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

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 ${\Phi }_X^G:T^{\ast }X\to {\mathfrak{𝔤}}^{\ast }$. 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 ${\mathbb{𝔾}}_a^n$. In particular, we will determine the pseudoeffective cone of $\mathbb{P}(T^{\ast }X)$ 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.