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Nov 4, 2012 at 17:15 comment added Chris Gerig You've actually cleared up all questions I (implicitly) had! At first I was thinking if when all the components of $Z$ were null-homologous then perhaps their is a canonical spin-c structure on $X$, somehow by considering the 'minimal surfaces' in $H_2(X,Z)$ with the components as boundary.
Nov 4, 2012 at 17:09 vote accept Chris Gerig
Nov 3, 2012 at 15:01 comment added Tim Perutz Rereading your question, perhaps what you're asking for a canonical spin-c structure on $X$. If so, it's not going to come in any natural way from the near-symplectic structure (unless you deform it to a symplectic structure).
Nov 3, 2012 at 14:56 history answered Tim Perutz CC BY-SA 3.0