$S^{1}$-invariant symplectic hypersurfaces in dimension $6$ and the Fano condition

TL;DR

This paper proves that symplectic Fano 6-manifolds with a Hamiltonian circle action are simply connected, satisfy a specific Chern class relation, and classifies fixed point sets using symplectic and Seiberg-Witten techniques.

Contribution

It establishes the topology and Chern class relations for symplectic Fano 6-manifolds with circle symmetry, and constructs invariant hypersurfaces analogous to Mori fibrations.

Findings

Any such manifold is simply connected.

The Chern class product c1 c2 equals 24.

Fixed submanifolds are diffeomorphic to del Pezzo surfaces, spheres, or points.

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 ${\cal F}(M)$ 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 $M$.