On moment map and bigness of tangent bundles of G-varieties

Liu, Jie

doi:10.2140/ant.2023.17.1501

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On moment map and bigness of tangent bundles of $G$-varieties

Jie Liu

Vol. 17 (2023), No. 8, 1501–1532

DOI: 10.2140/ant.2023.17.1501

Abstract

Let $G$ be a connected algebraic group and let $X$ be a smooth projective $G$ -variety. We prove a sufficient criterion to determine the bigness of the tangent bundle $T X$ using the moment map $Φ_{X}^{G} : T^{*} X \to 𝔤^{*}$ . As an application, the bigness of the tangent bundles of certain quasihomogeneous varieties are verified, including symmetric varieties, horospherical varieties and equivariant compactifications of commutative linear algebraic groups. Finally, we study in details the Fano manifolds $X$ with Picard number $1$ which is an equivariant compactification of a vector group $𝔾_{a}^{n}$ . In particular, we will determine the pseudoeffective cone of $ℙ (T^{*} X)$ and show that the image of the projectivised moment map along the boundary divisor $D$ of $X$ is projectively equivalent to the dual variety of the variety of minimal rational tangents of $X$ at a general point.

Keywords

$G$-variety, Fano manifold, tangent bundle, moment map, VMRT

Mathematical Subject Classification

Primary: 14M17

Secondary: 14J45, 14M27

Milestones

Received: 8 March 2022

Revised: 3 August 2022

Accepted: 14 September 2022

Published: 29 August 2023

Authors

Jie Liu
Institute of Mathematics Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing China

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Vol. 17, No. 8, 2023