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

Jie Liu

Vol. 17 (2023), No. 8, 1501–1532
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 TX using the moment map ΦXG : TX 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 𝔾an. In particular, we will determine the pseudoeffective cone of (TX) 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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