Adjunction for Varieties With a ℂ* Action

Abstract

Let X be a complex projective manifold, L an ample line bundle on X, and assume that we have a ℂ* action on (X;L). We classify such triples (X; L;ℂ*) for which the closure of a general orbit of the ℂ* action is of degree ≤ 3 with respect to L and, in addition, the source and the sink of the action are isolated fixed points, and the ℂ* action on the normal bundle of every fixed point component has weights ±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 ≥ 2.

High rank torus actions on contact manifolds

“…In the proof of the Theorem we will then assume that n ≥ 5. Note also that for the cases n = 5, 6, the statement has been proved in [22,Theorem 6.2] without the assumption on the extremal fixed points.…”

“…Step II and III contain the key point which allows us to extend the previous results of [6,22] to any dimension of the contact variety. In these steps we will make use of classification results of bandwith three and two varieties which have been recently obtained in [21].…”

“…We will also make use of the following complete classification of equalized bandwidth 3 actions with isolated extremal fixed points: Theorem 2.7 [22,Theorem 4.5], [21,Theorem 6.8] Let (X , L) be a polarized pair, where X is a projective manifold of dimension n ≥ 3 with a linearized action of C * of bandwidth three, such that its sink and source are isolated points. Assume in addition that the action is equalized at the sink and the source, and denote by Y i the union of the inner fixed point components of weight i, i = 1, 2.…”

“…The arguments in the proof of this result are about the action of a suitably chosen two dimensional subtorus in the Cartan torus in question. This downgrading of the torus action and suitable results about adjunction theory for varieties with a C * -action, [22], and about the birational geometry of small bandwidth C * -actions, [21], enable to identify the variety as the adjoint variety of a simple group.…”

“…22 Let G = G be two semisimple groups with Lie algebras g, g of type A 2 , B, D, E, F 4 , G 2 . If their Dynkin diagrams are different, then g, g are not isomorphic as M(H 2 )-graded vector spaces.Proof The dimensions of the Lie algebras appearing in the statement are:…”