The Donaldson invariant \(\Phi^X_{C,d}\) of a compact, oriented, simply connected, smooth 4-manifold \(X\) consists of a linear functional on polynomials in the homology of \(X\). In the case where \(b_+(X)=1\) the invariant is known to depend on the choice of the metric: thus there is a collection of invariants \(\Phi^{X,L}_{C,d}\), where the period point \(L\) determines the chamber in the space of metrics. The wall-crossing terms \(\delta^X_{\xi,d}\) depend on homotopy data of the manifold \(X\) (Kotschick-Morgan conjecture). This paper focuses on the case where the period point is at the boundary of the positive cone \(H^2(X,{\mathbb R})^+\). Such points may be in the closure of infinitely many chambers, in which case a renormalized sum is needed to define the corresponding invariant. With this definition, the difference of two invariants always satisfies a weaker (higher order) simple type condition and can be expressed in terms of modular forms.
The paper is organized as follows: after reviewing Donaldson invariants, wall-crossings, and some relevant computations with modular forms, the authors relate the wall-crossing terms to theta functions on lattices (in this case \(H_2(X,{\mathbb Z})\)) of signature \((r-1,1)\). For a unimodular lattice these can then be related to meromorphic Jacobi forms. The paper also discusses the blowup formula for these invariants. The difference of two invariants is then proved to satisfy the higher order simple type condition. Finally, in the case of rational algebraic surfaces, there is one period point where all the invariants vanish, hence the results obtained for the difference of two invariants can be rephrased as a structure theorem for Donaldson invariants.