Geometry & Topology, Vol. 8 (2004) Paper no. 20, pages 743--777.

Constructing symplectic forms on 4-manifolds which vanish on circles

David T Gay, Robion Kirby

Abstract. 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.

Keywords. Symplectic, 4-manifold, spin^C, almost complex, harmonic

AMS subject classification. Primary: 57R17. Secondary: 57M50, 32Q60.

DOI: 10.2140/gt.2004.8.743

E-print: arXiv:math.GT/0401186

Submitted to GT on 17 January 2004. (Revised 6 May 2004.) Paper accepted 16 May 2004. Paper published 18 May 2004.

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David T Gay, Robion Kirby
CIRGET, Universite du Quebec a Montreal, Case Postale 8888
Succursale centre-ville, Montreal (QC) H3C 3P8, Canada
and
Department of Mathematics, University of California
Berkeley, CA 94720, USA

Email: gay@math.uqam.ca, kirby@math.berkeley.edu

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