$S^1$-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 $\dim (M_{\min })=4$, 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 $6$-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 hypersurface $\mathcal{F}(M)$ 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 $M$.

Exposition improved