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

TL;DR

This paper classifies complex projective manifolds with a $ ext{C}^*$ action under specific orbit and fixed point conditions, relating it to adjunction theory, and applies results to the homogeneity of certain contact Fano manifolds.

Contribution

It provides a classification of $ ext{C}^*$-equivariant triples with low-degree orbits and isolated fixed points, linking to adjunction theory and proving homogeneity of certain contact Fano manifolds.

Findings

Classified triples $(X,L, ext{C}^*)$ with orbit degree ≤ 3 and fixed point conditions.

Established a connection between $ ext{C}^*$ actions and adjunction theory.

Proved contact Fano manifolds of dimensions 11 and 13 are homogeneous under certain automorphism conditions.

Abstract

Let $X$ be a complex projective manifold, $L$ an ample line bundle on $X$, and assume that we have a $\mathbb{C}^*$ action on $(X,L)$. We classify such triples $(X,L,\mathbb{C}^*)$ for which the closure of a general orbit of the $\mathbb{C}^*$ action 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 $\geq 2$.