ALGEBRAIC TORUS ACTIONS ON CONTACT MANIFOLDS

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.

Automated summary

We prove the correctness of the LeBrun-Salamon Conjecture in low dimensions, and specifically discuss the properties of contact Fano manifolds and positive quaternion-Kahler manifolds.