Adjunction for varieties with $\mathbb{C}^*$ action

Eleonora A. Romano, Jarosław A. Wiśniewski

Published 2019-04-03Version 1

Let $X$ be a complex projective manifold, $L$ an ample line bundle on $X$ and $H=\mathbb{C}^*$ an algebraic torus acting on $(X,L)$. We classify such triples $(X,L,H)$ for which the closure of a general orbit of the action of $H$ is of degree $\leq 3$ with respect to $L$ and, in addition, the source and the sink of the action are isolated fixed points and the $\mathbb{C}^*$ action on the normal bundle of every fixed point component has weights $\pm 1$. We treat this situation by relating it to the classical adjunction theory. As an application, we prove that contact Fano manifolds of dimension 11 and 13 are homogeneous if their group of automorphisms is reductive of rank $\geq 2$.