S1‐invariant symplectic hypersurfaces in dimension 6 and the Fano condition

Abstract

We prove that any Symplectic Fano $6$-manifold $M$ with a Hamiltonian $S^1$-action is simply connected and satisfies $c_1 c_2(M)=24$. This is done by showing that the fixed submanifold $M_{\\min}\\subseteq M$ 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 $\\dim(M_{\\min})=4$, the result follows from the fact that symplectic Fano $4$-manifolds are symplectomorphic to del Pezzo surfaces. The case when $\\dim(M_{\\min})=2$ involves a study of $6$-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 hypersuface ${\\cal F}(M)$ 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 $M$.