Donaldson Invariants of Non-simple Type 4-manifolds

D X : A(X) = Sym (H0(X)⊕H2(X)) → C, where w ∈ H(X ;Z). As for the grading of A(X), the elements in H2(X) have degree 2 and the point x ∈ H0(X) has degree 4. Let d0 = −w 2 − 32 (1 + b ), so that for homogeneous z ∈ A(X), D X(z) is non-zero only if 1 2 deg z ≡ d0 (mod 4). Set ℘ = x−4 ∈ A(X) as in [12]. By definition, X is of simple type if the condition D X(℘z) = 0 is satisfied for any z ∈ A(X). Also X is of finite type when there is some n ≥ 0 such that D X(℘ z) = 0, for any z ∈ A(X). The order of finite type is the minimum of such n. All 4-manifolds with b > 1 are of finite type [10] and the order of finite type is independent of w ∈ H(X ;Z) ([11, theorem 5]). The question about the existence of non-simple type 4-manifolds with b1 = 0 and b + > 1 is still open.