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Constructing symplectic forms on 4-manifolds which vanish on circles

Given a smooth, closed, oriented 4-manifold X and α ∈ H2(X, Z) such that α � α > 0, a closed 2-form ω is constructed, Poincare dual to α, which is symplectic on the complement of a finite set of unknotted circles Z. The number of circles, counted with sign, is given by d = (c1(s) 2 −3σ(X)−2χ(X))/4, where s is a certain spin C structure naturally associated to ω.