Algebraic torus actions on contact manifolds

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-Kähler 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 Białynicki-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 $\frac{1}{2} (\operatorname{dim}X+5)$ with $\operatorname{dim}$ 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 $3$.