{C}^{X,H}. In this section we extend the definition of these invariants to arbitrary H ∈ \overline{ℍ}_{X}. To do this we apply the wall-crossing formulas Theorem 2.8 (we also use the notations from there) and use the theta functions of Section 3.

Definition 4.1. Fix H ∈ ℍ_{X} with Hξ ≠ 0 for all ξ ∈ H^{2}(X, ℤ) + C/2. Let M ∈ ℍ_{X} and x ∈ H^{2}(X, ℂ), if M ∈ 𝕊_{X} and a primitive representative of M has even intersection with C, we also assume that xF ≠ 0. Denote again by h, f, c ∈ L the Poincaré duals of H, F, C. Then we put

Ψ_{C}^{X,M}(x · z, p^{r}) := Ψ_{C,r}^{X,H}(x · z, p^{r})
+ 1^{−3Q(c)/4}Coeff_{u(τ)}^{r+1}[\frac{2θ(τ)^{σ(X)}}{f(τ)} Θ_{L,c,c}^{m,h}(τ, \frac{xz}{2πif(τ)}) e^{2Q(x)G(τ)z^{2}/f(τ)^{2}}],  (4.1.1)
\overline{Ψ}_{C}^{X,M}(x · z, t) := ∑_{r≥0} Ψ_{C}^{X,M}(x · z, p^{r})t^{r+1}.

Here we view the expression in square brackets as a formal Laurent series in q^{1/8} and z (i.e. it is a Laurent series in z, and the coefficient of z^{n} is for every n a Laurent series in q^{1/8}) using the formulas (3.3.1) and (3.3.2) (in case M ∈ 𝕊_{X}) for Θ_{c,c}^{m,h}(τ, x), which, as we have seen, also make sense as formal power series. We also define Φ_{C}^{X,M}(α) for α ∈ A_{∗}(X) by putting Φ_{C}^{X,M}(x^{s}p^{r}) to be s! times the coefficient of z^{s} in Ψ_{C}^{X,M}(x · z, p^{r}) and extending linearly (this is compatible with our previous definition of Φ_{C}^{X,M}).

We check that \overline{Ψ}_{C}^{X,M}(x · z, t) is well defined. If M ∈ ℍ_{X} with Mξ ≠ 0 for all ξ ∈ H^{2}(X, ℤ) + C/2, we have to see that the definition coincides with (2.5.1). By Theorem 2.8 we get (with the old definition) that

Ψ_{C}^{X,M}(x · z, p^{r}) − Ψ_{C}^{X,H}(x · z, p^{r}) = ∑_{M·ξ>0>H·ξ} 1^{C^{2}/8}(−1)^{(ξ−C/2)C}Coeff_{u(τ)}^{r+1}[Δ_{ξ}^{X}(τ, x · z)],

the sum running through ξ ∈ L + c/2. As f(τ) and u(τ) are power series in q multiplied with q^{1/8} and q^{1/4} respectively we can replace [Δ_{ξ}^{X}(τ, x · z)] by [Δ_{ξ}^{X}(τ, x · z) − (−1)^{C^{2}}Δ_{−ξ}^{X}(τ, x · z)]/2, i.e. we get (4.1.1).

Now it follows immediately from the cocycle condition 3.4 that the definition of Ψ_{C}^{X,M}(x · z, p^{r}) above is independent of the choice of H.

Remark 4.2. The definition is motivated as follows:

1. For M ∈ ℍ_{X} lying on a wall defined by a class of type (C, d) the definition gives Φ_{C,d}^{X,M} = Φ_{C,d}^{X,H} + ∑ δ_{ξ,d}^{X}/2 for H in any chamber containing L in its closure, and ξ running through the classes of type (C, d) through M with ξH < 0; in other words, we take the average over all chambers which contain M in their closure.

2. If F ∈ 𝕊_{X}, and its primitive representative has odd intersection with C, then F lies in the closure of a unique chamber of type (C, d). Our definition gives Φ_{C,d}^{X,F} := Φ_{C,d}^{X,H} for H in this chamber.

3. If F ∈ 𝕊_{X}, and its primitive representative in L has even intersection with C, then F will in general lie on infinitely many walls defined by classes of type (C, d), for every d. We would formally get from the definition (3.3.1) that

Ψ_{C}^{X,F} (x · z, p^{r}) − Ψ_{C}^{X,H}(x · z, p^{r}) =

Coeff_{u(τ)}^{r+1}[ ∑_{ξ∈H^{2}(X,ℤ)+C/2}(μ(F · ξ) − μ(H · ξ))1^{C^{2}/8}(−1)^{(ξ−C/2)C}Δ_{ξ}^{X}(τ, x · z)],

if the sum in square brackets converged as a formal power series in z and q. Instead, we first make an analytic continuation. Therefore we can view the part of degree d − 2r in z of Ψ_{C}^{X,M}(x · z, p^{r}) as a “renormalized average” over the infinitely many chambers of type (C, d) having F in their closure.

Note that by (3.3.2) the function Ψ_{C}^{X,M}(x · z, p^{r}) will, in the case that both CF and C^{2} are even, usually be meromorphic in z (with a simple pole at z = 0). This is the main reason why we introduced the notation Ψ_{C}^{X,M}(x · z, p^{r}) instead of Φ_{C}^{X,M}(e^{xz}p^{r}) for the Donaldson invariants.

With this definition the connection between the difference of the Donaldson invariants at period points F, G ∈ 𝕊_{X} on the one hand and the theta functions Θ_{L,c,c}^{f,g} and also the associated function φ_{L,c}^{f,g} on the other becomes evident:

Corollary 4.3. Let F, G ∈ 𝕊_{X} and f, g Poincaré duals of representatives in H^{2}(X, ℤ). Then

Ψ_{C}^{X,F} (x · z, p^{r}) − Ψ_{C}^{X,G}(x · z, p^{r}) = Coeff_{u(τ)}^{r+1}[φ_{L,c}^{f,g}(τ, x · z)].

Proof. This is straightforward from definition 4.1, 3.11 and the cocycle condition 3.4.

4.2. Blowup formulas. The blowup formulas relate the Donaldson invariants of a 4-manifold Y and Ŷ = Y #\overline{ℙ}_{2}. We have already used a small part of them in (2.5.2). In the case b_{+}(Y) > 1, when the invariants do not depend on the chamber structure, they have been shown in the most general form in [F-S1]. In [T1] they are shown also to hold in the case b_{+}(Y) = 1, if one takes the chamber structure into account (see also [K-L]). We cite only a weakened form, also avoiding the concept of related chambers.

Theorem 4.4. ([F-S1],[T1]) There exist universal polynomials B_{k}(t), S_{k}(t) ∈ ℚ[t] (k = 0, 1, …) such that the following holds. Let X be a simply connected 4-manifold, let C ∈ H^{2}(X, ℤ) be not divisible by 2, and, in case b_{+}(X) = 1, let M ∈ H^{2}(X, ℝ)^{+} with Mξ ≠ 0 for all ξ ∈ H^{2}(X, ℤ)+C/2. Then for all α ∈ A_{∗}(X) we have

Φ_{C}^{X̂,M}(αe^{k}) = Φ_{C}^{X,M}(αB_{k}(p)), Φ_{C+E}^{X̂,M}(αe^{k}) = Φ_{C}^{X,M}(αS_{k}(p))  (4.4.1)

Here, as usual, p ∈ A_{∗}(X) denotes the class of a point. Using the notation of (2.5.3) and replacing α by e^{xz}p^{r}, and rewriting everything in terms of the generating series B(u, t) := ∑ B_{k}(u)t^{k}/k!, S(u, t) := ∑ S_{k}(u)t^{k}/k! ∈ ℚ[[u, t]], we can write the blowup formulas as

Ψ_{C}^{X̂,M}(xz + te, p^{r}) = Ψ_{C}^{X,M}(xz, B(p, t)p^{r}), Ψ_{C+E}^{X̂,M}(xz + te, p^{r}) = Ψ_{C}^{X,M}(xz, S(p, t)p^{r}) .(4.4.2)

We now show that these formulas are compatible with our extension of the Donaldson invariants to \overline{ℍ}_{X}, and give a formula for the power series B(u, t) and S(u, t) in terms of theta functions. [F-S1] also gave explicit formulas for these power series, but in terms of elliptic functions. It is a (not completely trivial) exercise in elliptic functions to show that these formulas are equivalent to ours. However, our formulation, which we derive directly, is more practical for our purposes.

Proposition 4.5. The power series B(u, t), S(u, t) are determined by

B(U(τ), t) = e^{t^{2}G(τ)/f(τ)} θ_{00}(τ, \frac{t}{2πif(τ)})/θ(τ),  (4.5.1)
S(U(τ), t) = 1^{−1/8} e^{t^{2}G(τ)/f(τ)} θ_{11}(τ, \frac{t}{2πif(τ)})/θ(τ).  (4.5.2)

Proof. We determine the formulas by making use of the universality. Let X be a 4-manifold with b_{+} = 1. We denote by X̂ the connected sum X#ℙ̄_{2} and by E the generator of H^{2}(ℙ̄_{2}, ℤ), similarly let X̃ := X̂#ℙ̄_{2}, and let F be the corresponding generator. Let C ∈ H^{2}(X, ℤ), C ∉ 2H^{2}(X, ℤ). Let H, M ∈ ℍ_{X} with Mξ ≠ 0 for all ξ ∈ H^{2}(X, ℤ) + C/2. By the proof of Lemma 4.1 of [G], Theorem 4.4 implies that for all x ∈ H_{2}(X, ℂ)

∑_{Hξ<0<Mξ}(−1)^{ξC}δ_{ξ}^{X}(e^{xz}p^{r}B(p, t)) = ∑_{Hξ<0<Mξ}(−1)^{ξC} ∑_{n∈ℤ} δX̂ ; ξ+nE(e^{xz+te}p^{r})
∑_{Hξ<0<Mξ}(−1)^{ξC}δ_{ξ}^{X}(e^{xz}p^{r}S(p, t)) = 1^{−1/8} ∑_{Hξ<0<Mξ}(−1)^{ξC} ∑_{n∈ℤ}(−1)^{n}δX̂ ; ξ+(n+1/2)E(e^{xz+te}p^{r}),

the sums running as usual over ξ ∈ H^{2}(X, ℤ) + C/2 (note again the different conventions from [G]). Now using Conjecture 2.5 in the same way as in the proof of Lemma 4.5 of [G], we see that we can remove the sums ∑_{Hξ<0<Mξ} on both sides in both equalities, so that

δ_{ξ}^{X}(e^{xz}p^{r}B(p, t)) = ∑_{n∈ℤ} δX̂ ; ξ+nE(e^{xz+te}p^{r})

for all ξ ∈ \frac{1}{2}H^{2}(X, ℤ) with ξ^{2} < 0, and similarly for S(p, t). Specializing this to x = 0 and applying Theorem 2.8, we get

Coeff_{u(τ)}^{r+1}[q^{−ξ^{2}/2}θ(τ)^{σ(X)}B(U(τ), t)/f(τ)]
= Coeff_{u(τ)}^{r+1}[q^{−ξ^{2}/2} ∑_{n∈ℤ} q^{n^{2}/2}e^{nt/f(τ)}e^{t^{2}G(τ)/f(τ)^{2}}θ(τ)^{σ(X)−1}/f(τ)] (4.5.3)
= Coeff_{u(τ)}^{r+1}[q^{−ξ^{2}/2}θ_{00}(τ, \frac{t}{2πif(τ)})e^{t^{2}G(τ)/f(τ)^{2}}θ(τ)^{σ(X)−1}/f(τ)]

If we assume that σ(X) < 0 (as we may, since the formula is supposed to be universal), then all negative integers appear as 4ξ^{2}, so this last formula holds for all r and with ξ^{2} replaced by −N/4 for any integer N > 0. As f(τ) and u(τ) are power series in q^{1/2} with nonvanishing constant term multiplied with q^{1/8} and q^{1/4}, respectively, and θ_{00}(τ, t) is even in t, this implies the identity (4.5.1). The same argument and the fact that θ_{11}(τ, t) is odd in t imply the identity (4.5.2).

Proposition 4.6. Let X be a simply connected 4-manifold with b_{+} = 1. Then (4.4.2) holds for all M ∈ \overline{ℍ}_{X} and all C ∈ H^{2}(X, ℤ).

Proof. First we want to remove the assumption that C is not congruent to 0 modulo 2. Let M ∈ ℍ_{X} with Mξ ≠ 0 for all ξ ∈ H^{2}(X, ℤ), fix d, k ∈ ℤ ≥ 0. By (4.4.1) with E replaced by F together with formula (2.5.2), we have for all α ∈ A_{d}(X) and all ϵ > 0 sufficiently small

Φ_{0}^{X,M}(αB_{k}(p)) = Φ_{E}^{X̂,M+ϵE}(eαB_{k}(p)) = Φ_{E}^{X̃,M+ϵE}(eαf^{k}) = Φ_{0}^{X̂,M}(αe^{k}),
Φ_{0}^{X,M}(αS_{k}(p)) = Φ_{E}^{X̂,M+ϵE}(eαS_{k}(p)) = Φ_{E+F}^{X̃,M+ϵE}(eαf^{k}) = Φ_{E}^{X̂,M}(αe^{k}).

To show that the blowup formulas hold for all M ∈ \overline{ℍ}_{X} amounts to showing the compatiblity with Definition 4.1. Let L be a unimodular lattice and h, m ∈ C_{L} ∪ S_{L}. Let L_{1} be the orthogonal direct sum L ⊕ ⟨e⟩, where Q(e) = 1/2. In view of Proposition 4.5, the compatibility with Definition 4.1 amounts to the identities

Θ_{L_{1},c,c}^{h,m}(τ, xz + te) = θ_{00}(τ, t) Θ_{L,c,c}^{h,m}(τ, xz), Θ_{L_{1},c+e,c+e}^{h,m}(τ, xz + te) = 1^{1/4}θ_{11}(τ, t) Θ_{L,c,c}^{H,M}(τ, xz)

for c ∈ L and x ∈ L, and these are obvious by the definition of Θ_{L,c,c}^{h,m} (eq. (3.3.1)).

4.3. The structure theorem for the differences. Let X be a simply connected 4-manifold with b_{+} = 1 and F, G ∈ 𝕊_{X}, and use the same letters for their primitive representatives in H^{2}(X, ℤ). As we want to describe the differences of the Donaldson invariants at F and at G we write

Φ_{C}^{X,F,G} := Φ_{C}^{X,F} − Φ_{C}^{X,G}, Ψ_{C}^{X,F,G} := Ψ_{C}^{X,F} − Ψ_{C}^{X,G}, \overline{Ψ}_{C}^{X,F,G} := \overline{Ψ}_{C}^{X,F} − \overline{Ψ}_{C}^{X,G}.

Definition 4.7. Let M = M(X, F, G) := max {W^{2} | W ∈ H^{2}(X, ℤ) characteristic , W F ≥ 0 ≥ W G}. Note that M ≤ 0, M ≡ σ(X) (mod 8). Put

B = B(X, F, G) := {W ∈ H^{2}(X, ℤ) characteristic | W^{2} > σ(X), (W F)(W G) ≤ 0},
B_{I} = B_{I}(X, F, G) := {W ∈ B | W F > 0 > W G},
B_{F} = B_{F} (X, F, G) := {W ∈ B | 0 = W F < W G ≤ 2F G},
B_{G} = B_{G}(X, F, G) := {W ∈ B | 0 = W G < W F ≤ 2F G}

(cf. Definition 3.12). The elements W ∈ B are the basic classes for (X, F, G). For (τ, x) ∈ ℌ × L_{ℂ} we put

Ω_{C}^{X,F,G}(τ, x) := − ∑_{W ∈B_{F}} \frac{(−1)^{C(W +C)/2}q^{W^{2}/8}e^{W x}}{1 − (−1)^{CF} e^{2F x}} + ∑_{W ∈B_{G}} \frac{(−1)^{C(W +C)/2}q^{W^{2}/8}e^{W x}}{1 − (−1)^{CG}e^{2Gx}} (4.7.1)

+ ∑_{W ∈B_{I}} q^{W^{2}/8}((−1)^{C(W +C)/2}e^{W x} − (−1)^{C(−W +C)/2}e^{−W x}).

For x ∈ L_{ℂ} we also write

O_{C}^{X,F,G}(x) := − ∑_{{W ∈B_{F} \atop W^{2}=M}} \frac{(−1)^{C(W +C)/2}e^{W x}}{1 − (−1)^{CF} e^{2F x}} + ∑_{{W ∈B_{G} \atop W^{2}=M}} \frac{(−1)^{C(W +C)/2}e^{W x}}{1 − (−1)^{CG}e^{2Gx}}  (4.7.2)

+ ∑_{{W ∈B_{I} \atop W^{2}=M}} ((−1)^{C(W +C)/2}e^{W x} − (−1)^{C(−W +C)/2}e^{W x}).

We put

A(τ, x · z) = θ_{10}^{0}(τ)^{σ(X)} \frac{4η(2τ)^{2}}{η(τ)^{4}} exp ( − Q(x)(4G_{2}(τ) + 2e_{1}(τ))\frac{η(2τ)^{4}}{η(τ)^{8}} z^{2}).

Theorem 4.8. (Structure theorem). Let x ∈ H_{2}(X, ℤ). If B_{F} ≠ ∅ and CF is even, assume that F x ≠ 0 and if B_{G} ≠ ∅ and CG is even, assume that Gx ≠ 0. Let k = (M − σ(X))/8.

1. Φ_{C}^{X,F,G} fulfills the k-th order simple type condition, i.e. Φ_{C}^{X,F,G} vanishes identically if k ≤ 0 and Φ_{C}^{X,F,G}(α(p^{2} − 4)^{k}) = 0 for all α ∈ A_{∗}(X) if k > 0.

\overline{Ψ}_{C}^{X,F,G}(x · z, t) = ∑_{n≥−1} (P_{n} (\frac{t}{1 − 2t}) − 1^{−(C^{2}+1+n)/4}P_{n} (\frac{−t}{1 + 2t})) z^{n}

for suitable polynomials P_{n}(y) (n ≥ −1) of degree ≤ k with no constant term.

3. The polynomials P_{n}(y) are determined by the fact that \overline{Ψ}_{C}^{X,F,G}(x · z, ũ(τ)) is the principal part in the development of A(τ, x · z)Ω_{C}^{X,F,G}(τ, xz \frac{η(2τ)^{2}}{η(τ)^{4}} ) as a Laurent series in Ũ(τ) − 2.

4. Ψ_{C}^{X,F,G}(x · z, (1 + p/2)(p^{2} − 4)^{k−1}) = 2^{1+M}O_{C}^{X,F,G}(x · z) e^{Q(x)z^{2}/2}.

Proof. We apply Theorem 3.13 to φ_{L,c}^{f,g}(τ, x·z) for L = H_{2}(X, ℤ) with the negative of the intersection form and f, g, c the Poincaré duals of F, G and C, respectively. We obtain

\overline{Ψ}_{C}^{X,F,G}(x · z, u(τ)) = φ_{L,c}^{f,g}(τ, x · z) − ∑_{n≥−1} R_{n}(U(τ)) z^{n},

where R_{n}(t) is the polynomial from Theorem 3.13. Therefore 2. and 3. follow directly from parts 2. and 3. of Theorem 3.13. To show 1. note that by 2.

Φ_{C}^{X,F,G}(p^{r}(p^{2} − 4)^{k}x^{n}/n!) = Coeff_{z^{n}t^{r+1}}[(t^{−2} − 4)^{k} \overline{Ψ}_{C}^{X,F,G}(x · z, t)]

= Coeff_{t^{r+1}} [(t^{−2} − 4)^{k} (P_{n} (\frac{1}{t^{−1} − 2}) − 1^{−(C^{2}+1+n)/4}P_{n} (\frac{−1}{t^{−1} + 2}))] ,

which is 0 because P_{n}(y) has degree ≤ k and has no constant term.

To prove 4. let a_{n} be the coefficient of the leading term in P_{n}(y) as in Theorem 3.13. Then we get

Ψ_{C}^{X,F,G}(x · z, (1 + p/2)(p^{2} − 4)^{k−1}) = \frac{1}{2}Coeff_{t}[(t^{−1} + 2)^{k}(t^{−1} − 2)^{k−1}

· ∑_{n≥−1} (P_{n} (\frac{1}{t^{−1} − 2}) − 1^{−(C^{2}+1+n)/4}P_{n} (\frac{−1}{t^{−1} + 2})) ]

= \frac{1}{2}Coeff_{t}[(t^{−1} + 2)^{k} ∑_{n≥−1} a_{n} (t^{−1} − 2)^{−1} z^{n}] = 2^{2k−1} ∑_{n≥−1} a_{n}z^{n}.

So the result follows by part 4. of Theorem 3.13.

Remark 4.9. 1. Now we view z as a complex variable. By the obvious identity

\frac{(−1)^{C(C+W)/2}e^{W xz}}{1 − (−1)^{CF} e^{2F xz}} = ∑_{n≥0}(−1)^{C(C+W +2nF)/2}e^{(W +2nF)xz} (ℜ(F xz) < 0)

we can (again for ℜ(F xz) < 0) write

Ω_{C}^{X,F,G}(τ, x · z) = ∑_{W ∈B}(−1)^{C(C+W)/2} ϵ_{W} q^{W^{2}/8} e^{W xz}, ϵ_{W} = \begin{cases}1 & W F > 0 ≥ W G \\ −1 & W F ≤ 0 < W G \\ 0 & otherwise. \end{cases}

Therefore we get that \overline{Ψ}_{C}^{X,F,G}(x · z, ũ(τ)) is the principal part of the development of

∑_{W ∈B}(−1)^{C(C+W)/2} ϵ_{W} q^{W^{2}/8} exp (W xz \frac{η(2τ)^{2}}{η(τ)^{4}} ) A(τ, x · z)

as a Laurent series in Ũ(τ) − 2, when the latter converges. So we can view the elements of B as basic classes in the sense of the Introduction. Note that in general the signs ϵ_{W} will depend on the signs of ℜ(F xz), ℜ(Gxz)

2. The principal part of the Laurent development of exp (W xz \frac{η(2τ)^{2}}{η(τ)^{4}} )A(τ, x · z) (i.e., up to a factor (−1)^{C(C+W)/2}ϵ_{W}, the contribution of the basic class W to \overline{Ψ}_{C}^{X,F,G}(x · z, ũ(τ))) is

e^{W xz}e^{Q(x)z^{2}/2}Q_{k_{W}} ((Ũ(τ) − 2)^{−1}, σ(X), z^{2}Q(x), W xz)

where k_{W} = (W^{2} − σ(X))/8 and Q_{k}(z_{1}, z_{2}, z_{3}, z_{4}) is a universal polynomial of degree k. This can be seen by writing

A(τ, x · z) = q^{(σ(X)−W^{2})/8}(q^{−1/8}θ_{10}^{0}(τ))^{σ(X)} \frac{4η(2τ)^{2}}{η(τ)^{4}} exp (− Q(x)(4G_{2}(τ) + 2e_{1}(τ)) \frac{η(2τ)^{4}}{η(τ)^{8}} z^{2}),

and using the developments

q = \frac{y}{64} − \frac{y^{2}}{512} + …,  q^{−1/8}θ_{10}^{0}(τ) = 2 + \frac{y}{32} − \frac{y^{2}}{256} + …,
\frac{η(2τ)^{2}}{η(τ)^{4}} = 1 + \frac{y}{16} − \frac{5y^{2}}{1024} + …, (4G_{2}(τ) + 2e_{1}(τ)) \frac{η(2τ)^{4}}{η(τ)^{8}} = \frac{1}{2} + \frac{y}{8} − \frac{y^{2}}{256} + ….

in y := Ũ(τ) − 2. The only term in the expression for A(τ, x · z) containing negative powers of y will then be q^{(σ(X)−W^{2})/8}. Writing

exp (bz(1 + ∑ a_{i}y^{i})) = e^{bz} · exp (bz( ∑ a_{i}y^{i}))

and expanding the second factor, the result follows. Using part 2. of Theorem 4.8 we can also rewrite this result as follows: Up to a factor (−1)^{C(C+W)/2}ϵ_{W} the contribution of the basic class W to \overline{Ψ}_{C}^{X,F,G}(x · z, t) is

e^{W xz}e^{Q(x)/2}Q_{k_{W}} ((t^{−1} − 2)^{−1}, σ(X), z^{2}Q(x), W xz)
− 1^{−C^{2}−1}e^{−iW xz}e^{−Q(x)/2}Q_{k_{W}} ((−t^{−1} − 2)^{−1}, σ(X), −z^{2}Q(x), −iW xz).

3. As we noted in Remark 2.7, Γ_{u} is conjugate to Γ(2) via a matrix M ∈ GL(2, ℤ) sending ∞, 1, −1 to 1, 0, ∞ respectively. Therefore we could reexpress our results in terms of developments of modular functions for Γ(2) in powers of the modular function ū(τ) = u(Mτ), thus getting a connection to the computations [W1], [W2] in theoretical physics.

Corollary 4.10. If σ(X) > −8, then \overline{Ψ}_{C}^{X,F} is independent of F ∈ 𝕊_{X}.

Proof. This is immediate from part 2. of Theorem 4.8, since k < 1.

We also get another version of the blowup formulas.

Corollary 4.11. Let F, G ∈ 𝕊_{X} and C ∈ H^{2}(X, ℤ) and x ∈ H_{2}(X, ℤ); we denote by the same letters the pullbacks to X̂ = X#\overline{ℙ}_{2}, and by E the class of the exceptional divisor. Let t be an indeterminate and set s = t η(2τ)^{2}/η(τ)^{4}. Then we have

1. With respect to the developments in powers of Ũ(τ) − 2, the function \overline{Ψ}_{C}^{X̂,F,G}(x · z + te, ũ(τ)) is the principal part of

\overline{Ψ}_{C}^{X,F,G}(x · z, ũ(τ)) exp(s^{2}(4G_{2}(τ) + 2e_{1}(τ))) cosh(s) ∏_{n>0} \frac{(1 + q^{n}e^{2s})(1 + q^{n}e^{−2s})}{(1 + q^{n})^{2}}

and \overline{Ψ}_{C+E}^{X̂,F,G}(x · z + te, ũ(τ)) is the principal part of

\overline{Ψ}_{C}^{X,F,G}(x · z, ũ(τ)) exp(s^{2}(4G_{2}(τ) + 2e_{1}(τ))) sinh(s) ∏_{n>0} \frac{(1 − q^{n}e^{2s})(1 − q^{n}e^{−2s})}{(1 + q^{n})^{2}} .

2. In particular, putting M := M(X, F, G) and k := (M −σ(X)/8), we get Φ_{D}^{X̂,F,G}(α(p^{2}−4)^{k}) = 0 for all α ∈ A_{∗}(X̂) and all D ∈ H^{2}(X̂, ℤ) and

Φ_{C}^{X̂,F,G}(e^{xz+te}(1 + p/2)(1 − p^{2}/4)^{k−1}) = Φ_{C}^{X,F,G}(e^{xz}(1 + p/2)(1 − p^{2}/4)^{k−1}) cosh(t) e^{−t^{2}/2} ,
Φ_{C+E}^{X̂,F,G}(e^{xz+te}(1 + p/2)(1 − p^{2}/4)^{k−1}) = Φ_{C}^{X,F,G}(e^{xz}(1 + p/2)(1 − p^{2}/4)^{k−1}) sinh(t) e^{−t^{2}/2} .

Proof. This follows from Propositions 4.5 and 4.6 by applying W and using the standard identities

(θ^{−1}θ_{00}|W )(τ, t) = cosh(πit) ∏_{n>0} \frac{(1 + q^{n}e^{2πit})(1 + q^{n}e^{−2πit})}{(1 + q^{n})^{2}} ,
(θ^{−1}θ_{11}|W )(τ, t) = 1^{1/8} sinh(πit) ∏_{n>0} \frac{(1 − q^{n}e^{2πit})(1 − q^{n}e^{−2πit})}{(1 + q^{n})^{2}} .

We omit the details, which are not difficult. Notice that the principle used here (as already in the proof of Theorem 3.13) is that if the expansions of two functions at ∞ coincide and if both functions are known to be modular, then their expansions at any other cusp also coincide.

4.4. Speculations about the relation to Seiberg-Witten theory. Theorem 4.8 is closely related to the expectations from Seiberg-Witten theory. The elements W ∈ B are characteristic elements of H^{2}(X, ℤ) and thus correspond to Spin^{c}-structures, and (W^{2} − σ(X))/8 − 1 is the expected dimension of the corresponding Seiberg-Witten moduli space. The SW -basic classes are those classes W, for which the Seiberg-Witten invariant is not zero. X is of SW -simple type if only classes for which the expected dimension is 0 give rise to nonzero invariants. At least in the case b_{+} > 1 the set of SW -basic classes together with the corresponding invariants and expected dimensions are conjectured [W1] to determine the Donaldson invariants of X by a universal formula, similar to Theorem 4.8. In case b_{+} > 1 there is only a finite number of SW -basic classes.

In the case of b_{+} = 1 this relationship should still be there, but obscured by the chamber structure both in Seiberg-Witten and Donaldson theory. According to the program of [P-T1], [P-T2] the Donaldson invariants should be determined from the Seiberg-Witten invariants after a large perturbation of the equation; the Donaldson invariants Φ_{C,d}^{X,g} for a metric should be more or less given in terms of Seiberg-Witten invariants for perturbed equations, with the perturbation depending on d. This also makes it possible that an infinite number of basic classes contribute to Φ_{C}^{X,g}. Our formulas suggest however that for period points F, G ∈ 𝕊_{X}, the situation again becomes easier. Denote by SW_{L}(W) the unperturbed Seiberg-Witten invariant for the Spin^{c}-structure with first Chern-class W, for the metric with period point L ∈ ℍ_{X}. It is well known that SW_{L}(W) is constant on both connected components of ℍ_{X} \W^{⊥}, and that for L_{+}W > 0 > L_{−}W we have SW_{L_{+}}(W) = SW_{L_{−}}(W)±1. This also shows that 4-manifolds with b_{+} = 1 are, for essentially trivial reasons, usually not of SW -simple type. Theorem 4.8 says that for each class W for which SW_{L}(W) changes when going from F to G we get a contribution to the difference \overline{Ψ}_{C}^{X,F,G}(x · z, t) given by a universal formula. Classes orthogonal to F and G are treated in a special way. Therefore we conjecture that our formula indeed gives the formula for the conjectured relation between Donaldson and Seiberg-Witten invariants, viz.: Conjecture 4.12. Let X a simply-connected 4-manifold with b_{+} = 1, F a primitive representative of a class in 𝕊_{X}, and B_{1} = {W ∈ H^{2}(X, ℤ)| W characteristic, W^{2} > σ(X), W F = 0}. Then there is a system R_{F} of representatives of B_{1} modulo ⟨2F ⟩ such that Ψ_{C}^{X,F} (ũ(τ), x · z) is the principal part in the development of A(τ, xz) Ω_{C}^{X,F} (τ, xz \frac{η(2τ)^{2}}{η(τ)^{4}} ) as a Laurent series in Ũ(τ) − 2, where

Ω_{C}^{X,F} (τ, x) := − ∑_{W ∈R_{F}} \frac{(−1)^{C(W +C)/2}q^{W^{2}/8}e^{W x}}{1 − (−1)^{CF} e^{2F x}} + ∑_{W} q^{W^{2}/8}(−1)^{C(W +C)/2}SW_{N_{W}} (W)e^{W x}.

In the second sum W runs through the characteristic elements of H^{2}(X, ℤ) and for each W the period point N_{W} ∈ ℍ_{X} is chosen in such a way that (W N_{W})(W F) > 0 (for a representative of N_{W} with N_{W} F > 0). If CF is odd and F x = 0, we can replace the first sum by − ∑_{W} (−1)^{C(W +C)/2}q^{W^{2}/8}e^{W x}/2, with W now running through any system of representatives of B_{1} modulo ⟨2F ⟩.

Note that this is compatible with the predictions of Witten in the simple type case. One can suspect that a modification of this formula should work in the case b_{+} > 1 (if 4-manifolds with b_{+} > 1 not of simple type do indeed exist).

5. The case of rational algebraic surfaces

For rational algebraic surfaces X we shall see that there are always some G ∈ 𝕊(X) such that Φ_{C}^{X,G} = 0 for all C ∈ H^{2}(X, ℤ). Therefore Theorem 4.8 will give us the structure of the Donaldson invariants Φ_{C}^{X,F} for F ∈ 𝕊_{X} instead of only the differences.

5.1. Donaldson invariants of ℙ_{1} ×ℙ_{1} and ℙ_{2}#ℙ̄_{2}. As a first application of Theorem 4.8 we want to study the Donaldson invariants of ℙ_{1} × ℙ_{1} and ℙ_{2}#ℙ̄_{2}, and thus of all rational ruled surfaces. We compute the limit of these Donaldson invariants for the period point going to the boundary of the positive cone. We also show that they satisfy certain relations, also for period points in the inside of the positive cone. We use the following elementary result from algebraic geometry: Lemma 5.1. [Q2] Let X ⟶ ℙ_{1} be a rational ruled surface, F ∈ H^{2}(X, Z) the class of a fibre and H the class of a section of the ruling. If C ∈ H^{2}(X, ℤ) fulfills CF odd, then the moduli space M_{NF +H}(C, c_{2}) of NF + H stable rank 2 sheaves on X is empty for all N ∈ ℤ_{>0} sufficiently large with respect to the second Chern class c_{2}. In particular, for any given d the invariant Φ_{C,d}^{X,F +ϵH} vanishes for all sufficiently small ϵ > 0.

Notation 5.2. 1. Let F and G be the Poincaré duals of the classes of the fibres of the projections of ℙ_{1} ×ℙ_{1} to its factors. We denote by ℙ̂_{2} the blowup of ℙ_{2} in a point and by E_{1} the class of the exceptional divisor. We denote by F̄ := H −E_{1} the Poincaré dual of the fibre of the ruling of ℙ̂_{2} and Ḡ := H +E_{1}, for H the pullback of the hyperplane class. Let σ : ℙ̃_{2} → ℙ̂_{2} be the blowup in a general point, with exceptional divisor E_{2}. Then there is a blowup ϵ : ℙ̃_{2} → ℙ_{1} ×ℙ_{1} in a point with exceptional divisor E such that ϵ^{∗}(F) = σ^{∗}(F̄), ϵ^{∗}(G) = σ^{∗}(H) − E_{2}, E = ϵ^{∗}(F) − E_{2}.

2. We denote by f, g, f̄, ḡ the Poincaré duals of F, G, F̄, Ḡ respectively.

3. For X = ℙ_{1} × ℙ_{1} or X = ℙ̂_{2} and L ∈ {F, G, F̄, Ḡ}, C ∈ H^{2}(X, ℤ) and α ∈ A_{d}(X) we denote by Φ_{C,d}^{X,L+}(α) := Φ_{C,d}^{X,L+ϵM}(α) for M an ample divisor and ϵ > 0 sufficiently small. As before we put Φ_{C}^{X,L+} := ∑_{d} Φ_{C,d}^{X,L+}. Note that if LC is odd then Φ_{C}^{X,L+} = Φ_{C}^{X,L}.

Theorem 5.3. 1. For X = ℙ_{1} × ℙ_{1} and L ∈ {F, G} or X = ℙ̂_{2}, and L ∈ {F̄, Ḡ} and for all

C ∈ H^{2}(X, ℤ), the Donaldson invariants Φ_{C}^{X,L} vanish.

2. For all r ≥ 0 and indeterminates s, t we have

Φ_{0}^{ℙ_{1}×ℙ_{1},F +}(e^{sf+tg}p^{r}) = −Coeff_{u(τ)^{r+1}}[ coth (\frac{t}{2f(τ)})e^{−2stG(τ)/f(τ)^{2}}/f(τ)],
Φ_{0}^{ℙ̂_{2},F̄+}(e^{sf̄+tḡ}p^{r}) = Φ_{0}^{ℙ_{1}×ℙ_{1},F +}(e^{sf+2tg}p^{r}),
Φ_{F}^{ℙ_{1}×ℙ_{1},F +}(e^{sf+tg}p^{r}) = −Coeff_{u(τ)^{r+1}}[\frac{e^{−2stG(τ)/f(τ)^{2}}}{sinh(\frac{t}{2f(τ)})f(τ)}],
Φℙ̂_{2},F̄+ ; F̄(e^{sf+tg}p^{r}) = Φ_{0}^{ℙ_{1}×ℙ_{1},F +}(e^{sf+2tg}p^{r}).

3. More generally we have for all a, b ∈ ℝ_{>0}

Φ_{F}^{ℙ_{1}×ℙ_{1},aF +bG}(e^{sf+tg}p^{r}) = Φ_{G}^{ℙ_{1}×ℙ_{1},aG+bF} (e^{sg+tf}p^{r}),
Φ_{0}^{ℙ_{1}×ℙ_{1},aF +bG}(e^{sf+tg}p^{r}) = Φ_{0}^{ℙ_{1}×ℙ_{1},aG+bF} (e^{sg+tf}p^{r}),
Φℙ̂_{2},aF̄+bḠ ; F̄(e^{sf̄+tḡ}p^{r}) = Φ_{F}^{ℙ_{1}×ℙ_{1},aF +2bG}(e^{sf+2tg}p^{r}) − Φ_{G}^{ℙ_{1}×ℙ_{1},2aF +bG}(e^{2sf+tg}p^{r})
= Φ_{0}^{ℙ_{1}×ℙ_{1},2aF +bG}(e^{2sf+tg}p^{r}) − Φ_{0}^{ℙ_{1}×ℙ_{1},aF +2bG}(e^{sf+2tg}p^{r}),
Φ_{0}^{ℙ̂_{2},aF̄+bḠ}(e^{sf̄+tḡ}p^{r}) = Φ_{0}^{ℙ_{1}×ℙ_{1},2aF +bG}(e^{2sf+tg}p^{r}) − Φ_{F}^{ℙ_{1}×ℙ_{1},aF +2bG}(e^{sf+2tg}p^{r}).

One can check that these formulas are compatible up to conventions with the known results (e.g. [L-Q],[K-L],[E-G2]). Note that part 2 of the theorem implies in particular Conjecture 6.2 from [E-G2].

Proof. 1. By Lemma 5.1 we get that

Φ_{F +G}^{ℙ_{1}×ℙ_{1},F} = Φ_{G}^{ℙ_{1}×ℙ_{1},F} = Φ_{F}^{ℙ_{1}×ℙ_{1},G} = Φ_{H}^{ℙ̂_{2},F̄} = Φ_{H}^{ℙ̂_{2},Ḡ} = Φ_{E_{1}}^{ℙ̂_{2},F̄} = Φ_{E_{1}}^{ℙ̂_{2},Ḡ} = 0.

Applying Corollary 4.10 it follows that Φ_{F}^{ℙ_{1}×ℙ_{1},F} = Φ_{G}^{ℙ_{1}×ℙ_{1},G} = 0. In the following let x ∈ H_{2}(ℙ_{1} × ℙ_{1}, ℂ) and y ∈ H_{2}(ℙ̂_{2}, ℂ). We apply Proposition 4.6 to obtain

Φ_{ϵ^{∗}(F)}^{ℙ̃_{2},ϵ^{∗}(F)}(e^{xz+te}p^{r}) = Φ_{F}^{ℙ_{1}×ℙ_{1},F} (e^{xz}B(p, t)p^{r}) = 0,

i.e. Φ_{ϵ^{∗}(F)}^{ℙ̃_{2},ϵ^{∗}(F)} = 0. Therefore we have by again using 4.6

Φℙ̂_{2},F̄ ; F̄(e^{yz}p^{r}) = Φ_{ϵ^{∗}(F)}^{ℙ̃_{2},ϵ^{∗}(F)}(e^{yz}p^{r}) = 0.

Furthermore we have by repeating essentially the same argument

Φ_{E_{2}}^{ℙ̃_{2},ϵ^{∗}(F)}(e^{xz+te}p^{r}) = −Φ_{F}^{ℙ_{1}×ℙ_{1},F} (e^{xz}S(p, t)p^{r}) = 0,

and thus Φ_{0}^{ℙ̂_{2},F̄}(e^{yz}p^{r}) = Φ_{E_{2}}^{ℙ̃_{2},ϵ^{∗}(F)}(e_{2}e^{yz}p^{r}) = 0. Finally we get

Φ_{E}^{ℙ̃_{2},ϵ^{∗}(F)}(e^{yz+te}p^{r}) = −Φ_{F}^{ℙ̂_{2},F} (e^{yz}S(p, t)p^{r}) = 0,

and thus Φ_{0}^{ℙ_{1}×ℙ_{1},F} (e^{xz}p^{r}) = Φ_{E}^{ℙ̃_{2},ϵ^{∗}(F)}(e^{xz}ep^{r}) = 0. The other cases follow by symmetry.

2. Putting X = ℙ_{1} × ℙ_{1} and L := aF + bG for a, b ∈ ℝ_{>0} and writing ν(t) := \begin{cases}1 & t > 0, \\ 1/2 & t = 0 \\ 0 & t < 0, \end{cases} we

get by part 1., Definition 4.1 and formula (3.3.2) the formulas

Φ_{0}^{ℙ_{1}×ℙ_{1},L}(e^{xz}p^{r}) = ∑_{n,m>0} ν(bn − am) δ_{nF −mG,d}^{ℙ_{1}×ℙ_{1}}(e^{xz}p^{r}) − Coeff_{u(τ)^{r+1}}[L_{0}(τ, xz)],

Φ_{F}^{ℙ_{1}×ℙ_{1},L}(e^{xz}p^{r}) = ∑_{n,m>0}(−1)^{m} ν(b(n − \frac{1}{2}) − am) δ_{(n−1/2)F −mG,d}^{ℙ_{1}×ℙ_{1}}(e^{xz}p^{r}) − Coeff_{u(τ)^{r+1}}[L_{F} (τ, xz)],

where we have put

L_{0}(τ, xz) = e^{−Q(x)z^{2}G(τ)/f(τ)^{2}} \frac{1 + e^{−F xz/f(τ)}}{f(τ)(1 − e^{−F xz/f(τ)})},
L_{F} (τ, xz) = 2\frac{e^{−Q(x)z^{2}G(τ)/f(τ)^{2}}}{f(τ)(e^{F xz/(2f(τ))} − e^{−F xz/(2f(τ))})}.

If a/b is sufficiently large, then for all n, m ocurring in the sums above the number nm is larger then (d + 3)/4 and thus δ_{nF −mG,d}^{ℙ_{1}×ℙ_{1}} = 0 and δ_{(n−1/2)F −mG,d}^{ℙ_{1}×ℙ_{1}} = 0. Therefore

Φ_{0}^{ℙ_{1}×ℙ_{1},F +}(e^{xz}p^{r}) = −Coeff_{u(τ)^{r+1}}[L_{0}(τ, xz)], Φ_{F}^{ℙ_{1}×ℙ_{1},F +}(e^{xz}p^{r}) = −Coeff_{u(τ)^{r+1}}[L_{F} (τ, xz)].

The argument for ℙ̂_{2} is analogous.

3. The first two identities are obvious by symmetry. Part 2. implies that, given d, the other formulas hold until degree d in s, t for a/b sufficiently large. Note that by Theorem 2.4

Φ_{0}^{ℙ_{1}×ℙ_{1},aF +bG} − Φ_{0}^{ℙ_{1}×ℙ_{1},F +} = ∑_{n,m>0} ν(bn − am)δ_{nF −mG}^{ℙ_{1}×ℙ_{1}},

Φ_{F}^{ℙ_{1}×ℙ_{1},aF +bG} − Φ_{F}^{ℙ_{1}×ℙ_{1},F +} = ∑_{n,m>0}(−1)^{m} ν(b(n − \frac{1}{2}) − am) δ_{(n−1/2)F −mG}^{ℙ_{1}×ℙ_{1}},

Φ_{0}^{ℙ̂_{2},aF̄+bḠ} − Φ_{0}^{ℙ̂_{2},F̄+} = ∑_{n,m>0} (ν(bn − am)δℙ̂_{2} ; nF̄−mḠ

+ ν(b(n − \frac{1}{2}) − a(m − \frac{1}{2})) δℙ̂_{2} ; (n−1/2)F̄−(m−1/2)Ḡ),

Φℙ̂_{2},aF̄+bḠ ; F̄ − Φℙ̂_{2},F̄+ ; F̄ = ∑_{n,m>0} (ν(b(n − \frac{1}{2}) − am) δℙ̂_{2} ; (n−1/2)F̄−mḠ

− ν(bn − a(m − \frac{1}{2})) δℙ̂_{2} ; nF̄−(m−1/2)Ḡ).

Using Theorem 2.8, this reduces the proof to the easy identities

∑_{n,m>0} (ν(b(n − \frac{1}{2}) − am) q^{(2n−1)m}e^{(2n−1)t−2ms} − ν(bn − a(m − \frac{1}{2})) q^{n(2m−1)}e^{2nt−(2m−1)s})

= ∑_{n,m>0} ((−1)^{m} ν(b(2n − 1) − am) q^{(n−1/2)m}e^{(2n−1)t−ms} −

(−1)^{m} ν(bn − a(2m − 1)) q^{(n−1/2)m}e^{nt−(2m−1)s})

= ∑_{n,m>0} (ν(bn − 2am) q^{nm}e^{nt−2ms} − ν(2bn − am) q^{nm} e^{2nt−ms}),

∑_{n,m>0} (ν(bn − am) q^{2nm} e^{2nt−2ms} + ν(b(n − \frac{1}{2}) − a(m − \frac{1}{2})) q^{(n−1/2)(2m−1)} e^{(2n−1)t−(2m−1)s})

= ∑_{n,m>0} (ν(bn − 2am) q^{nm} e^{nt−2ms} − (−1)^{m} ν(b(2n − 1) − am) q^{(n−1/2)m} e^{(2n−1)t−ms}).

5.2. The structure theorem for rational surfaces. We want to determine the structure of the Donaldson invariants of rational algebraic surfaces X at period points F in the boundary of the positive cone. We already know that all the Donaldson invariants Φ_{C}^{ℙ_{1}×ℙ_{1},F} for F ∈ 𝕊_{ℙ_{1}×ℙ_{1}} vanish. As the Donaldson invariants depend only on the diffeomorphism type of the pair (X, F), we can assume that X is ℙ_{2} blown up in N points.

Notation 5.4. Let X be the blowup of ℙ_{2} in N points. Let H ∈ H^{2}(X, ℤ) be the pullback of the hyperplane class from ℙ_{2}, and denote by E_{1}, …, E_{N} the classes of the exceptional divisors. A class D ∈ H^{2}(X, ℤ) is written as

D = (d_{0}, d_{1}, …, d_{N}) := d_{0}H + ∑_{i=1}^{N} d_{i}E_{i}

(uppercase and lowercase letters correspond). If d_{0} ≥ d_{1} ≥ … ≥ d_{N}, we write also D = (α_{0}^{n_{0}}, …, α_{k}^{n_{k}}), where the α_{j} are the different integers occuring in d_{0}, d_{1}, …, d_{N} with multiplicity n_{j} > 0.

Corollary 5.5. 1. Φ_{C}^{X,H+E_{j}} = 0 for all j ∈ {1 … N} and all C ∈ H^{2}(X, ℤ).

2. If N < 9, then Φ_{C}^{X,F} = 0 for all F ∈ 𝕊_{X}.

Proof. 1. holds by Theorem 5.3 in case N = 1 and follows by Proposition 4.6 in the general case.

2. follows from 1. by Corollary 4.10.

Definition 5.6. Let F ∈ H^{2}(X, ℤ) be a representative of an element in 𝕊_{X} , with f_{0} ≥ f_{1}. Let

B(X, F) := {W ∈ H^{2}(X, ℤ) | all w_{i} odd, W^{2} > −N + 1, (w_{0} − w_{1})W F ≤ 0},
B_{I}(X, F) := {W ∈ B(X, F) | w_{0} − w_{1} < 0 < W F },
B_{F} (X, F) := {W ∈ B(X, F) | 0 = W F < w_{0} − w_{1} < 2(f_{0} − f_{1})}, B_{G}(X, F) = ∅.

The nonnegative integer (W^{2} + N − 1)/8 is the order of W ∈ B(X, F). Usually we will drop (X, F) in the notation. We define Ω_{C}^{X,F} (τ, x) and O_{C}^{X,F} (x) by the formulas (4.7.1) resp. (4.7.2) replacing B_{I}(X, F, G), B_{F} (X, F, G), B_{G}(X, F, G), by B_{I}(X, F), B_{F} (X, F), B_{G}(X, F) respectively.

Theorem 5.7. For a rational surface X and F ∈ 𝕊_{X}, we can replace (X, F, G) by (X, F) everywhere in Theorem 4.8, Remark 4.9 and Corollary 4.11.

Proof. We put G := H + E_{1} in Theorem 4.8 and apply Corollary 5.5.

Corollary 5.8. Let X be a rational surface. Then Conjecture 4.12 holds with R_{F} = B_{F}. The set of characteristic elements W ∈ H^{2}(X, ℤ) with SW_{N_{W}} (W) ≠ 0 is B_{I} ∪ −B_{I}.

5.3. Examples. We want to finish the paper by giving a number of examples which illustrate our main results. Also they should make it clear that for any given pair (X, F) of a rational algebraic surface X and F ∈ 𝕊_{X} it is an elementary task to determine all the basic classes for (X, F) and their orders, and thus to compute the corresponding Donaldson invariants completely. We will always give the elements in B_{F} and −B_{I}.

Remark 5.9. The following observations simplify the task of determining the basic classes W for a given (X, F) with F ∈ 𝕊_{X}.

1. We can always assume that f_{0} ≥ f_{1} ≥ … ≥ f_{N} ≥ 0 (if necessary change the numbering of the E_{i}), and in this case an element W in B_{F} ∪ −B_{I} must satisfy w_{0} > 0 and w_{0} > w_{i} for all i > 0.

2. If N − 1 ≥ 8 is divisible by 8 and (X, F) is of strictly (N − 1)/8-th order simple type, then all the f_{i} are odd, and F is the only class in B_{F} ∪ −B_{I} of maximal order: any other such basic class V would have to satisfy V^{2} = 0 and this already determines F.

3. For W ∈ B_{F} also 2F − W ∈ B_{F}, and thus we can get all elements W ∈ B_{F} from those with 0 ≤ w_{0} ≤ f_{0} of the same order.

4. If W ∈ B_{F} ∪ −B_{I} would fulfill w_{0} − w_{1} = 2(f_{0} − f_{1}), then V := W − 2F would be a basic class orthogonal to G, which does not exist. If W ∈ −B_{I} with w_{0} − w_{1} > 2(f_{0} − f_{1}), then also (W − 2F) ∈ −B_{I}, (in fact of a higher order then W). Therefore we obtain all W ∈ B_{F} ∪ −B_{I} from those with 0 < w_{0} − w_{1} < 2(f_{0} − f_{1}).

5. If W ∈ −B_{I} with W F ≤ 2w_{i}f_{i} < 0 for some i, then V := W + 2w_{i}E_{i} ∈ B_{F} ∪ −B_{I}. If many of the f_{i} are 1, this reduces considerably the number of cases to consider; often one can exclude the existence of certain basic classes by excluding that of corresponding elements of B_{F}, which is easy by 3.

To determine the basic classes in the following examples we have used the above observations, and some additional elementary arguments.

(1) If F = (f_{0}, f_{1}, …, f_{s}, 0, …, 0) with s ≤ 8, then Φ_{C}^{X,F} = 0 for all C (this follows from Corollary 4.10 and Proposition 4.6).

(2) If f_{1} ≥ f_{0} − 1, then Φ_{C}^{X,F} = 0 for all C (use Remark 5.9, parts 3. and 4.).

(3) If all f_{i} are odd and N ≥ 9, then N − 1 is divisible by 8 and (X, F) is of strictly (N − 1)/8-th order simple type. The only element of B_{F} ∪ −B_{I} of maximal order is F. Therefore we get for if CF odd,

Φ_{C}^{X,F} (e^{xz}(1 + p/2)(1 − p^{2}/4)^{(N−9)/8}) = −(−1)^{C(C+F)/2}e^{Q(x)z^{2}/2}/ cosh(F xz)

A particular case is that of N = 9 and F = (3, 1^{9}), where (X, F) is of simple type. In this case the result was, for CF odd, brought to our attention by John Morgan, and later Zoltan Szabó informed us of some similar results in other cases (see [M-Sz]). This has been one of the principal motivations for this work.

(4) If f_{0} = 2n (with n ≥ 2) and ∑_{i≥1} f_{i} ≥ 4n^{2} − 2n, and all f_{i} are strictly positive, then (X, F) is strictly of \binom{n}{2}-th order simple type. (We omit the proof.) One basic class of maximal order is (2n − 1, 1^{N}).

(5) If F = (4, 2^{2}, 1^{8}) then (X, F) is of simple type (see (4)). B_{F} consists of W = (3, 1^{10}) and 2F −W and B_{I} = ∅. Therefore we obtain e.g.

Φ_{E_{10}}^{X,F} (e^{xz}(1 + p/2)) = e^{Q(x)z^{2}/2} \frac{cosh((H + E_{1} + E_{2})xz)}{cosh(F xz)}

(6) If F = (4, 2, 1^{12}) then (X, F) is of simple type (see (4)). B_{F} ∪ −B_{I} consists of

W_{1} := (3, 1^{13}), W_{1}^{i} := W_{1} − 2E_{i} (i = 2, …, 13), W_{2}^{i} := (5, 3, 1^{12}) + 2E_{i} (i = 2, …, 13).

Of these, all but W_{1} lie in B_{F}. So we obtain, e.g.,

Φ_{E_{2}}^{X,F} (e^{xz}(1 + p/2)) = \frac{e^{Q(x)z^{2}/2}}{8} (2 cosh(W_{1}xz) + ∑_{i=2}^{13}(1 − 2δ_{i,2}) \frac{cosh((H + E_{1} + 2E_{i})xz)}{cosh(F xz)}).

(7) F = (4, 1^{16}): Also here (X, F) is of simple type. B_{F} ∪ −B_{I} consists of

W_{1} := (3, 1^{16}), W_{1}^{i} := W_{1} − 2E_{i} (1 ≤ i ≤ 16),

W_{1}^{i,j} := W_{1} − 2E_{i} − 2E_{j} (1 ≤ i < j ≤ 16), W_{2}^{i,j} := W_{1} + 2H + 2E_{i} + 2E_{j} (1 ≤ i < j ≤ 16).

The W_{1}^{i,j} and the W_{2}^{i,j} lie in B_{F}.

(8) F = (5, 2^{5}, 1^{5}). Also here (X, F) is of simple type. The elements of B_{F} ∪ −B_{I} are W = (3, 1^{10}) and 2F − W.

(9) If F = (5, 3, 1^{16}) then (X, F) is of second order simple type. The unique class in B_{F} ∪ −B_{I} of order 2 is F. The classes of order 1 are

W_{1} := (3, 1^{17}), W_{1}^{i} := W_{1} − 2E_{i} (2 ≤ i ≤ 17), W_{1}^{i,j} := W_{1} − 2E_{i} − 2E_{j} (2 ≤ i < j ≤ 17),
W_{2}^{i} := F + 2E_{i} (2 ≤ i ≤ 17), W_{2}^{i,j} := F + 2E_{i} − 2E_{j} (2 ≤ i, j ≤ 17, i ≠ j),
W_{3}^{i,j} := F + 2H + 2E_{1} + 2E_{i} + 2E_{j} (2 ≤ i < j ≤ 17).

The classes in B_{F} are F, the W_{1}^{i,j}, the W_{2}^{i,j} and the W_{3}^{i,j}.

(10) If F = (5, 1^{25}) then (X, F) is of 3-rd order simple type. The unique class in B_{F} ∪ −B_{I} of order 3 is F. The classes of order 2 are

W_{1}^{i} := F + 2E_{i}, (1 ≤ i ≤ 25), W_{1}^{i,j} := F + 2E_{i} − 2E_{j}, (1 ≤ i, j ≤ 25, i ≠ j).

The classes of order 1 are

W_{2}^{J} := (3, 1^{25}) − ∑_{j∈J} 2E_{j} (J ⊂ {1, …, 25}, |J| ≤ 5),

W_{3}^{J_{1},J_{2}} := F + ∑_{j_{1}∈J_{1}} 2E_{j_{1}} − ∑_{j_{2}∈J_{2}} 2E_{j_{2}} (J_{1}, J_{2} ⊂ {1, …, 25} disjoint, |J_{1}| = 2, |J_{2}| ≤ 2),

W_{4}^{J} := (7, 1^{25}) + ∑_{j∈J} 2E_{j}, (J ⊂ {1, …, 25}, |J| = 5).

We finish by giving a list of the types of all boundary points F with 1 ≤ f_{0} ≤ 6 for which the Φ_{C}^{X,F} do not vanish identically. The verifications are elementary, though quite tedious. 1-st order: (3, 1^{9}), (4, 1^{16}), (4, 2, 1^{12}), (4, 2^{2}, 1^{8}), (5, 2^{3}, 1^{13}), (5, 2^{4}, 1^{9}), (5, 2^{5}, 1^{5}), (5, 3, 2, 1^{12}),

(5, 3, 2^{2}, 1^{8}), (5, 3^{2}, 1^{7}), (6, 2^{6}, 1^{12}), (6, 2^{7}, 1^{8}), (6, 2^{8}, 1^{4}), (6, 3, 2^{4}, 1^{11}), (6, 3, 2^{5}, 1^{7}), (6, 3, 2^{6}, 1^{3}), (6, 3^{2}, 2, 1^{14}), (6, 3^{2}, 2^{2}, 1^{10}), (6, 3^{2}, 2^{3}, 1^{6}), (6, 3^{3}, 1^{9}).

2-nd order: (5, 2, 1^{21}), (5, 2^{2}, 1^{17}), (5, 3, 1^{16}), (6, 2^{4}, 1^{20}), (6, 2^{5}, 1^{16}), (6, 3, 2, 1^{23}), (6, 3, 2^{2}, 1^{19}),

(6, 3, 2^{3}, 1^{15}), (6, 3^{2}, 1^{18}), (6, 4, 1^{20}), (6, 4, 2, 1^{16}). 3-rd order: (5, 1^{24}), (6, 1^{36}), (6, 2, 1^{32}), (6, 2^{2}, 1^{28}), (6, 2^{3}, 1^{24}), (6, 3, 1^{27}).

References

[Bo] Borcherds, R., Automorphic forms with singularities on Grassmannians, preprint 1996.

[Br] Brussee, R., The canonical class and C^{∞} properties of Kähler surfaces, preprint 1995.

[Do] Donaldson, S.K., Polynomial invariants for smooth four-manifolds, Topology 29 (1990), 257–315. [E-G1] Ellingsrud, G. and Göttsche, L., Variation of moduli spaces and Donaldson invariants under change of

polarisation, J. reine angew. Math. 467 (1995), 1–49.

[E-G2] Ellingsrud, G. and Göttsche, L., Wall-crossing formulas, Bott residue formula and the Donaldson invariants

of rational surfaces, preprint 1995.

[E-LP-S] Ellingsrud, G., Le Potier, J. and Strømme, S. A., Some Donaldson Invariants of ℂℙ^{2}, preprint 1995. [E-Z] Eichler, M., Zagier, D.,The theory of Jacobi forms, Progress in Math 55 Birkhäuser 1985. [F-S1] Fintushel, R., Stern, R.J., The blowup formula for Donaldson invariants, Annals of Math. 143 (1996),

529–546.

[F-S2] Fintushel, R., Stern, R.J., Donaldson invariants of 4-manifolds of simple type, J. Diff. Geom. 42 (1995), 577–633.

[F-S3] Fintushel, R., Stern, R.J., Rational blowdowns of smooth 4-manifolds, preprint 1995. [F-M] Friedman, R., Morgan, J., On the diffeomorphism types of certain algebraic surfaces, I, J. Diff. Geom. 27

(1988), 297–369.

[F-M2] Friedman, R., Morgan, J., Algebraic surfaces and Seiberg-Witten invariants, preprint 1995. [F-Q] Friedman, R., Qin, Z., Flips of moduli spaces and transition formulas for Donaldson polynomial invariants

of rational surfaces, Commun. in Analysis and Geometry 3 (1995), 11–83.

[G] Göttsche, L., Modular forms and Donaldson invariants for 4-manifolds with b_{+} = 1, J. Amer. Math. Soc. 9 (1996), 827–843.

[H-B-J] Hirzebruch, F., Berger, T., Jung, R., Manifolds and Modular forms, Vieweg, Braunschweig-Wiesbaden, 1992.

[K1] Kotschick, D., SO(3)-invariants for 4-manifolds with b^{+} = 1, Proc. London Math. Soc. 63 (1991), 426–448.

[K2] Kotschick, D., On manifolds homeomorphic to CP^{2}#8\overline{CP}^{2}, Invent. Math. 95 (1989), 591–600. [K-L] Kotschick, D., Lisca, P., Instanton Invariants of ℂℙ^{2} via Topology, Math. Ann. 303 (1995), 345–371. [K-M] Kotschick, D., Morgan, J., SO(3)-invariants for 4-manifolds with b^{+} = 1 II, J. Diff. Geom. 39 (1994),

433–456.

[Kr-M1] Kronheimer, P., Mrowka, T., Recurrence relations and asymptotics for four-manifold invariants, Bull. Amer.

Math. Soc. 30 (1994), 215–221.

[Kr-M2] Kronheimer, P., Mrowka, T., Embedded surfaces and the structure of Donaldson’s Polynomial invariants, J.

Diff. Geom. 33 (1995), 573-734.

[L-Q] Li, W.P., Qin, Z., Lower-degree Donaldson polynomials of rational surfaces, J. Alg. Geom. 2 (1993), 413–442. [M-O] Morgan, J., Ozsváth, P., private communication. [M-Sz] Morgan, J., Szabó, Z., Embedded tori in four-manifolds, preprint 1996. [O-V] Okoneck, C., Van de Ven, A., Γ-type invariants associated to P U(2)-bundles and the differentiable structure

of Barlow’s surface, Invent. Math. 95 (1989), 601–614.

[O] Ozsváth, P., Some Blowup formulas for SU(2) Donaldson Polynomials, J. Diff. Geom. 40 (1994), 411–447. [P-T1] Pidstrigach, V.Y. Tyurin, A.N., Localization of the Donaldson invariants along Seiberg-Witten classes,

preprint 1995.

[P-T2] Pidstrigach, V.Y. Tyurin, A.N., in preparation.

[Q1] Qin, Z., Equivalence classes of polarizations and moduli spaces of sheaves, J. Diff. Geom. 37 (1993), 397–413.

[Q2] Qin, Z., Moduli of stable sheaves on ruled surfaces and their Picard groups, J. reine angew. Math. 433 (1992), 201–219.

[R] Rankin, R.A., Modular forms and functions, Cambridge University Press, Cambridge, 1977. [S-W] Seiberg, N., Witten, E., Electric-Magnetic Duality, Monopole Condensation, and Confinement in N = 2

Supersymmetric Yang-Mills Theory, Nucl. Phys. B279 (1994)

[T1] Taubes, C.H., in preparation.

[T2] Taubes, C.H., The Seiberg-Witten and the Gromov invariants, Math. Research Letters 2 (1995), 221–238.

[We] Weil, A., Elliptic functions according to Eisenstein and Kronecker, Grundlehren 88, Springer, Berlin 1976.

[W1] Witten, E., Monopoles and Four-Manifolds, Math. Res. Lett. 1 (1994).

[W2] Witten, E., On S-Duality in Abelian Gauge Theory, Selecta Math. 1 (1995) 383–410.

[Z] Zagier, D., Periods of modular forms and Jacobi theta functions, Invent. math. 104 (1991), 449-465.

Institut Mittag Leffler, Auravägen 17, S-18262 Djursholm, Sweden

   E-mail address: gottsche@ml.kva.se

Max–Planck–Institut für Mathematik, Gottfried–Claren–Straße 26, D-53225 Bonn, Germany

   E-mail address: zagier@mpim-bonn.mpg.de