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We classify four-dimensional manifolds endowed with symplectic pairs admitting embedded symplectic spheres with non-negative self-intersection, following the strategy of McDuffs classification of rational and ruled symplectic four manifolds.
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We study the geometry of manifolds carrying symplectic pairs consisting of two closed 2-forms of constant ranks, whose kernel foliations are complementary. Using a variation of the construction of Boothby and Wang we build contact-symplectic and contact pairs from symplectic pairs.
We discuss a correspondence between certain contact pairs on the one hand, and certain locally conformally symplectic forms on the other. In particular, we characterize these structures through suspensions of contactomorphisms. If the contact pair is endowed with a normal metric, then the corresponding lcs form is locally conformally Kaehler, and, in fact, Vaisman. This leads to classification results for normal metric contact pairs. In complex dimension two we obtain a new proof of Belguns classification of Vaisman manifolds under the additional assumption that the Kodaira dimension is non-negative. We also produce many examples of manifolds admitting locally conformally symplectic structures but no locally conformally Kaehler ones.
We prove Gray--Moser stability theorems for complementary pairs of forms of constant class defining symplectic pairs, contact-symplectic pairs and contact pairs. We also consider the case of contact-symplectic and contact-contact structures, in which the constant class condition on a one-form is replaced by the condition that its kernel hyperplane distribution have constant class in the sense of E. Cartan.
Given a symplectic three-fold $(M,omega)$ we show that for a generic almost complex structure $J$ which is compatible with $omega$, there are finitely many $J$-holomorphic curves in $M$ of any genus $ggeq 0$ representing a homology class $beta$ in $H_2(M,Z)$ with $c_1(M).beta=0$, provided that the divisibility of $beta$ is at most 4 (i.e. if $beta=nalpha$ with $alphain H_2(M,Z)$ and $nin Z$ then $nleq 4$). Moreover, each such curve is embedded and 4-rigid.
The paper was withdrawn due to a gap in the proof of Lemma 3.
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