Algebraic torus actions on contact manifolds

TL;DR

This paper proves the LeBrun-Salamon Conjecture for low-dimensional contact Fano manifolds, showing they are homogeneous and linking them to symmetric spaces, by analyzing torus actions and equivariant theorems.

Contribution

It introduces new methods for recognizing homogeneous spaces via algebraic torus actions in high complexity T-varieties with contact structures.

Findings

Contact Fano manifolds of dimension 2n+1 with certain automorphism groups are homogeneous.

Any positive quaternion-Kahler manifold of dimension at most 16 is a Wolf space.

The study extends torus action analysis to high complexity T-varieties with contact structures.

Abstract

We prove the LeBrun-Salamon Conjecture in low dimensions. More precisely, we show that a contact Fano manifold X of dimension 2n+1 that has reductive automorphism group of rank at least n-2 is necessarily homogeneous. This implies that any positive quaternion-Kahler manifold of real dimension at most 16 is necessarily a symmetric space, one of the Wolf spaces. A similar result about contact Fano manifolds of dimension at most 9 with reductive automorphism group also holds. The main difficulty in approaching the conjecture is how to recognize a homogeneous space in an abstract variety. We contribute to such problem in general, by studying the action of algebraic torus on varieties and exploiting Bialynicki-Birula decomposition and equivariant Riemann-Roch theorems. From the point of view of T-varieties (that is, varieties with a torus action), our result is about high complexity T-manifolds. The complexity here is at most 1/2(dim X+5) with dim X arbitrarily high, but we require this special (contact) structure of X. Previous methods for studying T-varieties in general usually only apply for complexity at most 2 or 3.