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June 2022 Algebraic torus actions on contact manifolds
Jarosław Buczyński, Jarosław A. Wiśniewski, Andrzej Weber
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J. Differential Geom. 121(2): 227-289 (June 2022). DOI: 10.4310/jdg/1659987892


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 $3$.

With an appendix by Andrzej Weber.

Dedicated to Andrzej Szczepan Białynicki-Birula.


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Jarosław Buczyński. Jarosław A. Wiśniewski. Andrzej Weber. "Algebraic torus actions on contact manifolds." J. Differential Geom. 121 (2) 227 - 289, June 2022.


Received: 20 May 2018; Accepted: 6 December 2019; Published: June 2022
First available in Project Euclid: 9 August 2022

Digital Object Identifier: 10.4310/jdg/1659987892

Primary: 14L30
Secondary: 14J45 , 14M17 , 22E46 , 53C26 , 53D10

Keywords: adjoint action , algebraic torus action , complex contact manifolds , Fano manifolds , homogeneous spaces , localization in K-theory , quaternion-Kähler manifolds

Rights: Copyright © 2022 Lehigh University


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Vol.121 • No. 2 • June 2022
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