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

TL;DR

This paper establishes a criterion for when the tangent bundle of a smooth projective G-variety is big, using the moment map, and applies it to various classes of varieties including symmetric, horospherical, and certain Fano manifolds.

Contribution

It introduces a new sufficient criterion for the bigness of tangent bundles of G-varieties based on the moment map, and applies it to classify bigness in several important cases.

Findings

Criteria for bigness of tangent bundles using the moment map.

Verification of bigness for symmetric, horospherical, and equivariant compactifications.

Determination of the pseudoeffective cone for certain Fano manifolds.

Abstract

Let $G$ be a connected algebraic group and let $X$ be a smooth projective $G$-variety. In this paper, we prove a sufficient criterion to determine the bigness of the tangent bundle $TX$ using the moment map $\Phi_X^G:T^*X\rightarrow \mathfrak{g}^*$. As an application, the bigness of the tangent bundles of certain quasi-homogeneous 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 $\mathbb{G}_a^n$. In particular, we will determine the pseudoeffective cone of $\mathbb{P}(T^*X)$ 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 VMRT of $X$.