| Summary: | In this thesis, we study symplectic manifolds satisfying certain co-homological conditions, in the presence of a Hamiltonian group action. In the case of 6-dimensional symplectic manifolds, the assumption of a Hamiltonian circle action allows us to use tools of 4-dimensional symplectic geometry, such as Seiberg-Witten theory, to prove restric-tions on their geometry and topology. The connection occurs primarily through symplectic reduction, and additionally when one considers col-lections of 4-dimensional symplectic submanifolds (i.e. fixed submani-folds, isotropy submanifolds or the symplectic fibre). The cohomological conditions of prime interest to us are the sym-plectic Fano condition and its relative version. The symplectic Fano condition is analogous to the Fano condition from algebraic geometry. In particular, symplectic manifolds satisfying this condition contain the class of smooth Fano varieties over the complex numbers. It is an open question to determine the extent to which these manifolds sat-isfy the boundedness of their algebraic counterparts in real dimensions 6 ≤ 2n ≤ 10. In this thesis, we give some positive progress towards a conjecture of Fine and Panov regarding the 6-dimensional case (see Conjecture 1.2). In particular, we prove that a symplectic Fano 6-manifold with a Hamiltonian circle action is simply connected and satisfies c1c2 = 24 (this result appeared in a joint work of the author and Panov [41]). In addition, we are able to show that there is at most one fixed surface of positive genus in such 6-manifolds, thus allowing us to bound the third Betti number in certain cases. Finally, we are able to show that these manifolds are symplectically birational to CP3, using the method of symplectic cuts.
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