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 ${\mathbb{C}}^{\ast }$, then $X$ is contained in one of the families ${\mathbb{P}}^4,Q^4,W_5,X_8^m$. All of the families contain members that have a semi-free ${\mathbb{C}}^{\ast }$-action.

44 pages. In the symplectic result Theorem C the assumption that M is simply connected is needed. Since smooth Fano varieties are simply connected, the proof of the algebraic results Theorem A and Theorem B are not affected. In addition, some misprints are corrected and minor expositional changes are made