Constructing symplectic forms on 4-manifolds which vanish on circles

David T. Gay, Robion Kirby

Published 2004-01-15, updated 2004-05-20Version 2

Given a smooth, closed, oriented 4-manifold X and alpha in H_2(X,Z) such that alpha.alpha > 0, a closed 2-form w is constructed, Poincare dual to alpha, which is symplectic on the complement of a finite set of unknotted circles. The number of circles, counted with sign, is given by d = (c_1(s)^2 -3sigma(X) -2chi(X))/4, where s is a certain spin^C structure naturally associated to w.