Prime Fano $4$-folds with semi-free torus actions

Abstract

Let $X$ be a smooth complex prime Fano fourfold having a semi-free action of $\mathbb{C}^*$, then $X$ is contained in one of the families $\mathbb{P}^4,Q^4,W_5, X^{m}_{8}$. All of the families contain members that have a semi-free $\mathbb{C}^*$-action. This is mostly proved by an analysis of closed symplectic $8$-manifolds with $b_{2}(M)=1$ with a semi-free Hamiltonian $S^1$-action using methods of equivariant symplectic geometry. In this setting some notable restrictions are obtained: the index is greater than $1$, $b_{4}(M)$ is bounded above and all fixed components are shown to be symplectomorphic to products of projective spaces. The proof of the main result appeals in the last step to the classification of prime Fano $4$-folds with index greater than $1$.

Theorems & Definitions (69)

• Theorem A
• Theorem B
• Theorem C
• Definition 2.1
• Theorem 2.2
• Theorem 2.3
• Lemma 2.4
• proof
• Theorem 2.5
• Lemma 2.6