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Abstract  We prove that any symplectic Fano 6‐manifold   M   with a Hamiltonian    S  1   ‐action is simply connected and satisfiesc  1    c  2    (  M  )   =  24   . This is done by showing that the fixed submanifoldM  min   ⊆  M    on which the Hamiltonian attains its minimum is diffeomorphic to either a del Pezzo surface, a 2‐sphere or a point. In the case when    dim  (   M  min   )  =  4   , we use the fact that symplectic Fano 4‐manifolds are symplectomorphic to del Pezzo surfaces. The case when    dim  (   M  min   )  =  2    involves a study of   six  ‐dimensional Hamiltonian    S  1   ‐manifolds with    M  min    diffeomorphic to a surface of positive genus. By exploiting an analogy with the algebro‐geometric situation we construct in each such 6‐manifold an    S  1   ‐invariant symplectic hypersurface    F  (  M  )    playing the role of a smooth fibre of a hypothetical Mori fibration over    M  min   . This relies upon applying Seiberg–Witten theory to the resolution of symplectic 4‐orbifolds occurring as the reduced spaces of   M  .

S 1 ‐invariant symplectic hypersurfaces in dimension 6 and the Fano condition