  We consider product 4--manifolds S^1 X M, where M is a closed, connected and
oriented 3-manifold. We prove that if S^1 X M admits a complex structure or a
Lefschetz or Seifert fibration, then the following statement is true: S^1 X M
admits a symplectic structure if and only if M fibers over S^1, under the
additional assumption that M has no fake 3-cells. We also discuss the
relationship between the geometry of M and complex structures and Seifert
fibrations on S^1 X M.
